{"id":39911,"date":"2025-01-21T15:53:33","date_gmt":"2025-01-21T15:53:33","guid":{"rendered":"https:\/\/peraltafinancing.com\/agriculture\/theory-solved-example-and-demonstration-in-agri-analyze\/"},"modified":"2025-01-21T15:53:33","modified_gmt":"2025-01-21T15:53:33","slug":"theory-solved-example-and-demonstration-in-agri-analyze","status":"publish","type":"post","link":"https:\/\/fivemor.com\/?p=39911","title":{"rendered":"Theory, Solved Example and Demonstration in Agri Analyze"},"content":{"rendered":"<p> <br \/>\n<\/p>\n<div id=\"post-body-7692458841938354709\">\n<p><i>\u00a0The blog discuss in details about theory of F test, its use cases, solved example (manually) and a demonstration using online tool Agri Analyze (Reading time 10 min)\u00a0<\/i><\/p>\n<p><strong>Introduction<\/strong><\/p>\n<p>The F-test is a statistical method used to compare the variances of two samples or the ratio of variances across multiple samples. It assesses whether the data follow an F-distribution under the null hypothesis, assuming standard conditions for the error term (\u03b5). The test statistic, denoted as F, is commonly used to compare fitted models to determine which best represents the underlying population. F-tests are frequently employed in models fitted using least squares. The test is named after Ronald Fisher, who introduced the concept as the &#8220;variance ratio&#8221; in the 1920s, with George W. Snedecor later naming the test in Fisher\u2019s honor.<\/p>\n<p><strong>Definition<\/strong><\/p>\n<p>An F-test uses the F-statistic to evaluate whether the variances of two samples (or populations) are equal. The test assumes that the population follows an F-distribution and that the samples are independent. If the F-test yields statistically significant results, the null hypothesis of equal variances is rejected; otherwise, it is not.<\/p>\n<p><strong>Use of F-Test in Statistics<\/strong><\/p>\n<p>The F-test is a statistical tool used to compare variances and determine if there are significant differences between two populations or samples. It is commonly applied in regression analysis, statistical inference, model fitting, and analysis of variance (ANOVA) to identify the best-fitting statistical model or assess differences across groups.<\/p>\n<ul>\n<li><strong>Regression analysis<\/strong><\/li>\n<li><strong>Statistical inference<\/strong><\/li>\n<li><strong>Model fitting<\/strong><\/li>\n<li><strong>Analysis of variance (ANOVA)<\/strong><\/li>\n<\/ul>\n<div>\n<p><strong>Assumptions<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify;\"><strong>Independence:<\/strong> The observations within each group must be independent, meaning there should be no relationship between observations across samples.<\/li>\n<li style=\"text-align: justify;\"><strong>Normality:<\/strong> Data in each group should follow a normal distribution. For large sample sizes, this assumption can be relaxed based on the Central Limit Theorem.<\/li>\n<li style=\"text-align: justify;\"><strong>Homogeneity of variances:<\/strong> The variances across groups being compared should be approximately equal.<\/li>\n<\/ul>\n<div>\n<p><strong>Important Notes on the F-Test<\/strong><\/p>\n<ul>\n<li>The F-test is used to assess whether the variances of two populations are equal by comparing them using an F distribution.<\/li>\n<li>The F-test statistic is calculated as <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mo>=<\/mo><mfrac><msubsup><mi>\u03c3<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><msubsup><mi>\u03c3<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">F = \\frac{\\sigma_1^2}{\\sigma_2^2}<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"\/><\/span><\/span><\/span><\/span>, where <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c3<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\sigma_1^2<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c3<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\sigma_2^2<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> are the sample variances.<\/li>\n<li>The null hypothesis is evaluated using a critical value, which determines whether to reject the hypothesis.<\/li>\n<li>A common application of the F-test is the one-way ANOVA, which assesses variability between group means and within group observations.<\/li>\n<\/ul>\n<div>\n<p><strong>Selection Criteria for \u03c3\u2081\u00b2 and \u03c3\u2082\u00b2 in an F-Test<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify;\">For a <strong>right-tailed<\/strong> or <strong>two-tailed F-test<\/strong>, the variance with the greater value is placed in the numerator, making the sample corresponding to \u03c3\u2081\u00b2 the first sample. The smaller variance (\u03c3\u2082\u00b2) is the denominator for the second sample.<\/li>\n<li style=\"text-align: justify;\">For a <strong>left-tailed test<\/strong>, the smaller variance is in the numerator (sample 1), while the larger variance is in the denominator (sample 2).<\/li>\n<\/ul>\n<div>\n<p><strong>Hypotheses<\/strong><br \/><strong>Left-Tailed Test:<\/strong><\/p>\n<ul>\n<li>Null Hypothesis (H\u2080): \u03c3\u2081\u00b2 = \u03c3\u2082\u00b2<\/li>\n<li>Alternative Hypothesis (H\u2081): \u03c3\u2081\u00b2 <\/li>\n<li><strong>Decision Criteria<\/strong>: Reject H\u2080 if the F-statistic <\/li>\n<\/ul>\n<p><strong>Right-Tailed Test:<\/strong><\/p>\n<ul>\n<li>Null Hypothesis (H\u2080): \u03c3\u2081\u00b2 = \u03c3\u2082\u00b2<\/li>\n<li>Alternative Hypothesis (H\u2081): \u03c3\u2081\u00b2 &gt; \u03c3\u2082\u00b2<\/li>\n<li><strong>Decision Criteria<\/strong>: Reject H\u2080 if the F-statistic &gt; F-critical value.<\/li>\n<\/ul>\n<div>\n<p><strong>Two-Tailed Test:<\/strong><\/p>\n<ul>\n<li>Null Hypothesis (H\u2080): \u03c3\u2081\u00b2 = \u03c3\u2082\u00b2<\/li>\n<li>Alternative Hypothesis (H\u2081): \u03c3\u2081\u00b2 \u2260 \u03c3\u2082\u00b2<\/li>\n<\/ul>\n<div>\n<h3>Procedure for Conducting an F-Test:<\/h3>\n<ol>\n<li>\n<p><strong>Define Hypotheses<\/strong><\/p>\n<ul>\n<li>Null Hypothesis (H\u2080): The variances of the groups are equal.<\/li>\n<li>Alternative Hypothesis (H\u2081): The variances of the groups are not equal.<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>Collect Data<\/strong><br \/>Gather sample data from the groups being compared.<\/p>\n<\/li>\n<li>\n<p><strong>Calculate Sample Variances<\/strong><br \/>For each group, compute the sample variance (S\u00b2) using the formula:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>S<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mo>\u2211<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>\u2212<\/mo><mover accent=\"true\"><mi>x<\/mi><mo>\u02c9<\/mo><\/mover><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S^2 = \\frac{\\sum (x_i &#8211; \\bar{x})^2}{n &#8211; 1}<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"\/><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>where <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>x<\/mi><mi>i<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">x_i<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"strut\"\/><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"pstrut\"\/><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> represents the observations, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>x<\/mi><mo>\u02c9<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\bar{x}<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"strut\"\/><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"pstrut\"\/><span class=\"mord mathnormal\">x<\/span><span class=\"pstrut\"\/><span class=\"accent-body\"><span class=\"mord\">\u02c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the sample mean, and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"strut\"\/><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> is the sample size.<\/p>\n<\/li>\n<li>\n<p><strong>Calculate F-Statistic<\/strong><br \/>Compute the F-statistic as follows:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>F<\/mi><mo>=<\/mo><mfrac><msubsup><mi>S<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><msubsup><mi>S<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">F = \\frac{S_1^2}{S_2^2}<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mord\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"\/><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>where <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>S<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">S_1^2<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0is the larger variance and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>S<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">S_2^2<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the smaller variance.<\/p>\n<\/li>\n<li>\n<p><strong>Determine Degrees of Freedom<\/strong><br \/>Calculate degrees of freedom for each group:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mtext>df<\/mtext><mn>1<\/mn><\/msub><mo>=<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"\/><mtext>(numerator)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{df}_1 = n_1 &#8211; 1 \\quad \\text{(numerator)}<\/annotation><\/semantics><\/math><\/span><br \/><\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mtext>\u00a0 \u00a0df<\/mtext><mn>2<\/mn><\/msub><mo>=<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"\/><mtext>(denominator)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{df}_2 = n_2 &#8211; 1 \\quad \\text{(denominator)}<\/annotation><\/semantics><\/math><\/span><br \/><\/span><\/span><\/li>\n<li>\n<p><strong>Find the Critical Value<\/strong><br \/>Using an F-distribution table, locate the critical value for your chosen significance level (e.g., <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mn>0.05<\/mn><\/mrow><\/semantics><\/math><\/span><\/span>) based on <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>df<\/mtext><mn>1<\/mn><\/msub><\/mrow><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>df<\/mtext><mn>2<\/mn><\/msub><\/mrow><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<li>\n<p><strong>Make a Decision<\/strong><\/p>\n<ul>\n<li>If <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mo>&gt;<\/mo><mtext>Critical\u00a0Value<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">F &gt; \\text{Critical Value}<\/annotation><\/semantics><\/math><\/span><\/span>, reject the null hypothesis, indicating significant differences in variances.<\/li>\n<li>If <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mo>\u2264<\/mo><mtext>Critical\u00a0Value<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">F \\leq \\text{Critical Value}<\/annotation><\/semantics><\/math><\/span><\/span>, fail to reject the null hypothesis, suggesting no significant variance differences.<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>Conclusion<\/strong><\/p>\n<\/li>\n<\/ol>\n<ul>\n<li>Reject the null hypothesis if <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0exceeds the critical value, indicating significant variance between groups.<\/li>\n<li>Fail to reject the null hypothesis if <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0is less than or equal to the critical value, implying insufficient evidence for variance differences.<\/li>\n<\/ul>\n<div>\n<p class=\"MsoNormal\" style=\"text-align: justify;\"><b><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt; line-height: 107%;\">Example: &#8211; <\/span><\/b><\/p>\n<p class=\"MsoNormal\" style=\"text-align: justify; text-indent: 0.5in;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt; line-height: 107%;\">Life<br \/>\nexpectancy in 9 regions of Brazil in 1900 and 11 regions of Brazil in 1970 was<br \/>\nas given in the table below:<\/span><\/p>\n<table border=\"1\" cellpadding=\"0\" cellspacing=\"0\" class=\"MsoTableGridLight\" style=\"border-collapse: collapse; border: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-yfti-tbllook: 1184;\">\n<tbody>\n<tr>\n<td rowspan=\"2\" style=\"border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><b><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">Region<\/span><\/b><\/p>\n<\/td>\n<td colspan=\"2\" style=\"border-left: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; padding: 0in 5.4pt; width: 300.55pt;\" width=\"401\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><b><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">Life<br \/>\n  expectancy (year)<\/span><\/b><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><b><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">1900<\/span><\/b><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><b><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">1970<\/span><\/b><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">1<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">42.7<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">54.2<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">2<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">43.7<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">50.4<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">3<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">34.0<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">44.2<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">4<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">39.2<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">49.7<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">5<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">46.1<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">55.4<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">6<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">48.7<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">57.0<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">7<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">49.4<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">58.2<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">8<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">45.9<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">56.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">9<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">55.3<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">61.9<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">10<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">\u00a0<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">57.5<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border-top: none; border: 1pt solid rgb(191, 191, 191); mso-border-alt: solid #BFBFBF .5pt; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">11<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.25pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">\u00a0<\/span><\/p>\n<\/td>\n<td style=\"border-bottom: 1pt solid rgb(191, 191, 191); border-left: none; border-right: 1pt solid rgb(191, 191, 191); border-top: none; mso-border-alt: solid #BFBFBF .5pt; mso-border-bottom-themecolor: background1; mso-border-bottom-themeshade: 191; mso-border-left-alt: solid #BFBFBF .5pt; mso-border-left-themecolor: background1; mso-border-left-themeshade: 191; mso-border-right-themecolor: background1; mso-border-right-themeshade: 191; mso-border-themecolor: background1; mso-border-themeshade: 191; mso-border-top-alt: solid #BFBFBF .5pt; mso-border-top-themecolor: background1; mso-border-top-themeshade: 191; padding: 0in 5.4pt; width: 150.3pt;\" width=\"200\">\n<p align=\"center\" class=\"MsoNormal\" style=\"line-height: normal; margin-bottom: 0in; text-align: center;\"><span style=\"font-family: &quot;Times New Roman&quot;,serif; font-size: 12pt;\">53.4<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<p>We aim to determine whether the variation in life expectancy across different regions in 1900 and 1970 is the same. Assuming the populations in 1900 and 1970 follow normal distributions, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>\u03bc<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msubsup><mi>\u03c3<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N(\\mu_1, \\sigma_1^2)<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>\u03bc<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msubsup><mi>\u03c3<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N(\\mu_2, \\sigma_2^2)<\/annotation><\/semantics><\/math><\/span><\/span>, the hypotheses can be formulated as:<\/p>\n<ul>\n<li>Null Hypothesis <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>H<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">H_0<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c3<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\sigma_1^2 = \\sigma_2^2<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0(the variances are equal)<\/li>\n<li>Alternative Hypothesis <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>H<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">H_1<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c3<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo mathvariant=\"normal\">\u2260<\/mo><msubsup><mi>\u03c3<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\sigma_1^2 \\neq \\sigma_2^2<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span> (the variances are different)<\/li>\n<\/ul>\n<p>The F-test is applied to evaluate these hypotheses.<\/p>\n<ol>\n<li>\n<p><strong>Calculate Sample Variances:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>S<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mn>1<\/mn><mn>8<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><munderover><mo>\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mn>9<\/mn><\/munderover><msubsup><mi>x<\/mi><mrow><mn>1<\/mn><mi>i<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><mfrac><msup><mrow><mo fence=\"true\">(<\/mo><munderover><mo>\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mn>9<\/mn><\/munderover><msub><mi>x<\/mi><mrow><mn>1<\/mn><mi>i<\/mi><\/mrow><\/msub><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mn>9<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_1^2 = \\frac{1}{8} \\left( \\sum_{i=1}^9 x_{1i}^2 &#8211; \\frac{\\left( \\sum_{i=1}^9 x_{1i} \\right)^2}{9} \\right)<\/annotation><\/semantics><\/math><\/span><\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>S<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mn>1<\/mn><mn>8<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mn>18527.78<\/mn><mo>\u2212<\/mo><mfrac><mrow><mn>40<\/mn><msup><mn>5<\/mn><mn>2<\/mn><\/msup><\/mrow><mn>9<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>302.78<\/mn><mn>8<\/mn><\/mfrac><mo>=<\/mo><mn>37.848<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">S_1^2 = \\frac{1}{8} \\left( 18527.78 &#8211; \\frac{405^2}{9} \\right) = \\frac{302.78}{8} = 37.848<\/annotation><\/semantics><\/math><\/span><br \/><\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>S<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mn>1<\/mn><mn>10<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><munderover><mo>\u2211<\/mo><mrow><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mn>11<\/mn><\/munderover><msubsup><mi>x<\/mi><mrow><mn>2<\/mn><mi>j<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><mfrac><msup><mrow><mo fence=\"true\">(<\/mo><munderover><mo>\u2211<\/mo><mrow><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mn>11<\/mn><\/munderover><msub><mi>x<\/mi><mrow><mn>2<\/mn><mi>j<\/mi><\/mrow><\/msub><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mn>11<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_2^2 = \\frac{1}{10} \\left( \\sum_{j=1}^{11} x_{2j}^2 &#8211; \\frac{\\left( \\sum_{j=1}^{11} x_{2j} \\right)^2}{11} \\right)<\/annotation><\/semantics><\/math><\/span><span aria-hidden=\"true\" class=\"katex-html\"><span class=\"base\"><span class=\"minner\"><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\"\/><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>S<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mn>1<\/mn><mn>10<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mn>32799.91<\/mn><mo>\u2212<\/mo><mfrac><mrow><mn>598.<\/mn><msup><mn>5<\/mn><mn>2<\/mn><\/msup><\/mrow><mn>11<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>236.07<\/mn><mn>10<\/mn><\/mfrac><mo>=<\/mo><mn>23.607<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">S_2^2 = \\frac{1}{10} \\left( 32799.91 &#8211; \\frac{598.5^2}{11} \\right) = \\frac{236.07}{10} = 23.607<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<li>\n<p><strong>Calculate the F-Statistic:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>F<\/mi><mo>=<\/mo><mfrac><msubsup><mi>S<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><msubsup><mi>S<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mfrac><mo>=<\/mo><mfrac><mn>37.848<\/mn><mn>23.607<\/mn><\/mfrac><mo>=<\/mo><mn>1.603<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">F = \\frac{S_1^2}{S_2^2} = \\frac{37.848}{23.607} = 1.603<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<li>\n<p style=\"text-align: justify;\"><strong>Conclusion:<\/strong><br \/>\nThe critical values from the F-distribution table at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mn>0.05<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = 0.05<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0for a two-tailed test with degrees of freedom (8, 10) are <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mn>0.025<\/mn><\/msub><mo>=<\/mo><mn>3.85<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">F_{0.025} = 3.85<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mn>0.975<\/mn><\/msub><mo>=<\/mo><mn>0.233<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">F_{0.975} = 0.233<\/annotation><\/semantics><\/math><\/span><\/span>. Since the calculated F-value (1.603) is less than 3.85 and greater than 0.233, we fail to reject the null hypothesis. This indicates that there is no significant difference in the variances of life expectancy between 1900 and 1970 across the regions of Brazil.\u00a0 \u00a0<\/p>\n<\/li>\n<\/ol>\n<h3 style=\"text-align: left;\">F test Demonstration in Agri Analyze<\/h3>\n<\/div>\n<p><iframe loading=\"lazy\" title=\"F test Analysis using Agri Analyze\" width=\"840\" height=\"473\" src=\"https:\/\/www.youtube.com\/embed\/wTGBFxnfUiI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p><b>Video Demo for F-test in Agri Analyze<\/b><\/p>\n<div>Sample Data File: The data is same as shown in the above example. <a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1Y6g2BjGtkfPQuJSNVywlaLcoq7hfcBpnDN3LCCvHe2Q\/edit?usp=sharing\" target=\"_blank\">File Link<\/a><\/div>\n<p><b>Step1:<\/b> Prepare data file and save in a csv format<\/p>\n<div><b>Step2: Register on Agri Analyze<\/b>\u00a0(Only first time) <a href=\"https:\/\/www.agrianalyze.com\/Register.aspx\" target=\"_blank\">Link<\/a><\/div>\n<div><b>Step3: Go to Analytical Tool -&gt; Hypothesis testing -&gt; F-test <\/b><a href=\"https:\/\/www.agrianalyze.com\/FTest.aspx\" target=\"_blank\">Link<\/a><\/div>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgVTVHywIw9BuJiKWKxNBG-AykYSSwl19AigLElpuoCPQ3U6n28isggt3xUzlpMDjtWvMnOfQKnYS8tm20qY-vqXbWV0gl92zuhAJyzP5w5Ediag7Hgy0iUA-7zDlbKGnUkeHwxgQTBZYWyMiafzk5fRRGkJ2-5E_l-Nneq6kuA3ERrMKnppQxqMC6U58w\/s1410\/F%20test.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"653\" data-original-width=\"1410\" height=\"296\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgVTVHywIw9BuJiKWKxNBG-AykYSSwl19AigLElpuoCPQ3U6n28isggt3xUzlpMDjtWvMnOfQKnYS8tm20qY-vqXbWV0gl92zuhAJyzP5w5Ediag7Hgy0iUA-7zDlbKGnUkeHwxgQTBZYWyMiafzk5fRRGkJ2-5E_l-Nneq6kuA3ERrMKnppQxqMC6U58w\/w640-h296\/F%20test.png\" width=\"640\"\/><\/a><\/div>\n<p><b>Step4:\u00a0<\/b><\/p>\n<div>\n<ul style=\"text-align: left;\">\n<li>Upload the file\u00a0<\/li>\n<li>Select level of significance (for 5% write 0.05)\u00a0<\/li>\n<li>Variable name: Life Expectancy<\/li>\n<li>Category Type: Time Frame<\/li>\n<\/ul>\n<p><b>Step5:\u00a0<\/b><b>Click on Submit and Download<\/b><\/p>\n<h3 style=\"text-align: left;\"><b>Output<\/b><\/h3>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgG5PWwaUCwB5y3YZKjAQO_-Fz0C6crwe-yKK_PlztGL9J1NxSew35Mz-9U6_gfCenPe7zYB4Gj-gzuAh2qhNHBw2IvmL423xX-skJT9WgeGKjHAQ2VnMXGmPHwmHhWD28KGfEdZfOGD1pNw7_lo5l2ZBFlkHfgvO_373kSnP5hlGgaCQTxL1ve8XrbvQ0\/s800\/F%20test1.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"402\" data-original-width=\"800\" height=\"322\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgG5PWwaUCwB5y3YZKjAQO_-Fz0C6crwe-yKK_PlztGL9J1NxSew35Mz-9U6_gfCenPe7zYB4Gj-gzuAh2qhNHBw2IvmL423xX-skJT9WgeGKjHAQ2VnMXGmPHwmHhWD28KGfEdZfOGD1pNw7_lo5l2ZBFlkHfgvO_373kSnP5hlGgaCQTxL1ve8XrbvQ0\/w640-h322\/F%20test1.png\" width=\"640\"\/><\/a><\/div>\n<p><\/p>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEhtSSiFM05dDIZyGtQ-psjwJlztGHCA7qfxIJQuWgKb_j-eHLU0a5-ea3QDd7zssOBgYPxXhCwIkdrLHwf5lUwkmbcOqC1wcSJvlKB7AbmMcH5ONIhT37aJgPXjqRaETTziu2lWZkZfBzG21mCWA5khafSQdqqiEYzXadzvW5O169-QPXV-Bwol-dNXoe8\/s788\/F%20test2.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"650\" data-original-width=\"788\" height=\"528\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEhtSSiFM05dDIZyGtQ-psjwJlztGHCA7qfxIJQuWgKb_j-eHLU0a5-ea3QDd7zssOBgYPxXhCwIkdrLHwf5lUwkmbcOqC1wcSJvlKB7AbmMcH5ONIhT37aJgPXjqRaETTziu2lWZkZfBzG21mCWA5khafSQdqqiEYzXadzvW5O169-QPXV-Bwol-dNXoe8\/w640-h528\/F%20test2.png\" width=\"640\"\/><\/a><\/div>\n<p><\/p>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgVKUKwjc2Ruo9oDwPro5WnAgaHYB-MUDunj9SimGAGakCrqe0xYCf7lTGPsf5RUeGN65CsOAFhglH41grYk_E67G-CDipKYezfdG0SSlQ8iXqSM45BYH8mxTLJq3mssEez8IGIDkVL97UKkPkfWLqUYdzAbRvRYsB28TBV7CRO6jgBDiVi3k47LRB6Few\/s821\/F%20test3.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"656\" data-original-width=\"821\" height=\"512\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEgVKUKwjc2Ruo9oDwPro5WnAgaHYB-MUDunj9SimGAGakCrqe0xYCf7lTGPsf5RUeGN65CsOAFhglH41grYk_E67G-CDipKYezfdG0SSlQ8iXqSM45BYH8mxTLJq3mssEez8IGIDkVL97UKkPkfWLqUYdzAbRvRYsB28TBV7CRO6jgBDiVi3k47LRB6Few\/w640-h512\/F%20test3.png\" width=\"640\"\/><\/a><\/div>\n<p><\/p>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEjNVwMVvFcmqYGPSXhhoQTi5P3AW_pYB_16ygCCNRtM3bAv47H-Ij3FYKBxhVfr2iu88Ei1DTiRYZtC1ifNmOjFObah7eg5vbrMKFzXnI6C1FLFyQ9_cMTDWLbq6kVh84pcC0yZ91Kw_vo-Ct3wWRBN_EIYCL0FQlCy6I-dKg87AeXxiHdk1QlDEcUASNg\/s817\/F%20test4.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"526\" data-original-width=\"817\" height=\"412\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEjNVwMVvFcmqYGPSXhhoQTi5P3AW_pYB_16ygCCNRtM3bAv47H-Ij3FYKBxhVfr2iu88Ei1DTiRYZtC1ifNmOjFObah7eg5vbrMKFzXnI6C1FLFyQ9_cMTDWLbq6kVh84pcC0yZ91Kw_vo-Ct3wWRBN_EIYCL0FQlCy6I-dKg87AeXxiHdk1QlDEcUASNg\/w640-h412\/F%20test4.png\" width=\"640\"\/><\/a><\/div>\n<p><\/p>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEjkgoOINaKP9yys-iVBIKAgj6x8L_wsF1Hr_FwqPJtSsz7BhgrG9r_scjLnNQmqzdyO8rWRsGhGWWx6Tjb6bdFW3v0xBb7WEzbn2KXPJypE_sHdFFUALwpjy1glLsDlGqdKNLCzjtg629H_rrNrneus3-PZbws1lA1e823Zw_9JVOToNo0m7HvNE6yJSOo\/s842\/F%20test5.png\" style=\"margin-left: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"550\" data-original-width=\"842\" height=\"418\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEjkgoOINaKP9yys-iVBIKAgj6x8L_wsF1Hr_FwqPJtSsz7BhgrG9r_scjLnNQmqzdyO8rWRsGhGWWx6Tjb6bdFW3v0xBb7WEzbn2KXPJypE_sHdFFUALwpjy1glLsDlGqdKNLCzjtg629H_rrNrneus3-PZbws1lA1e823Zw_9JVOToNo0m7HvNE6yJSOo\/w640-h418\/F%20test5.png\" width=\"640\"\/><\/a><\/div>\n<h3 style=\"text-align: left;\">Other Related Topics:<\/h3>\n<div>\n<div>The blog is written with great effort and due research by\u00a0<a href=\"https:\/\/www.linkedin.com\/in\/uttam-baladaniya-463676164\/\" target=\"_blank\">Uttam Baladaniya<\/a>\u00a0(PhD Scholar, Department of Agricultural Statistics, Anand Agricultural University)<\/div>\n<div class=\"separator\" style=\"clear: both; text-align: center;\"><a href=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEirDYZmCMLeTRyD3ODG5ffzXe5V7-Y7_SX4wD0HSRID1-0ZTwbvb2TkioUdf0cAZQtpb08T571GoPX6YMTdXbmZ4hx9s9PsYx6NCfFmEpo87UZfGv_Uh3Xa7-_KG36zV2o6SjKyqLQhVI4JbgSZhibZV03YW36pnhyphenhyphen825Qd_zQBJx3kvrMINhhTEJ4Xt1k\/s401\/uttam-photoaidcom-cropped.jpg\" style=\"clear: left; float: left; margin-bottom: 1em; margin-right: 1em;\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"401\" data-original-width=\"401\" height=\"148\" src=\"https:\/\/blogger.googleusercontent.com\/img\/b\/R29vZ2xl\/AVvXsEirDYZmCMLeTRyD3ODG5ffzXe5V7-Y7_SX4wD0HSRID1-0ZTwbvb2TkioUdf0cAZQtpb08T571GoPX6YMTdXbmZ4hx9s9PsYx6NCfFmEpo87UZfGv_Uh3Xa7-_KG36zV2o6SjKyqLQhVI4JbgSZhibZV03YW36pnhyphenhyphen825Qd_zQBJx3kvrMINhhTEJ4Xt1k\/w148-h148\/uttam-photoaidcom-cropped.jpg\" width=\"148\"\/><\/a><\/div>\n<p><\/div>\n<\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>\u00a0The blog discuss in details about theory of F test, its use cases, solved example (manually) and a demonstration using online tool Agri Analyze (Reading time 10 min)\u00a0 Introduction The F-test is a statistical method used to compare the variances of two samples or the ratio of variances across multiple samples. It assesses whether the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":39912,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12026],"tags":[25333,24554,23134,25332,2359],"dealstore":[],"offerexpiration":[],"class_list":["post-39911","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-agriculture","tag-agri","tag-analyze","tag-demonstration","tag-solved","tag-theory"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Theory, Solved Example and Demonstration in Agri Analyze - Som2ny Network<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/fivemor.com\/?p=39911\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Theory, Solved Example and Demonstration in Agri Analyze - Som2ny Network\" \/>\n<meta property=\"og:description\" content=\"\u00a0The blog discuss in details about theory of F test, its use cases, solved example (manually) and a demonstration using online tool Agri Analyze (Reading time 10 min)\u00a0 Introduction The F-test is a statistical method used to compare the variances of two samples or the ratio of variances across multiple samples. 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