{"id":191726,"date":"2025-02-15T10:36:52","date_gmt":"2025-02-15T10:36:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191726"},"modified":"2025-02-15T10:36:54","modified_gmt":"2025-02-15T10:36:54","slug":"a-porester-studving-diameter-growth-of-red-pine-belleves-that-the-mean-diameter-growth-will-be-different-from-the-known-theatment-growth-of-1-25in-year-if-a-fertilization-treatment-is-applied-to-the","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/15\/a-porester-studving-diameter-growth-of-red-pine-belleves-that-the-mean-diameter-growth-will-be-different-from-the-known-theatment-growth-of-1-25in-year-if-a-fertilization-treatment-is-applied-to-the\/","title":{"rendered":"A porester studving diameter growth of red pine belleves that the mean diameter growth will be different from the known theatment growth of 1.25in\/ year if a fertilization treatment is applied to the stand"},"content":{"rendered":"\n<p>A porester studving diameter growth of red pine belleves that the mean diameter growth will be different from the known theatment growth of 1.25in\/ year if a fertilization treatment is applied to the stand. He condua his experiment collects data from 32 plots \\&amp; gets sample mean of 1.4 in per year The population standard deviation For this is .46 .<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>This scenario describes a hypothesis test for comparing the mean diameter growth of red pine trees, in the presence of a fertilization treatment, to the known treatment growth of 1.25 inches per year. Let&#8217;s break down the process.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Define the Hypotheses<\/h3>\n\n\n\n<p>In hypothesis testing, we first define the null hypothesis (H\u2080) and the alternative hypothesis (H\u2081). Here:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>H\u2080: \u03bc = 1.25 (the mean diameter growth is equal to 1.25 inches per year).<\/li>\n\n\n\n<li>H\u2081: \u03bc \u2260 1.25 (the mean diameter growth is different from 1.25 inches per year).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Identify Known Values<\/h3>\n\n\n\n<p>We have the following:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Population standard deviation (\u03c3) = 0.46 inches<\/li>\n\n\n\n<li>Sample mean (x\u0304) = 1.4 inches<\/li>\n\n\n\n<li>Sample size (n) = 32 plots<\/li>\n\n\n\n<li>Hypothesized population mean (\u03bc\u2080) = 1.25 inches<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Conduct the Z-Test<\/h3>\n\n\n\n<p>Since the population standard deviation is known, we can perform a Z-test for the mean. The formula for the Z-test statistic is:<\/p>\n\n\n\n<p>[<br>Z = \\frac{x\u0304 &#8211; \u03bc\u2080}{\\frac{\u03c3}{\\sqrt{n}}}<br>]<\/p>\n\n\n\n<p>Plugging in the values:<\/p>\n\n\n\n<p>[<br>Z = \\frac{1.4 &#8211; 1.25}{\\frac{0.46}{\\sqrt{32}}}<br>]<\/p>\n\n\n\n<p>Let&#8217;s calculate this.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Determine the Critical Value and Make the Decision<\/h3>\n\n\n\n<p>The critical value for a two-tailed test at the 0.05 significance level is \u00b11.96. If the calculated Z-value is beyond this range, we reject the null hypothesis.<\/p>\n\n\n\n<p>I&#8217;ll compute the Z-test statistic and make the final conclusion.<\/p>\n\n\n\n<p>Let&#8217;s go ahead and calculate this now.<\/p>\n\n\n\n<p>The calculated Z-test statistic is approximately 1.845.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Compare the Z-Score with the Critical Value<\/h3>\n\n\n\n<p>The critical Z-value for a two-tailed test at a significance level of 0.05 is \u00b11.96. Since 1.845 is less than 1.96, it does not fall in the rejection region. Therefore, we <strong>fail to reject<\/strong> the null hypothesis.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>There is not enough evidence to conclude that the fertilization treatment has a significant effect on the mean diameter growth of red pine trees compared to the known treatment growth of 1.25 inches per year.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>In hypothesis testing, the Z-test is used when the population standard deviation is known, and we are comparing a sample mean to a population mean. The Z-test statistic helps us determine how far our sample mean is from the population mean in terms of standard deviations. If the calculated Z-value is greater than the critical value (1.96), we reject the null hypothesis. In this case, because the Z-value of 1.845 is less than 1.96, we fail to reject the null hypothesis, meaning the fertilization treatment did not significantly change the growth rate.<\/p>\n\n\n\n<p>Let me also generate the visualization for this.<\/p>\n\n\n\n<p>Here is the visualization showing the Z-test diagram with the critical value regions shaded. The Z-value of 1.845 is within the non-rejection region, which confirms the decision to fail to reject the null hypothesis. Let me know if you need further clarification!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-641.png\" alt=\"\" class=\"wp-image-191727\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A porester studving diameter growth of red pine belleves that the mean diameter growth will be different from the known theatment growth of 1.25in\/ year if a fertilization treatment is applied to the stand. He condua his experiment collects data from 32 plots \\&amp; gets sample mean of 1.4 in per year The population standard [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-191726","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191726","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=191726"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191726\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191726"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191726"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191726"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}