{"id":1152,"date":"2021-06-23T10:16:37","date_gmt":"2021-06-23T14:16:37","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=1152"},"modified":"2024-02-08T14:34:09","modified_gmt":"2024-02-08T19:34:09","slug":"11-1-introduction","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/11-1-introduction\/","title":{"raw":"11.1 Introduction","rendered":"11.1 Introduction"},"content":{"raw":"Could you design an experiment to check whether a coin is unbalanced? A coin is said to be balanced if each of its two faces is equally likely to occur when the coin is tossed. Hence, if we define [latex]p[\/latex] as the proportion of times that a head occurs among an infinite number of coin tosses, it follows that [latex]p=0.5[\/latex] if the coin is balanced, while [latex]p \\neq 0.5[\/latex] if the coin is unbalanced. Therefore, in order to test whether the coin is unbalanced, we may toss the coin [latex]n[\/latex] times, record the number of heads observed, and then perform a one-proportion z test to test [latex]H_0: p=0.5[\/latex] versus [latex]H_a: p \\neq 0.5[\/latex].\r\n\r\nHow about an experiment to test whether a die is unbalanced? A 6-sided die is considered balanced if each of its six faces is equally likely to occur when the die is rolled. Hence, define [latex]p_i[\/latex] as the proportion of times that face [latex]i[\/latex] occurs among an infinite number of rolls. If the die is balanced, then [latex]p_i = \\frac{1}{6}[\/latex], for [latex]i=1, 2, 3, 4, 5, 6[\/latex]; if the die is unbalanced, then [latex]p_i \\neq \\frac{1}{6}[\/latex] for at least one of the faces. Therefore, in order to test whether the die is unbalanced, we may roll the die [latex]n[\/latex] times and compute the sample proportions [latex]\\hat{p}_1[\/latex] through [latex]\\hat{p}_6[\/latex]; there is evidence that the die is unbalanced, if any [latex]\\hat{p}_i[\/latex] is significantly different from [latex]\\frac{1}{6}[\/latex].\r\n\r\nThe question arises: how do we conduct a hypothesis test when there are six proportions of interest? The na\u00efve approach is to perform six one-proportion z tests. However, this approach is problematic for two main reasons. First, it is time-consuming to conduct six consecutive hypothesis tests; a single test would be more efficient. Second, when several hypothesis tests are performed in succession, the overall type I error rate increases (this is called the multiple comparisons problem). The solution to these problems is to perform a single hypothesis test, with hypotheses\r\n

[latex]H_0: p_1 = p_2 = p_3 = p_4 = p_5 = p_6 = \\frac{1}{6}[\/latex] versus [latex]H_a: p_i \\neq \\frac{1}{6}[\/latex] for at least one [latex]i = 1,2,3,4,5,6[\/latex].<\/p>\r\nTo test such hypotheses, we rely on new types of tests based on the chi-square distribution. The tests are referred to as chi-square tests.","rendered":"

Could you design an experiment to check whether a coin is unbalanced? A coin is said to be balanced if each of its two faces is equally likely to occur when the coin is tossed. Hence, if we define [latex]p[\/latex] as the proportion of times that a head occurs among an infinite number of coin tosses, it follows that [latex]p=0.5[\/latex] if the coin is balanced, while [latex]p \\neq 0.5[\/latex] if the coin is unbalanced. Therefore, in order to test whether the coin is unbalanced, we may toss the coin [latex]n[\/latex] times, record the number of heads observed, and then perform a one-proportion z test to test [latex]H_0: p=0.5[\/latex] versus [latex]H_a: p \\neq 0.5[\/latex].<\/p>\n

How about an experiment to test whether a die is unbalanced? A 6-sided die is considered balanced if each of its six faces is equally likely to occur when the die is rolled. Hence, define [latex]p_i[\/latex] as the proportion of times that face [latex]i[\/latex] occurs among an infinite number of rolls. If the die is balanced, then [latex]p_i = \\frac{1}{6}[\/latex], for [latex]i=1, 2, 3, 4, 5, 6[\/latex]; if the die is unbalanced, then [latex]p_i \\neq \\frac{1}{6}[\/latex] for at least one of the faces. Therefore, in order to test whether the die is unbalanced, we may roll the die [latex]n[\/latex] times and compute the sample proportions [latex]\\hat{p}_1[\/latex] through [latex]\\hat{p}_6[\/latex]; there is evidence that the die is unbalanced, if any [latex]\\hat{p}_i[\/latex] is significantly different from [latex]\\frac{1}{6}[\/latex].<\/p>\n

The question arises: how do we conduct a hypothesis test when there are six proportions of interest? The na\u00efve approach is to perform six one-proportion z tests. However, this approach is problematic for two main reasons. First, it is time-consuming to conduct six consecutive hypothesis tests; a single test would be more efficient. Second, when several hypothesis tests are performed in succession, the overall type I error rate increases (this is called the multiple comparisons problem). The solution to these problems is to perform a single hypothesis test, with hypotheses<\/p>\n

[latex]H_0: p_1 = p_2 = p_3 = p_4 = p_5 = p_6 = \\frac{1}{6}[\/latex] versus [latex]H_a: p_i \\neq \\frac{1}{6}[\/latex] for at least one [latex]i = 1,2,3,4,5,6[\/latex].<\/p>\n

To test such hypotheses, we rely on new types of tests based on the chi-square distribution. The tests are referred to as chi-square tests.<\/p>\n","protected":false},"author":19,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1152","chapter","type-chapter","status-publish","hentry"],"part":1148,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1152","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":10,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1152\/revisions"}],"predecessor-version":[{"id":4878,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1152\/revisions\/4878"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/1148"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1152\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1152"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=1152"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1152"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}