{"id":33897,"date":"2025-06-23T12:04:26","date_gmt":"2025-06-23T12:04:26","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=33897"},"modified":"2025-06-23T12:04:33","modified_gmt":"2025-06-23T12:04:33","slug":"the-production-manager-of-a-manufacturer-wants-to-evaluate-a-modified-ball-bearing-production-process","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-production-manager-of-a-manufacturer-wants-to-evaluate-a-modified-ball-bearing-production-process\/","title":{"rendered":"The production manager of a manufacturer wants to evaluate a modified ball bearing production process"},"content":{"rendered":"\n<p>The production manager of a manufacturer wants to evaluate a modified ball bearing production process. When the process is operating properly, the process produces ball bearings whose weights have a mean of 5 ounces and a standard deviation of 0.1 ounce. A new raw-material supplier was used for a recent production run, and the manager wants to know if that change has caused any problem. If there was a problem, the mean weight of a ball bearing would be different. The manager will test the following hypotheses and From a random sample of 100 ball bearings, the sample mean was 5.03. The significance level is specified as 5%. What is the probability of type I errors in this hypothesis testing? Question 9 options: a) 5.03% b) 1.00% c) 1.96% d) 0.03% e) 5.00%<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is: <strong>e) 5.00%<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>In hypothesis testing, the <strong>probability of a Type I error<\/strong> is the probability of rejecting the null hypothesis when it is actually true. This probability is also known as the <strong>significance level<\/strong>, denoted by <strong>\u03b1 (alpha)<\/strong>.<\/p>\n\n\n\n<p>Here is the context:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The null hypothesis H0H_0 states that the mean weight of ball bearings is 5 ounces.<\/li>\n\n\n\n<li>The alternative hypothesis HaH_a states that the mean weight is different from 5 ounces (two-tailed test).<\/li>\n\n\n\n<li>The significance level \u03b1 is given as 5%, or 0.05.<\/li>\n\n\n\n<li>The sample size is 100, and the sample mean observed is 5.03 ounces.<\/li>\n<\/ul>\n\n\n\n<p>The significance level \u03b1 of 5% means that there is a 5% chance that the test will wrongly reject the null hypothesis, even if the true mean is indeed 5 ounces. This error is a <strong>Type I error<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>Type I error<\/strong> probability is chosen by the researcher before conducting the test.<\/li>\n\n\n\n<li>It does not depend on the sample data but is fixed by the significance level.<\/li>\n\n\n\n<li>In this problem, the significance level is explicitly stated as 5%, so the Type I error probability is 5%.<\/li>\n<\/ul>\n\n\n\n<p>The other answer choices are percentages that might confuse with p-values or test statistics, but the question specifically asks for the probability of Type I error, which directly corresponds to the significance level.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Type I error probability = significance level \u03b1<\/li>\n\n\n\n<li>\u03b1 = 0.05 or 5%<\/li>\n\n\n\n<li>Therefore, the probability of Type I error in this hypothesis test is <strong>5.00%<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>This means if the process is operating correctly, there is a 5% chance that the manager would incorrectly conclude the new supplier has caused a change in the mean weight.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The production manager of a manufacturer wants to evaluate a modified ball bearing production process. When the process is operating properly, the process produces ball bearings whose weights have a mean of 5 ounces and a standard deviation of 0.1 ounce. A new raw-material supplier was used for a recent production run, and the manager [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-33897","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33897","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=33897"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33897\/revisions"}],"predecessor-version":[{"id":33898,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33897\/revisions\/33898"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=33897"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=33897"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=33897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}