{"id":5314,"date":"2022-08-19T17:25:07","date_gmt":"2022-08-19T17:25:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5314"},"modified":"2022-08-19T17:52:18","modified_gmt":"2022-08-19T17:52:18","slug":"11b-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/11b-coreq\/","title":{"raw":"11B Coreq","rendered":"11B Coreq"},"content":{"raw":"In the next preview assignment, you will need to calculate and interpret standardized\u00a0 scores, use a normal distribution to describe the sampling variability of a sample\u00a0 proportion, and use the Empirical Rule to identify unusual values in a normal\u00a0 distribution.\r\n\r\nStandardized Scores\r\n\r\nIn In-Class Activity 4.E, you learned to compute and interpret standardized scores by\u00a0 subtracting the mean and dividing by the standard deviation.\r\n\r\n[latex] standardized~error = \\frac{data~value - mean}{standard~deviation} [\/latex]\r\n\r\nA standardized score describes how far a data value is from the mean in terms of the standard deviation. For example, a standardized score of -2 means that the data value is 2 standard deviations below the mean.\r\n\r\nStandardized scores are useful for deciding whether a value is unusual relative to its distribution. They are also useful for comparing values that are on different scales.\r\n
\r\n

Question 1<\/h3>\r\nSuppose Brett took a test where the class average was a 75 with a standard deviation of 10. He made a 93 on the test. Calculate and interpret his standardized score.\r\n\r\n<\/div>\r\n
\r\n

Question 2<\/h3>\r\nSuppose Isabel took a test where the class average was a 70 with a standard\u00a0 deviation of 6. She made a 90 on the test. Calculate and interpret her standardized\u00a0 score. Round to two decimal places.\r\n\r\n<\/div>\r\n
\r\n

Question 3<\/h3>\r\nWhich of the previous test scores is more unusual, relative to its own distribution?\r\n\r\n<\/div>\r\nSampling Distribution of a Sample Proportion\r\n\r\nIn In-Class Activity 9.C, you learned that a normal distribution can be used to describe\u00a0 the sampling distribution of a sample proportion as long as the sample size is large enough:\r\n\r\n[latex]np \\geq 10 [\/latex] and [latex]n(1-p) \\geq 10 [\/latex]\r\n\r\nFurther, the center of the sampling distribution will be equal to the value of the population proportion, [latex] p[\/latex], and the standard error formula can be used to describe the spread of the sampling distribution. (When the population proportion is known, the standard error calculated using this formula is the standard deviation of the sampling distribution.)\r\n\r\n[latex] SE = \\sqrt{\\frac{p(1-p)}{n}}[\/latex]\r\n\r\nExample: Out of over 5,000 students admitted to the University of Georgia in a recent year, 16% came from outside the state of Georgia. How many out-of-state students would we expect to see in a random sample of 100 freshmen the following year?\r\n
\r\n

Question 4<\/h3>\r\nWould you expect the proportion of out-of-state students to be exactly 0.16 in every random sample of size 100? Explain.\r\n\r\n<\/div>\r\n
\r\n

Question 5<\/h3>\r\nWhat would be more common: a sample proportion close to 0.16 or a sample proportion far from 0.16?\r\n\r\n<\/div>\r\nThe following graph shows the sampling distribution for this scenario.\r\n\r\n\"\"\r\n
\r\n

Question 6<\/h3>\r\nAdd a label to the x-axis of the sampling distribution.\r\n\r\n<\/div>\r\n
\r\n

Question 7<\/h3>\r\nIs it reasonable to describe the sampling distribution using a normal distribution?\u00a0 Justify your answer by checking the sample size condition.\r\n\r\n<\/div>\r\n
\r\n

Question 8<\/h3>\r\nWhat is the mean of this sampling distribution? In other words, where is the\u00a0 distribution centered?\r\n\r\n<\/div>\r\n
\r\n

Question 9<\/h3>\r\nCalculate the standard error. In other words, calculate the standard deviation of the\u00a0 sampling distribution of the sample proportion. Round to three decimal places.\r\n\r\n<\/div>\r\n
\r\n

Question 10<\/h3>\r\nBy the Empirical Rule, about 95% of sample proportions fall between ___ and ___.\r\n\r\n<\/div>\r\nPractice\r\n\r\nIn a recent year, 58% of male students at Millsaps College were members of a\u00a0 fraternity. How many fraternity members would we expect to see in a random sample of\u00a0 30 male students from Millsaps College? The following graph shows the sampling\u00a0 distribution for this scenario.\r\n\r\n \r\n\r\n\"\"\r\n
\r\n

Question 11<\/h3>\r\nIs it reasonable to describe the sampling distribution using a normal distribution?\u00a0 Justify your answer by checking the sample size condition.\r\n\r\n<\/div>\r\n
\r\n

Question 12<\/h3>\r\nWhat is the mean of this sampling distribution? In other words, where is the\u00a0 distribution centered?\r\n\r\n<\/div>\r\n
\r\n

Question 13<\/h3>\r\nCalculate the standard error. In other words, calculate the standard deviation of the\u00a0 sampling distribution of the sample proportion. Round to three decimal places.\r\n\r\n<\/div>\r\n
\r\n

Question 14<\/h3>\r\nAbout 99.7% of sample proportions would fall between ___ and ___. Use the\u00a0 Empirical Rule.\r\n\r\n<\/div>","rendered":"

In the next preview assignment, you will need to calculate and interpret standardized\u00a0 scores, use a normal distribution to describe the sampling variability of a sample\u00a0 proportion, and use the Empirical Rule to identify unusual values in a normal\u00a0 distribution.<\/p>\n

Standardized Scores<\/p>\n

In In-Class Activity 4.E, you learned to compute and interpret standardized scores by\u00a0 subtracting the mean and dividing by the standard deviation.<\/p>\n

[latex]standardized~error = \\frac{data~value - mean}{standard~deviation}[\/latex]<\/p>\n

A standardized score describes how far a data value is from the mean in terms of the standard deviation. For example, a standardized score of -2 means that the data value is 2 standard deviations below the mean.<\/p>\n

Standardized scores are useful for deciding whether a value is unusual relative to its distribution. They are also useful for comparing values that are on different scales.<\/p>\n

\n

Question 1<\/h3>\n

Suppose Brett took a test where the class average was a 75 with a standard deviation of 10. He made a 93 on the test. Calculate and interpret his standardized score.<\/p>\n<\/div>\n

\n

Question 2<\/h3>\n

Suppose Isabel took a test where the class average was a 70 with a standard\u00a0 deviation of 6. She made a 90 on the test. Calculate and interpret her standardized\u00a0 score. Round to two decimal places.<\/p>\n<\/div>\n

\n

Question 3<\/h3>\n

Which of the previous test scores is more unusual, relative to its own distribution?<\/p>\n<\/div>\n

Sampling Distribution of a Sample Proportion<\/p>\n

In In-Class Activity 9.C, you learned that a normal distribution can be used to describe\u00a0 the sampling distribution of a sample proportion as long as the sample size is large enough:<\/p>\n

[latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex]<\/p>\n

Further, the center of the sampling distribution will be equal to the value of the population proportion, [latex]p[\/latex], and the standard error formula can be used to describe the spread of the sampling distribution. (When the population proportion is known, the standard error calculated using this formula is the standard deviation of the sampling distribution.)<\/p>\n

[latex]SE = \\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/p>\n

Example: Out of over 5,000 students admitted to the University of Georgia in a recent year, 16% came from outside the state of Georgia. How many out-of-state students would we expect to see in a random sample of 100 freshmen the following year?<\/p>\n

\n

Question 4<\/h3>\n

Would you expect the proportion of out-of-state students to be exactly 0.16 in every random sample of size 100? Explain.<\/p>\n<\/div>\n

\n

Question 5<\/h3>\n

What would be more common: a sample proportion close to 0.16 or a sample proportion far from 0.16?<\/p>\n<\/div>\n

The following graph shows the sampling distribution for this scenario.<\/p>\n

\"\"<\/p>\n

\n

Question 6<\/h3>\n

Add a label to the x-axis of the sampling distribution.<\/p>\n<\/div>\n

\n

Question 7<\/h3>\n

Is it reasonable to describe the sampling distribution using a normal distribution?\u00a0 Justify your answer by checking the sample size condition.<\/p>\n<\/div>\n

\n

Question 8<\/h3>\n

What is the mean of this sampling distribution? In other words, where is the\u00a0 distribution centered?<\/p>\n<\/div>\n

\n

Question 9<\/h3>\n

Calculate the standard error. In other words, calculate the standard deviation of the\u00a0 sampling distribution of the sample proportion. Round to three decimal places.<\/p>\n<\/div>\n

\n

Question 10<\/h3>\n

By the Empirical Rule, about 95% of sample proportions fall between ___ and ___.<\/p>\n<\/div>\n

Practice<\/p>\n

In a recent year, 58% of male students at Millsaps College were members of a\u00a0 fraternity. How many fraternity members would we expect to see in a random sample of\u00a0 30 male students from Millsaps College? The following graph shows the sampling\u00a0 distribution for this scenario.<\/p>\n

 <\/p>\n

\"\"<\/p>\n

\n

Question 11<\/h3>\n

Is it reasonable to describe the sampling distribution using a normal distribution?\u00a0 Justify your answer by checking the sample size condition.<\/p>\n<\/div>\n

\n

Question 12<\/h3>\n

What is the mean of this sampling distribution? In other words, where is the\u00a0 distribution centered?<\/p>\n<\/div>\n

\n

Question 13<\/h3>\n

Calculate the standard error. In other words, calculate the standard deviation of the\u00a0 sampling distribution of the sample proportion. Round to three decimal places.<\/p>\n<\/div>\n

\n

Question 14<\/h3>\n

About 99.7% of sample proportions would fall between ___ and ___. Use the\u00a0 Empirical Rule.<\/p>\n<\/div>\n","protected":false},"author":574340,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5314","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5314","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5314\/revisions"}],"predecessor-version":[{"id":5326,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5314\/revisions\/5326"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5314\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5314"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5314"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5314"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}