{"id":1261,"date":"2023-12-24T08:24:09","date_gmt":"2023-12-24T08:24:09","guid":{"rendered":"https:\/\/fblog.quizplus.com\/blog\/?p=1261"},"modified":"2023-12-24T08:24:10","modified_gmt":"2023-12-24T08:24:10","slug":"critical-value-in-hypothesis-testing-steps-examples","status":"publish","type":"post","link":"https:\/\/quizplus.com\/blog\/critical-value-in-hypothesis-testing-steps-examples\/","title":{"rendered":"Critical Value in Hypothesis Testing: Steps & Examples"},"content":{"rendered":"\n

A critical value is a point in the distribution of test statistics that defines the boundary between the acceptance and rejection regions. It plays a crucial role in determining whether to reject the null hypothesis by comparing it to the calculated test statistic. <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

The null hypothesis (H0<\/sub>) is rejected in favor of the alternative hypothesis (H1<\/sub>) if the value of the test statistic is more extreme than the critical value. The critical value varies according to the statistical test being used and the level of significance chosen.<\/p>\n\n\n\n

The purpose of this post is to explore the concept of critical values in more detail. We will learn how to find out these values, compare them with calculated test statistics, and identify whether to reject the null hypothesis. Multiple examples will be covered to help our readers gain confidence in finding critical values.<\/p>\n\n\n\n

Let\u2019s start this topic by defining its basic definition. <\/p>\n\n\n\n

Definition of the Critical Value <\/strong><\/h2>\n\n\n\n

In the hypothesis test<\/a>, the critical value is a particular value that separates the acceptance and rejection regions. The null hypothesis is rejected if the test statistic value falls within the rejection region; otherwise, it will not be rejected. The critical value can be for a one-tailed (left or right tail) or a two-tailed test. <\/p>\n\n\n\n

Two-tailed tests require two critical values whereas one-tailed tests require only one. The critical value is determined based on the significance level of the test and the degrees of freedom (if applicable). It is used in many hypothesis tests, such as z-tests, t-tests, F-tests, and chi-square tests.<\/p>\n\n\n\n

How to Calculate a Critical Value?<\/strong><\/h2>\n\n\n\n

The method to calculate critical value<\/a> depends on the distribution of the test statistic. Critical value can be calculated using the confidence interval or the significance level (or alpha).<\/p>\n\n\n\n

    \n
  1. Calculate the significance level using the confidence interval:<\/li>\n<\/ol>\n\n\n\n

    Significance level = \u03b1 = 1 \u2013 (confidence level \/ 100)<\/p>\n\n\n\n

      \n
    1. Identify whether the test is one-tailed or two-tailed since the critical values and distribution tables differ for each type. The alpha value does not change in a one-tailed test. However, in a two-tailed test, 2 divides the alpha value. <\/li>\n\n\n\n
    2. Critical values are unique to each test. A distribution table can be used to determine the critical value for a test based on its alpha value.<\/li>\n<\/ol>\n\n\n\n

      The explanation for the step three process will be presented in the upcoming section.<\/p>\n\n\n\n

      T-Critical Value <\/strong><\/h2>\n\n\n\n

      T-critical value helps us make decisions about the difference between a sample mean and a population mean. The t-critical value is used when the sample size is small (less than 30) and\/or the population standard deviation is unknown. <\/p>\n\n\n\n

      Step to find t-critical value<\/strong><\/p>\n\n\n\n

        \n
      1. Determine the alpha from the confidence level. <\/li>\n\n\n\n
      2. Calculate degrees of freedom df by subtracting 1 from sample size n.  <\/li>\n\n\n\n
      3. Use a one-tailed t-distribution table for the one-tailed hypothesis test. For a two-tailed hypothesis test, use a two-tailed t-distribution table.<\/li>\n\n\n\n
      4. Find the value of calculated degrees of freedom in the column and look alpha value on the top row. Where both meet inside the t-table is the t-critical value.<\/li>\n<\/ol>\n\n\n\n

        Decision Criteria:<\/strong><\/p>\n\n\n\n

          \n
        • Reject if t-critical value < test statistic (Right-Tailed Test)<\/li>\n\n\n\n
        • Reject if t-critical value > test statistic (Left-Tailed Test)<\/li>\n\n\n\n
        • Reject if the test statistic either falls beyond the upper or lower critical values (Two-Tailed Test)<\/li>\n<\/ul>\n\n\n\n

          This criterion is the same for other hypothesis tests. Just the value of test statistic and critical value will change.<\/p>\n\n\n\n

          Z-Critical value<\/strong><\/h2>\n\n\n\n

          Z-critical value can be used if the standard deviation of the population is known and\/or the sample size n is large (greater or equal to 30). It is found from the standard normal distribution or z-distribution. <\/p>\n\n\n\n

          The z-critical value is evaluated by using the following method:<\/strong><\/p>\n\n\n\n

            \n
          1. Determine the alpha level based on the desired confidence level.<\/li>\n\n\n\n
          2. Identify whether the test is one or two-tailed. For a two-tailed test, subtract \u03b1 from 1; for a one-tailed test, subtract \u03b1 from 0.5.<\/li>\n\n\n\n
          3. Locate this resulting value within the z-table.<\/li>\n\n\n\n
          4. Add the values from the upper row and the leftmost column where the intersection occurs. For a left-tailed hypothesis test, include a negative sign.<\/li>\n<\/ol>\n\n\n\n

            Example of the Critical Value<\/strong><\/h2>\n\n\n\n

            Example 1:<\/strong><\/p>\n\n\n\n

            Given a sample mean of 73, a sample standard deviation of 17.5, and a sample size of 12, test the hypothesis that the value of the population mean is 60 against the alternative that is more than 60 using a 0.01 level of significance. <\/p>\n\n\n\n

            Solution: <\/strong><\/p>\n\n\n\n

            Given:<\/p>\n\n\n\n

            x\u0304 = 73, n = 12, s =17.5, \u03bc =60, \u03b1 = 0.01<\/p>\n\n\n\n

            (To test the hypothesis that the population mean is greater than 60, we will use a one-sample t-test since the population standard deviation is unknown)<\/p>\n\n\n\n

            Step 1:<\/strong> State the Null and Alternative Hypotheses: <\/p>\n\n\n\n

            H0<\/sub>:  \u03bc = 60<\/p>\n\n\n\n

            H1<\/sub>:  \u03bc > 60<\/p>\n\n\n\n

            Step 2:<\/strong> Calculate the t-test Statistic for one mean.<\/p>\n\n\n\n

            \u2234 T-statistic test = (x\u0304 – \u03bc) \/ (s \/ \u221an)<\/p>\n\n\n\n

            = (73 \u2013 60) \/ (17.5\/\u221a12)<\/p>\n\n\n\n

            = 13\/5.05<\/p>\n\n\n\n

            = 2.57<\/p>\n\n\n\n

            Step 3:<\/strong> Calculate the t-Critical Value. <\/p>\n\n\n\n

             \u03b1 = 0.01<\/p>\n\n\n\n

            \u2234 df = n \u2013 1 <\/p>\n\n\n\n

            df = 12 \u2013 1 = 11<\/p>\n\n\n\n

            Find the value of calculated degrees of freedom (11) in the column and look alpha value (0.01) on the top row. <\/p>\n\n\n\n

            \"\"\/<\/figure>\n\n\n\n

            As both meet at 2.718, thus, that is a t-critical value. <\/p>\n\n\n\n

            Step 4:<\/strong> Draw a Conclusion: <\/p>\n\n\n\n

            The calculated t-statistic (2.57) is less than the critical value (2.718), so we fail to reject the null hypothesis.<\/p>\n\n\n\n

            Summary <\/strong><\/h2>\n\n\n\n

            In this article, we discussed the definition of critical values and explained how to find them. We covered the methods for determining both t and z critical values along with formulas to calculate their test statistics. In the example section, we solved problems to provide a clear understanding. You will be able to calculate z and t tests easily and quickly after reading this article.<\/p>\n","protected":false},"excerpt":{"rendered":"

            A critical value is a point in the distribution of test statistics that defines the boundary between the acceptance and rejection regions. It plays a crucial role in determining whether to reject the null hypothesis by comparing it to the calculated test statistic.  The null hypothesis (H0) is rejected in favor of the alternative hypothesis […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"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":"var(--ast-global-color-4)","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":[1],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/posts\/1261"}],"collection":[{"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/comments?post=1261"}],"version-history":[{"count":1,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/posts\/1261\/revisions"}],"predecessor-version":[{"id":1262,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/posts\/1261\/revisions\/1262"}],"wp:attachment":[{"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/media?parent=1261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/categories?post=1261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quizplus.com\/blog\/wp-json\/wp\/v2\/tags?post=1261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}