LAB ACTIVITY 9 Due Friday, Oct. 21 at 11:59pm (Use 'GSS-93.mtw' dataset) In this lab, you will need Minitab for both normal distribution probabilities and binomial distribution probabilities. They can both be found by selecting Graph Probability distribution plot View Probability OK. But you must be able to choose the correct settings and enter the correct parameters to obtain the answers you need. With a bit of experimentation, you should be able to figure out how this works. Activity 1 (Requires Minitab): 1. According to an older study, 82% of college students use smartphones. Say we take a random sample of 600 students at Penn State. In this question we will find an answer to 'What is the probability that more than 500 students in our sample have smartphones?' a. Let X = the number of students in our sample who have smartphones. X is a binomial random variable. What are n and p? (You may assume for the sake of this exercise that the results of the older study give the exact population parameter.) b. What is the expected value of X? What is the standard deviation? c. Is it proper to use the normal approximation for X? Why? d. Find the z-score for X=500 and use this result to find the normal approximation to P(X>500). e. Now repeat part d using the continuity correction (check the book or Lecture #15 for information on the continuity correction). Hint: The z-score should be a bit larger and the probability a bit smaller when you use the continuity correction properly. f. Use Minitab to find the exact binomial probability that X>500. (Careful: If you use Minitab to look at the right tail probability, you will need to use the value 501 instead of 500 since the discreteness of X implies that P(X>500) is the same as P(X501).) Which normal approximation result (part d or part e) is closer to the exact value? 2. To be eligible for a certain job, women need to be at least 62 inches tall, and 87% of women meet this criterion. In a random sample of 2000 women, let X = # who qualify for the job based on height. a. Use the normal approximation to the binomial distribution to approximate P(X1700), the probability that 1700 or fewer women from the sample have the necessary height to qualify for the job. You do not need to use the continuity correction but you are welcome to do so if you want. Activity 2 (Requires Minitab): Distribution of p-hat. For this activity we use the data set GSS-93.mtw available in the data folder on Angel. For this exercise, assume that these respondents comprise the entire population. In this case, the data set is considered a census. The variable 'gunlaw` is whether a respondent favors or opposes stronger gun control laws. 1. Follow the Minitab sequence stat tables tally individual variables select 'gunlaw` OK. The result table gives the counts and sample size. What is p, the proportion of respondents that favor stronger gun control laws? 2. Suppose a random sample of size n=60 is drawn from this population and used to calculate p-hat. Remember that p-hat has a sampling distribution. a. What is the mean of the sampling distribution for p-hat? b. What is the standard deviation of the sampling distribution for p-hat? 3. Now suppose we draw a random sample of size 200 from this population and use it to calculate a new p-hat. a. What is the mean of the sampling distribution for p-hat? b. What is the standard deviation of the sampling distribution for p-hat? 4. Based on your previous answers, which statement below completes the following sentence correctly? When we increase the sample size... a. the mean of p-hat increases and the standard deviation of p-hat decreases. b. the mean of p-hat increases and the standard deviation of p-hat does not change c. the mean of p-hat does not change and the standard deviation of p-hat increases d. the mean of p-hat does not change and the standard deviation of p-hat decreases Activity 3 (Requires Minitab): The normal approximation versus the exact distribution. Your weekly quizzes each have 5 questions. Assume that for a particular week all 5 are multiple choice questions with 4 possible answers, and that you randomly choose the answer for each question. Let X=the number of answers you get correct. 1. X is a binomial random variable. What are n and p? 2. Could we be justified in using the normal approximation to test a hypothesis about p? Why or why not? 3. Say a student takes this quiz. Using the exact distribution, what is the probability that she gets 3 or more questions correct if she is randomly guessing for each answer? 4. If we choose to use a normal approximation (even if conditions aren't met), what would the mean and standard deviation of the appropriate normal distribution be? 5. Now using the normal distribution, what would be the probability that she gets 3 or more answers correct by random guessing? Use the continuity correction to find this probability. Activity 4 (Does not require Minitab). Construction of a confidence interval for one population proportion. 1. We would like to know: "What percentage of college students read the college newspaper every day?" In a random sample of 500 students, 231 of them said they read the college newspaper every day. Use these data to build a 95% confidence interval for the percentage of college students that read the college newspaper. a. First, we phrase this question slightly differently and build a confidence interval for the population proportion. Using the sample data, identify n = ______ p-hat = _______ b. Can we use the normal approximation to construct this confidence interval? c. Using the Z-multiplier of 2 (for 95%), calculate the margin of error using p-hat. d. Now, using your answer from part c, complete the construction of the confidence interval. e. Write a sentence interpreting this confidence interval for this particular situation. (Hint: Your sentence should avoid talking about the probability that this particular interval contains the true value of p. Instead, talk about the process that created the interval.) Activity 5 (Does not require Minitab). What can change the margin of error and how? 1. Right after the Republican party convention in 1992, four different polls reported on the support for President George H. W. Bush. On the same day, each conducted a 95% confidence interval for the proportion of all Americans who supported Bush. The CNN/USA Today poll found that .42.04 would vote for Bush, the Newsweek poll found .39.04, the Los Angeles Times reported .41.03, and the Washington Post reported .40.04. (i) Should we be surprised that the four polls gave four different proportions? (A) Yes, the population proportion is always the same, so all the sample proportions should be the same. (B) No, the population proportion is always the same, but each sample is different and by random chance might have a different sample proportion. (C) Yes, I expected major polls like this to be a lot more accurate and all give the same results. (D) No, for each sample there is a different population proportion. (ii) Why would the margin of error be .03 in one of the polls and .04 in the other polls? (A) Different sample sizes result in different margins of error. (B) Some polls asked different kinds of people. 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