Rec 13 - t-distribution

Recitation 13 t-Distribution Part 1: Use your lecture notes to help you answer the following questions. 1. Suppose X has a normal distribution. If you do not know the population standard deviation and you want to estimate the mean of the population, what do you use? a. Use the sample standard deviation. b. Use the t-distribution c. Both a and b are correct 2. Suppose you want to make a confidence interval for the mean and X has a normal distribution, but you don’t know the population standard deviation. What is the penalty you have to pay for using the SAMPLE standard deviation and not the POPULATION standard deviation? You use a t- value that is ____________ than Z, making the confidence interval ______________. a. Bigger, wider b. Smaller, narrower c. Bigger, narrower d. Smaller, wider 3. There is only one t-distribution and you use it for all different values of n. a. True b. False 4. The t-distribution is flatter than the Z-distribution. a. True b. False 5. For the same area you will have a/the __________________ t-value, compared to the Z-value. a. Bigger b. Smaller c. Same 6. The larger ___________is, the closer the t-distribution looks to the one and only Z-distribution. a. n b. p c. s d. µ 7. To estimate the population mean when you don’t know the population standard deviation, use the formula that contains t with ________________ degrees of freedom. a. n b. n+1 c. n-1 d. 0

8. Which value of t do you need on the t-table if you are doing a 95% confidence interval for the population mean, you don’t know the population standard deviation, and n is 20? a. 2.0860 b. 2.0796 c. 2.0930 d. None of these answers is correct. 9. If you do a hypothesis test for the mean and you used a t-test instead of a test involving Z, and everything else remained the same, your p-value for the t-test would be ____________ compared to the p-value for the test involving Z. a. Smaller b. The same c. Larger 10. Suppose you did a ONE TAILED t-test and the test statistic was 1.50, and the degrees of freedom for the test was 10 – 1 = 9, your p-value is between what two numbers, according to the t-table? a. .1 and .05 b. .05 and .025 c. .15 and .10 d. None of these answers is correct. 11. It is harder to reject Ho when you use a t-distribution than if you use a Z-distribution using the same data set. Note: Which one you use depends on your data. a. True b. False 12. The t-distribution has ______________standard deviation than the Z-distribution. a. A Larger b. A Smaller c. The same 13. The t-distribution has _________________mean than the Z distribution a. The same mean b. A larger mean c. A smaller mean 14. What value of t do you use in the confidence interval formula if you have 95% confidence and n = 10? 15. What values does the last row of the t-table represent?

c. The significance level would: increase decrease stay the same d. The p-value would: increase decrease stay the same e. Your decision about whether or not to reject H 0 would: change stay the same f. Your conclusion would: change stay the same #4-7 Movie Box Office Data under Recitation 5 in Module 5: 4. Use StatCrunch to calculate a 95% confidence interval for the opening weekend revenue for movies in this data set. Interpret your results. Use StatCrunch instructions from Recitation 5 to Import this Excel Data Set into StatCrunch. Then use STAT/T-STATS/WITH DATA and take it from there. 5. Use StatCrunch to calculate a 95% CI for the total U.S. revenue for movies in the dataset. Interpret your results. 6. Explain why the CI for total U.S. revenue is so much wider than the CI for opening weekend revenue. 7. According to yoexpert.com , the average budget for a U.S. movie now is approximately $139 million. What is your best estimate (use a CI) of the budget of a U.S. movie from our dataset, and is the average budget now higher, lower, or about the same compared to our dataset? Use your CI to decide. For questions #8-11: We are interested in exploring the average amount spent on marketing in the last quarter by retail outlets in the US, which we assume are normally distributed. An article in the Wall Street Journal reported that the average amount spent on marketing in the last quarter by retail outlets in the US was $551,715.69. Suppose the firm you are working for randomly samples 800 retail outlets and finds that the sample average is $547,195 with a sample standard deviation of $82,681 .

8. Estimate the average amount spent on marketing in the last quarter by all retail outlets using your data. Show any formulas needed, show all work and use proper units. Remember that ALL estimates should include an appropriate interpretation! 9. Suppose your partner Bob works for a different firm, which conducts their own analysis by randomly sampling 15 retail outlets and finds that the sample average is $583,425.16 and the sample standard deviation was found to be $85,221. What is your partner firm’s estimate of the average amount spent on marketing in the last quarter by all retail outlets, using their data? USE STAT CRUNCH. 10. A. Looking at the previous problem, if $85,221 had been the population standard deviation, σ, and not the sample standard deviation, s, recalculate your estimate USING STAT CRUNCH (Don’t use T-STATS, what do you use?) B. Why is your new confidence interval narrower than your answer to the previous problem? 11. Your partner Bob’s firm believed, before gathering data, that the true average amount spent on marketing is different than the amount reported by the Wall Street Journal. Is there enough evidence to find that the partner firm is correct? A. Do this by hand first. That means show all the steps, formulas, and notation necessary to answer this question. B. Now use Stat Crunch to perform a “t-test” (This is the name for hypothesis tests using t) Doing a t-test in StatCrunch --STAT/T-STATS/ONE SAMPLE/WITH SUMMARY --Enter the relevant information (mean, SD, sample size) --Under PERFORM: Click Hypothesis test for µ. --COMPUTE!