{"id":244424,"date":"2025-07-05T06:39:20","date_gmt":"2025-07-05T06:39:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244424"},"modified":"2025-07-05T06:39:22","modified_gmt":"2025-07-05T06:39:22","slug":"determine-the-null-hypothesis-h-subscript-0-and-the-alternative-hypothesis-h-subscript-1","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/determine-the-null-hypothesis-h-subscript-0-and-the-alternative-hypothesis-h-subscript-1\/","title":{"rendered":"Determine the null hypothesis, H Subscript 0, and the alternative hypothesis, H Subscript 1"},"content":{"rendered":"\n<p>Determine the null hypothesis, H Subscript 0, and the alternative hypothesis, H Subscript 1. Let males be population 1 and females be population 2. H Subscript 0: mu 1 minus mu 2 equals \u2000\u2000 0 H Subscript 1: mu 1 minus mu 2 not equals \u2000\u2000 0 (Type integers or decimals. Do not round.) Part 2 Compute the test statistic.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s break down the problem step by step.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Null and Alternative Hypotheses<\/h3>\n\n\n\n<p>In hypothesis testing, we compare two populations (in this case, males and females), which are represented by <strong>Population 1<\/strong> (males) and <strong>Population 2<\/strong> (females). The null hypothesis (H\u2080) and the alternative hypothesis (H\u2081) are statements that we test in a statistical procedure.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Null Hypothesis (H\u2080):<\/strong> This is a statement of no difference between the two populations. It assumes that the difference in the population means is zero. H0:\u03bc1\u2212\u03bc2=0H_0: \\mu_1 &#8211; \\mu_2 = 0H0\u200b:\u03bc1\u200b\u2212\u03bc2\u200b=0 Here, \u03bc1\\mu_1\u03bc1\u200b is the mean of the population of males, and \u03bc2\\mu_2\u03bc2\u200b is the mean of the population of females. The null hypothesis suggests that there is no difference in the means of the two populations.<\/li>\n\n\n\n<li><strong>Alternative Hypothesis (H\u2081):<\/strong> This is the opposite of the null hypothesis, stating that there is a difference between the two population means. This hypothesis could indicate either an increase or a decrease, and is typically expressed as a two-tailed test. H1:\u03bc1\u2212\u03bc2\u22600H_1: \\mu_1 &#8211; \\mu_2 \\neq 0H1\u200b:\u03bc1\u200b\u2212\u03bc2\u200b\ue020=0 This indicates that the difference between the means is not zero, i.e., the means of males and females are significantly different.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Test Statistic<\/h3>\n\n\n\n<p>To compute the test statistic, we often use a <strong>t-test<\/strong> or <strong>z-test<\/strong> depending on the sample size and whether the population standard deviations are known. The formula for the test statistic when comparing two sample means (assuming equal variances) is: t=X\u02c91\u2212X\u02c92s12n1+s22n2t = \\frac{\\bar{X}_1 &#8211; \\bar{X}_2}{\\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}}t=n1\u200bs12\u200b\u200b+n2\u200bs22\u200b\u200b\u200bX\u02c91\u200b\u2212X\u02c92\u200b\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>X\u02c91\\bar{X}_1X\u02c91\u200b and X\u02c92\\bar{X}_2X\u02c92\u200b are the sample means of the two populations (males and females).<\/li>\n\n\n\n<li>s1s_1s1\u200b and s2s_2s2\u200b are the sample standard deviations of the two populations.<\/li>\n\n\n\n<li>n1n_1n1\u200b and n2n_2n2\u200b are the sample sizes of the two populations.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Steps to Compute the Test Statistic:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Determine the sample means<\/strong> X\u02c91\\bar{X}_1X\u02c91\u200b and X\u02c92\\bar{X}_2X\u02c92\u200b for males and females, respectively.<\/li>\n\n\n\n<li><strong>Find the sample standard deviations<\/strong> s1s_1s1\u200b and s2s_2s2\u200b for each population.<\/li>\n\n\n\n<li><strong>Calculate the sample sizes<\/strong> n1n_1n1\u200b and n2n_2n2\u200b for each population.<\/li>\n\n\n\n<li><strong>Substitute these values<\/strong> into the formula for the test statistic to get the value of ttt.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p>If we had sample data like this:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Males: X\u02c91=75\\bar{X}_1 = 75X\u02c91\u200b=75, s1=10s_1 = 10s1\u200b=10, n1=30n_1 = 30n1\u200b=30<\/li>\n\n\n\n<li>Females: X\u02c92=70\\bar{X}_2 = 70X\u02c92\u200b=70, s2=12s_2 = 12s2\u200b=12, n2=30n_2 = 30n2\u200b=30<\/li>\n<\/ul>\n\n\n\n<p>Substitute into the formula: t=75\u22127010230+12230t = \\frac{75 &#8211; 70}{\\sqrt{\\frac{10^2}{30} + \\frac{12^2}{30}}}t=30102\u200b+30122\u200b\u200b75\u221270\u200b t=510030+14430t = \\frac{5}{\\sqrt{\\frac{100}{30} + \\frac{144}{30}}}t=30100\u200b+30144\u200b\u200b5\u200b t=53.33+4.8t = \\frac{5}{\\sqrt{3.33 + 4.8}}t=3.33+4.8\u200b5\u200b t=58.13t = \\frac{5}{\\sqrt{8.13}}t=8.13\u200b5\u200b t=52.85\u22481.75t = \\frac{5}{2.85} \\approx 1.75t=2.855\u200b\u22481.75<\/p>\n\n\n\n<p>Thus, the test statistic would be approximately <strong>1.75<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Interpretation:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Once you calculate the test statistic, you would compare it to the critical value from the t-distribution (for a t-test) or z-distribution (for a z-test) at a specific significance level (e.g., 0.05).<\/li>\n\n\n\n<li>If the absolute value of the test statistic is greater than the critical value, you reject the null hypothesis. If it&#8217;s smaller, you fail to reject the null hypothesis.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-453.jpeg\" alt=\"\" class=\"wp-image-244425\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Determine the null hypothesis, H Subscript 0, and the alternative hypothesis, H Subscript 1. Let males be population 1 and females be population 2. H Subscript 0: mu 1 minus mu 2 equals \u2000\u2000 0 H Subscript 1: mu 1 minus mu 2 not equals \u2000\u2000 0 (Type integers or decimals. Do not round.) Part [&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-244424","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244424","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=244424"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244424\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244424"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244424"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}