{"id":187352,"date":"2025-02-04T08:54:50","date_gmt":"2025-02-04T08:54:50","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=187352"},"modified":"2025-02-04T08:54:52","modified_gmt":"2025-02-04T08:54:52","slug":"the-school-nurse-thinks-the-average-height-of-7th-graders-has-increased","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/04\/the-school-nurse-thinks-the-average-height-of-7th-graders-has-increased\/","title":{"rendered":"The school nurse thinks the average height of 7th graders has increased"},"content":{"rendered":"\n

The school nurse thinks the average height of 7th graders has increased. The average height of a 7th grader five years ago was 145 cm. She takes a random sample of 101 students and finds that the average height of her sample is 147 cm with a standard deviation of 20 cm. Determine the critical value for testing whether the 7th graders now are taller than they were before? Use a = 0.05.
-1.984
1.984
1.660
-1.660
2.132<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To determine the critical value<\/strong> for testing whether the average height of 7th graders has increased, we need to conduct a one-tailed hypothesis test<\/strong> using a t-distribution<\/strong> because the population standard deviation is unknown, and we have a sample size of 101.<\/p>\n\n\n\n

Step 1: Define the Hypotheses<\/h3>\n\n\n\n
    \n
  • Null Hypothesis ((H_0))<\/strong>: The average height of 7th graders has not increased. That is:
    [
    H_0: \\mu = 145 \\text{ cm}
    ]<\/li>\n\n\n\n
  • Alternative Hypothesis ((H_a))<\/strong>: The average height of 7th graders has increased.
    [
    H_a: \\mu > 145 \\text{ cm}
    ]
    Since we are testing for an increase, this is a right-tailed test<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    Step 2: Identify Significance Level and Degrees of Freedom<\/h3>\n\n\n\n

    The significance level is given as ( \\alpha = 0.05 ).<\/p>\n\n\n\n

    Degrees of freedom (( df )) for a t-test<\/strong> is given by:
    [
    df = n – 1 = 101 – 1 = 100
    ]<\/p>\n\n\n\n

    Step 3: Find the Critical Value<\/h3>\n\n\n\n

    Since this is a right-tailed test<\/strong>, we need to find the critical value of ( t )<\/strong> for ( df = 100 )<\/strong> at ( \\alpha = 0.05 )<\/strong>.<\/p>\n\n\n\n

    From a t-table<\/strong>, the critical t-value for ( df = 100 ) at ( \\alpha = 0.05 ) (right-tailed test)<\/strong> is:
    [
    t_{0.05, 100} = 1.660
    ]<\/p>\n\n\n\n

    Step 4: Conclusion<\/h3>\n\n\n\n

    The correct answer is 1.660<\/strong>.<\/p>\n\n\n\n

    Explanation:<\/h3>\n\n\n\n
      \n
    • The t-distribution<\/strong> is used because the population standard deviation (( \\sigma )) is unknown.<\/li>\n\n\n\n
    • The right-tailed test<\/strong> is chosen since we are testing for an increase<\/strong> in height.<\/li>\n\n\n\n
    • Using the t-table<\/strong>, we find that the critical value at 100 degrees of freedom and ( \\alpha = 0.05 )<\/strong> is 1.660<\/strong>.<\/li>\n\n\n\n
    • If the calculated test statistic exceeds<\/strong> 1.660, we reject the null hypothesis<\/strong> and conclude that the average height has significantly increased.<\/li>\n\n\n\n
    • If the test statistic is less than<\/strong> 1.660, we fail to reject the null hypothesis<\/strong>, meaning we do not have enough evidence to support that 7th graders are taller today.<\/li>\n<\/ul>\n\n\n\n

      Thus, the correct answer is 1.660<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

      The school nurse thinks the average height of 7th graders has increased. The average height of a 7th grader five years ago was 145 cm. She takes a random sample of 101 students and finds that the average height of her sample is 147 cm with a standard deviation of 20 cm. Determine the critical […]<\/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-187352","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187352","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=187352"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187352\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}