Rec 14 - CI and HT for Proportions

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1430

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Apr 3, 2024

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STAT 1430 Recitation 14 CI and HT for Proportions PART 1: NOTE: ALL CONFIDENCE LEVELS (for confidence intervals) ARE 95% UNLESS OTHERWISE STATED. 1. A poll of 5000 American adults asked about what profession they would prefer their marriage partner to have. The choice receiving the highest percent, 754 of the responses, was “doctors, nurses, and other health care professionals.” A. What is the value of ^ p ? B. Calculate the standard error of ^ p (that’s the part that comes after the Z value) C. Estimate the proportion of all American adults who would prefer a doctor, nurse, and other health care professional as their marriage partners. Use the formula, verify with StatCrunch. Using StatCrunch to do a Confidence Interval for a proportion with a summary of the data: -Open Stat Crunch -STATS/Proportion Stats/One Sample/With Summary/ -Fill in the # successes (# yeses) and # observations (sample size) -Click “Confidence Interval for p” and COMPUTE! 2. In a survey of 1,430 undergrads, 687 reported that they have one or more credit cards. Estimate the proportion of all college undergrads who have at least one credit card. A. Do this work by hand first and show the work. Make sure you always interpret your results! B. Use StatCrunch to confirm your results. 3. For a study of dieting among college students, 267 college women aged 18-25 years were surveyed. Of these, 60% reported that they have been on a diet sometime during the past year. Estimate the proportion of college women aged 18-25 years in this population who dieted last year. A. To put this in StatCrunch, you need to know the # of yeses. Calculate that here: B. Use StatCrunch to find your estimate. Interpret the results.

STAT 1430 Recitation 14 CI and HT for Proportions 4. As part of the quality improvement program, your mail-order company is studying the process of filling customer orders. According to company standards, an order is shipped on time if it is sent within 2 working days of the time it is received. You select a random sample of 150 of the 5000 orders received in the past month for an audit. The audit reveals that 124 of them were shipped on time. Estimate the true proportion of the entire month’s order that were shipped on time. A. Use StatCrunch to get the results: 5. Of the 500 U.S. households that responded to your survey, 58% of them said they were from rural areas. A. Use this information to estimate the proportion of all U.S. households that are from rural areas. USE STATCRUNCH. B. The U.S. Census Bureau reports that 56% of U.S. households are from rural areas. Based on your previous answer, what do you conclude about the Census report? 6. Using the Movie Box Office Data from Recitation 5 (Part 1): A. Examine the variable “ Rating1 ”- whether or not there are age restrictions for the movie (1=yes,0=no) . Use StatCrunch to find a 95% confidence interval for the percentage of all the movies that have age restrictions for that year. Interpret your results. B. Using StatCrunch, find a 95% confidence interval for the proportion of movies that year that made a profit. Note: There is a profit variable in the data set. 7. The margin of error is sadly missing in this USA TODAY snapshot below but you can supply it using what you learned in class. Calculate the exact margin of error in this case using the formula and create a 95% confidence interval (and interpret it).

STAT 1430 Recitation 14 CI and HT for Proportions 8. A national opinion poll wants to know what proportion of all American adults agree that parents should be given vouchers good for education at any public or private school of their choice. Using the ‘quick and dirty’ method from lecture , how large of a sample is required to obtain a margin of error of ± 0.035? 9. Land’s Beginning sells merchandise through the mail. It is considering buying a list of addresses from a magazine. The magazine claims that many of its subscriber have high incomes. Land’s Beginning will verify this claim by taking a random sample of people on the magazine’s subscribers’ mailing list and finding a confidence interval for the proportion of subscribers that have high incomes. They would like the margin of error of 0.05 or less. What is a good sample size, based on the ‘quick and dirty method’ from lecture? PART 2: HT for Proportions 1. Of the 500 households that responded to the Christmas tree marketing survey, 38% of them were from rural areas and 62% of them were from the urban areas. According to the census, 36% of the Indiana households are in rural areas and the remaining 64% are in urban areas. Let p be the proportion of the rural respondents. Set up hypotheses about p 0 and perform a test of significance to examine how well the sample represents the state in regard to rural versus urban residence. Summarize your results. A. Set up your Ho and Ha for a hypothesis test. B. What is your test statistic? C. What is your p-value? D. What conditions did you have to check in order to solve this problem? List them and check them. E. What is your decision regarding Ho? F. What is your conclusion in the context of the problem? G. What type of error could you be committing here, Type I or Type II? What would be the impact of that error in terms of the Christmas tree marketing survey? 2. Explain what is wrong with the following: H 0 : ^ p = 0.6 and H a : ^ p > 0.6 3. Land’s Beginning sells merchandise through the mail. It is considering buying a list of addresses from a magazine. The magazine claims that at least 25% of its subscribers have high incomes. You believe it’s less. You take a random sample of 1,000 subscribers and find that 23% have high incomes. What do you conclude about the claim, based on your data?

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STAT 1430 Recitation 14 CI and HT for Proportions A. Set up your Ho and Ha for a hypothesis test. B. What is your test statistic? C. What is your p-value? D. What conditions did you have to check in order to solve this problem? List them and check them. E. What is your decision regarding Ho? F. What is your conclusion in the context of the problem? G. What type of error could you be committing here, Type I or Type II? What would be the impact of that error in terms what Land’s End believes? 4. Consumer Reports magazine says 60% of their subscribers are men. You believe the actual percentage is higher than that. Your sample of 100 subscribers shows 65% are men. What do you conclude? A. Set up your Ho and Ha for a hypothesis test. B. What is your test statistic ? USE STATCRUNCH. Using StatCrunch to do a Hypothesis Test for proportion with a summary of the data: -Open Stat Crunch -STATS/Proportion Stats/One Sample/With Summary/ -Fill in the # successes (# yeses) and # observations (sample size) -Click “Hypothesis Test for p” and fill out the Ho and Ha / COMPUTE! C. What is your p-value ? USE STATCRUNCH. D. What conditions did you have to check in order to solve this problem? List them and check them. E. What is your decision regarding Ho? F. What is your conclusion in the context of the problem? G. What type of error could you be committing here, Type I or Type II? What would be the impact of that error on Consumer Reports? 5. A recent report by the Smart Brain Institute claims that 54% of 4-6 year-olds prefer watching TV over spending time with their parents. You believe the true percentage is lower than that. You take a survey of 200 randomly chosen children in the 4-6 years old range and find that 105 of them would prefer TV over their parents. What do you conclude about the Smart Brain Institute’s claim based on your data? A. Set up your Ho and Ha for a hypothesis test. B. What is your test statistic? USE STATCRUNCH C. What is your p-value? USE STATCRUNCH

STAT 1430 Recitation 14 CI and HT for Proportions D. What conditions did you have to check in order to solve this problem? List them and check them. E. What is your decision regarding Ho? F. What is your conclusion in the context of the problem? G. What type of error could you be committing here, Type I or Type II? What would be the impact of that error in terms of the Smart Brain Institute? 6. The media reports that Person X is the leading Democratic Party candidate in an election, and that he or she is projected to receive 20% of the vote over his or her Democratic Party opponents. You believe the percentage is higher than that. A survey of 1,018 adults finds that 23% would vote for Person X as the Democratic Party candidate. What do you conclude about the media report, based on your data? STATCRUNCH NOTE: DO NOT PUT .23 IN FOR THE NUMBER OF SUCCESSES. THAT IS THE PERCENTAGE OF SUCCESSES. MULTIPLY BY THE SAMPLE SIZE TO GET THE NUMBER OF SUCCESSES. A. Set up your Ho and Ha for a hypothesis test. B. What is your test statistic? USE STATCRUNCH. C. What is your p-value? USE STATCRUNCH. D. What is your decision regarding Ho? E. What is your conclusion in the context of the problem? F. What conditions did you have to check in order to solve this problem? List them and check them. G. What type of error could you be committing here, Type I or Type II? What would be the impact of that error in terms of the media report?

Rec 14 - CI and HT for Proportions

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