{"id":937,"date":"2021-05-29T16:38:14","date_gmt":"2021-05-29T20:38:14","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=937"},"modified":"2024-02-08T14:07:07","modified_gmt":"2024-02-08T19:07:07","slug":"8-3-main-idea-behind-hypothesis-tests","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/8-3-main-idea-behind-hypothesis-tests\/","title":{"raw":"8.3 Main Idea Behind Hypothesis Tests for \u03bc","rendered":"8.3 Main Idea Behind Hypothesis Tests for \u03bc"},"content":{"raw":"The main idea of a hypothesis test is to use the data as evidence to disprove the null [latex]H_0[\/latex]<\/strong> and thus prove that the alternative [latex]H_a[\/latex] is true<\/strong>. The idea behind a hypothesis test for the population mean is as follows:\r\n\r\nCollect from the population a simple random sample: [latex]x_1, x_2, \\dots, x_n[\/latex] and calculate the sample mean [latex]\\bar{x} = \\frac{x_1 + x_2 + \\dots + x_n}{n}[\/latex]. Our \u201cevidence\u201d stems from the discrepancy between the point estimate [latex]\\bar{x}[\/latex] and the hypothesized population mean [latex]\\mu_0[\/latex].\r\n\r\nReject the null hypothesis [latex]H_0[\/latex] if the sample mean [latex]\\bar{x}[\/latex] does not support the null [latex]H_0[\/latex]. That is, we should reject [latex]H_0[\/latex] if [latex]\\bar{x}[\/latex] is too extreme. The word \u201cextreme\u201d means contradicting the null [latex]H_0[\/latex] in favour of the alternative [latex]H_a[\/latex].<\/a>\r\n\r\n[caption id=\"attachment_3582\" align=\"aligncenter\" width=\"1024\"]\"Three Figure 8.2<\/strong>: Rejection Region Based on Sample Mean. [Image Description (See Appendix D Figure 8.2)<\/a>][\/caption]In order to quantify how the data (our evidence) contradict the null hypothesis, we first assume the null hypothesis [latex]H_0[\/latex] is true and calculate the chance of observing a sample mean at least as extreme as the observed <\/strong>[latex]\\bar{x}[\/latex].<\/strong> Reject the null [latex]H_0[\/latex] if the chance is small; otherwise, fail to reject [latex]H_0[\/latex]. Recall that for a normal population or a large sample size, the sample mean [latex]\\bar{X} \\sim N \\left( \\mu, \\frac{\\sigma}{\\sqrt{n}} \\right) \\Longrightarrow Z = \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}} \\sim N(0,1)[\/latex] and [latex]t = \\frac{\\bar{X - \\mu}}{s \/ \\sqrt{n}} \\sim t[\/latex] distribution with [latex]df = n-1.[\/latex] We call the variables [latex]Z = \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}}[\/latex] or [latex]t = \\frac{\\bar{X - \\mu}}{s \/ \\sqrt{n}}[\/latex] the test statistics. We should reject the null hypothesis [latex]H_0[\/latex] if the observed test statistic [latex]z_o = \\frac{\\bar{x} - \\mu_0}{\\sigma \/ \\sqrt{n}}[\/latex] or [latex]t = \\frac{\\bar{x} - \\mu_0}{s \/ \\sqrt{n}}[\/latex] is too extreme.","rendered":"

The main idea of a hypothesis test is to use the data as evidence to disprove the null [latex]H_0[\/latex]<\/strong> and thus prove that the alternative [latex]H_a[\/latex] is true<\/strong>. The idea behind a hypothesis test for the population mean is as follows:<\/p>\n

Collect from the population a simple random sample: [latex]x_1, x_2, \\dots, x_n[\/latex] and calculate the sample mean [latex]\\bar{x} = \\frac{x_1 + x_2 + \\dots + x_n}{n}[\/latex]. Our \u201cevidence\u201d stems from the discrepancy between the point estimate [latex]\\bar{x}[\/latex] and the hypothesized population mean [latex]\\mu_0[\/latex].<\/p>\n

Reject the null hypothesis [latex]H_0[\/latex] if the sample mean [latex]\\bar{x}[\/latex] does not support the null [latex]H_0[\/latex]. That is, we should reject [latex]H_0[\/latex] if [latex]\\bar{x}[\/latex] is too extreme. The word \u201cextreme\u201d means contradicting the null [latex]H_0[\/latex] in favour of the alternative [latex]H_a[\/latex].<\/a><\/p>\n

\"Three
Figure 8.2<\/strong>: Rejection Region Based on Sample Mean. [Image Description (See Appendix D Figure 8.2)<\/a>]<\/figcaption><\/figure>\n

In order to quantify how the data (our evidence) contradict the null hypothesis, we first assume the null hypothesis [latex]H_0[\/latex] is true and calculate the chance of observing a sample mean at least as extreme as the observed <\/strong>[latex]\\bar{x}[\/latex].<\/strong> Reject the null [latex]H_0[\/latex] if the chance is small; otherwise, fail to reject [latex]H_0[\/latex]. Recall that for a normal population or a large sample size, the sample mean [latex]\\bar{X} \\sim N \\left( \\mu, \\frac{\\sigma}{\\sqrt{n}} \\right) \\Longrightarrow Z = \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}} \\sim N(0,1)[\/latex] and [latex]t = \\frac{\\bar{X - \\mu}}{s \/ \\sqrt{n}} \\sim t[\/latex] distribution with [latex]df = n-1.[\/latex] We call the variables [latex]Z = \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}}[\/latex] or [latex]t = \\frac{\\bar{X - \\mu}}{s \/ \\sqrt{n}}[\/latex] the test statistics. We should reject the null hypothesis [latex]H_0[\/latex] if the observed test statistic [latex]z_o = \\frac{\\bar{x} - \\mu_0}{\\sigma \/ \\sqrt{n}}[\/latex] or [latex]t = \\frac{\\bar{x} - \\mu_0}{s \/ \\sqrt{n}}[\/latex] is too extreme.<\/p>\n","protected":false},"author":19,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-937","chapter","type-chapter","status-publish","hentry"],"part":889,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/937","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":15,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/937\/revisions"}],"predecessor-version":[{"id":4808,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/937\/revisions\/4808"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/889"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/937\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=937"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=937"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=937"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=937"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}