Rec 14 CI and HT for Proportions

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Feb 20, 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, 805 of the responses, was “doctors, nurses, and other health care professionals.” A. What is the value of ? 805/5000 =0.161 B. Calculate the standard error of (that’s the part that comes after the Z value) Square root of ((.161(1-.161))/(5000) = 0.0052 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. The proportion of American adults who would prefer a doctor, nurse, and other health care professional as their partner is between 15.1% and 17.1% 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! p ˆ p ˆ

STAT 1430 Recitation 14 CI and HT for Proportions 2. In a survey of 1,430 undergrads, 1087 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! P-hat = 1087/1430 =.76 .76 +- 1.96 Square root( .76(1-.76) / (1430)) = (0.74, 0.78) Using our data we can say with 95% certainty the proportion of all undergrads that have at least one credit card is between 74% and 78% B. Use StatCrunch to confirm your results. 3. For a study of unhealthy eating behavior, 267 college women aged 18-25 years were surveyed. Of these, 69% 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: P-Hat = 0.69 0.69 +- 1.96 square root((0.69(1-0.69))/(267))= (0.635, 0.745) Use StatCrunch to find your estimate. Interpret the results. One sample proportion summary confidence interval: p : Proportion of successes Method: Standard-Wald 95% confidence interval results: Proportio n Coun t Tota l Sample Prop. Std. Err. L. Limit U. Limit p 1087 1430 0.76013986 0.01129166 0.73800861 0.78227111

STAT 1430 Recitation 14 CI and HT for Proportions From our data we see that with 95% confidence that the proportion of college woman from 18-25 years old in the population who dieted last year is between 63.5% and 74.5% 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 SRS 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. Show the work by hand: 125/150 =.83 0.83+-1.96 square root((0.38 (1-0.83))/ (150) ) = (0.770, 0.890) From our data we can determine with a 95% certainty that the true proportion of this month’s order that were shipped on time between 77% and 89% B. Use StatCrunch to confirm the results:

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STAT 1430 Recitation 14 CI and HT for Proportions 5. Of the 500 U.S. households that responded to your survey, 38% 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. 0.38 +- 1.96 Square root((0.38(1-0.38))/(500)) = (0.337, 0.890) Using our data we can determine with 95% confidence that the proportion of U.S. households that are from rural areas is between 33.7% and 42.3% B. The U.S. Census Bureau reports that 36% of U.S. households are from rural areas. Based on your previous answer, what do you conclude about the Census report? This census report is accurate or very likely to be as it falls between 33.7 and 42.3 6. Using the Movie Box Office Data from Recitation 5A: A. Examine the variable “ Ra#ng1”- whether or not there are age restric#ons 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.

STAT 1430 Recitation 14 CI and HT for Proportions 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. 1.96 square root((0.9(1-0.9))/(1139)) =0.17 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? (1/square root n) = 0.035, (1/0.035)^2 = 816.33 You would need a sample size of 817 to obtain the MOE 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? (1/.05)^2 =400 You need a sample size of 400 for the MOE PART 2: ±

STAT 1430 Recitation 14 CI and HT for Proportions Stat 1430 Rec HT for Propor3ons B HT for Propor3ons 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 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. HO: p=0.36 HA: px36 B. What is your test statistic? 0.93 C. What is your p-value? 0.352 D. What conditions did you have to check in order to solve this problem? List them and check them. 500(1-0.36) > 180 E. What is your decision regarding Ho? We reject it F. What is your conclusion in the context of the problem? It represents the Urban vs Rural residence percentage 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? Type 1. It could be advertising to the wrong people 2. Explain what is wrong with the following: : = 0.6 and : > 0.6 We should not be using P-hat and instead p. P is a population parameter we would not use a sample statistic 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 subscriber 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? A. Set up your Ho and Ha for a hypothesis test. HA: p<0.25 B. What is your test statistic? -1.96 p 0 H 0 ^ p H a ^ p

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STAT 1430 Recitation 14 CI and HT for Proportions C. What is your p-value? 0.0721 D. What conditions did you have to check in order to solve this problem? List them and check them. N * P =1000 *0.25 =250>10 1000(1-0.25)= 750 >10 E. What is your decision regarding Ho? Fail to reject Ho F. What is your conclusion in the context of the problem? Evidence shows that 25% of subscribers have a high income 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? Type 2 because we have failed to reject. We conclude that 25% have high incomes but it would in reality be lower. 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. Ho: p=0.60 Ha: p<0.60 B. What is your test statistic? USE STATCRUNCH. One sample proportion summary hypothesis test: p : Proportion of successes H 0 : p = 0.6 H A : p > 0.6 Hypothesis test results: Proportio n Count Tota l Sample Prop. Std. Err. Z-Stat P- value p 65 100 0.65 0.048989795 1.0206207 0.1537

STAT 1430 Recitation 14 CI and HT for Proportions 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. 0.1537 see statcrunch sheet above D. What conditions did you have to check in order to solve this problem? List them and check them. N p0=100*0.60 =60>10 n(1-p0) =100*(1-0.60)=40>10 E. What is your decision regarding Ho? We fail to reject it F. What is your conclusion in the context of the problem? There is enough data at significance level of 5% to say that more than 60% of the subscribers are male 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? Type 2 it could cause them to begin to more heavily advertise to men more than they would otherwise 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 Ho: p=0.54 Ha: p<0.54 B. What is your test statistic? USE STATCRUNCH

STAT 1430 Recitation 14 CI and HT for Proportions 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? We fail to reject Ho F. What is your conclusion in the context of the problem? We are not sure that the percent is under 54% One sample proportion summary hypothesis test: p : Proportion of successes H 0 : p = 0.54 H A : p < 0.54 Hypothesis test results: Proportio n Count Tota l Sample Prop. Std. Err. Z-Stat P- value p 105 200 0.525 0.03524202 -0.42562827 0.3352 One sample proportion summary hypothesis test: p : Proportion of successes H 0 : p = 0.54 H A : p < 0.54 Hypothesis test results: Proportio n Count Tota l Sample Prop. Std. Err. Z-Stat P- value p 105 200 0.525 0.03524202 -0.42562827 0.3352

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STAT 1430 Recitation 14 CI and HT for Proportions 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? Type 2 error and it would make it below 54% 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. Ho: p=0.20 Ha: p>0.20 B. What is your test statistic? USE STATCRUNCH. Square root ((0.23-0.20)/(0.20(1-0.20))/(1018)) =0.03/0.0125 =2.39 C. What is your p-value? USE STATCRUNCH. D. What is your decision regarding Ho? We reject it E. What is your conclusion in the context of the problem? One sample proportion summary hypothesis test: p : Proportion of successes H 0 : p = 0.2 H A : p > 0.2 Hypothesis test results: Proportio n Count Tota l Sample Prop. Std. Err. Z-Stat P- value p 234.14 1018 0.23 0.012536783 2.3929584 0.0084

STAT 1430 Recitation 14 CI and HT for Proportions There is enough evidence at a significance level of 5% to conclude that the proportion of people will vote for person X is more than 20% F. What conditions did you have to check in order to solve this problem? List and check. N p0=1018*0.20=203.6>10 N(1-p0)=1018*(1-0.20)=814.4>10 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? Type 1 it could incorrectly inform the media as to who is leading the race for the democratic nominee 7. SPAM is a meat substance that’s been around for years. Some people like it, many people do not. Suppose you hear it is reported that 25% of the American population likes SPAM and wants more of it in their lives. You believe the actual percentage is less than that. The data are found in your textbook, Chapter 8, with the title “SPAM.” (Go to MyStatLab / StatCrunch / “View the Data Sets from your Textbook” / Chapter 8 / SPAM to find the data.) Is there enough evidence in the sample from the data set to be able to say you’re right? Do all the a-f from the above problems. Told to disregard by professor!!! Using StatCrunch to do a Hypothesis Test for proportion from a data set: -Open Stat Crunch -STATS/Proportion Stats/One Sample/With Data/ -Click on the variable you are using (“MORE SPAM”) -Fill in Success = YES -Click “Hypothesis Test for p” and fill out the Ho and Ha / COMPUTE!

Rec 14 CI and HT for Proportions

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