Name: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%.

Transcription

1 Chapter 7-8 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. Please indicate which program or distribution used where applicable!! List the applicable formula, even if a calculator program was used, and list all quantities/variables. 1. Finding the Minimum Sample Size (2 points each) (a) Find the minimum sample size required to estimate an unknown population mean µ, with a margin of error of $57, confidence level 98%, and σ = $326. (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%. 2. Critical Values (3 points each) (a) Find the critical value when estimating population mean µ, for a sample size n = 37 and a confidence level of 96%, when σ is known. (b) Find the critical value when estimating a population proportion for a sample size n = 50 and a confidence level of 99%.

2 3. Margin of Error, E and point estimate (2 points each) (a) The following confidence interval is obtained for a population proportion: < p < Find the margin of error, E and a point estimate for p, ˆp. E = ˆp = (b) Find the margin of error, E for a 90% confidence interval for µ, when n = 100, x = 525, and s = 45. E = Page 2

3 4. Confidence Intervals (6 points each) I Assume all requirements needed to construct the confidence interval are met. II List the applicable formula, even if a calculator program was used. III List the calculator program used, and all quantities/variables. IV Construct the confidence interval. V Be sure to interpret each Confidence Interval and answer all questions! (a) Medical Malpractice An important issue facing Americans is the large number of medical malpractice lawsuits and the expenses that they generate. In a study of 1228 randomly selected medical malpractice lawsuits, it is found that 856 of them were later dropped or dismissed (based on data from the Physician Insurers Association of America). (i) Construct a 99% confidence interval estimate of the proportion of medical malpractice lawsuits that are dropped or dismissed. (ii) Does it appear that the majority of such suits are dropped or dismissed? (b) Thirty randomly selected students took the statistics final exam. The standard deviation for such exams is known to be 6.5 points. If the sample mean was 78.5 points and the sample standard deviation was 7 points, construct a 96% confidence interval for the mean score of all statistics students. Page 3

4 (c) Eat your spinach Six measurements were made of the mineral content (in percent) of spinach, with the results listed below. It is reasonable to assume that the population is approximately normal. (i) Construct a 95% confidence interval for the mean mineral content. (ii) Based on the confidence interval, is it reasonable to believe that the mean mineral content of spinach may be greater than 21%? Explain. (Source: Journal of Nutrition 66:5566) State and explain whether the hypothesis test involves a sampling distribution that is a Normal distribution, Student s t distribution, Chi-square distribution or neither. Assume all samples are simple random samples. Work without explanation is incorrect. (6 points) (a) Claim: µ = 7.6. Sample data: n = 28, x = 13.1, s = 3.2. The sample data appear to come from a normally distributed population. (b) Claim: µ > 122. Sample data: n = 17, x = 135, s = The sample data appear to come from a population with a very skewed distribution. (c) Claim: σ Sample data: n = 33, s = The sample data comes from a normally distributed population. Page 4

5 6. State the hypotheses Express the claim into a null or alternative hypothesis, and state the other hypothesis as well. Indicate which of the two hypotheses represents the claim. (3 points each) (a) Kids with cell phones: A marketing manager for a cell phone company claims that more than 35% of children aged 1011 have cell phones. H 0 : H 1 : claim is represented by H 0 / H 1 (circle one) (b) Big fish: Scientists want to perform a hypothesis test to determine how strong the evidence is that the mean weight of a certain species of flounder differs from 20 grams. H 0 : H 1 : claim is represented by H 0 / H 1 (circle one) 7. Critical Values (4 points each) (a) Assume that the data has a normal distribution and the number of observations is 36. Find the critical value(s), z used to test a null hypothesis µ = 0.73, α = 0.04 for a two-tailed test. (b) Find the critical value(s), χ 2 L and/or χ 2 R used to test H 1 : σ < 7.5, n = 18, and α = Test Statistic Use the given information to find the value of the test statistic. Make sure you name it correctly. (4 points each) (a) Carbon Monoxide Detectors The claim is that less than 1 of adults in the 2 United States have carbon monoxide detectors. A KRC Research survey of 1005 adults resulted in 462 who have carbon monoxide detectors. Page 5

6 (b) Find the value of the test statistic, where H 1 : µ > 17, x = 20.5, n = 7, and s = P -value Use the given information to find the P-value. (4 points each) (a) The test statistic in a left-tailed test is z = 1.67 (b) The test statistic in a two-tailed test is t = Type I and Type II Errors A publisher of a local newspaper claims that the percentage of year-olds who read a daily newspaper is greater than 15%. Assume that a hypothesis test of the given claim will be conducted. Identify the Type I and Type II errors for the given claim. (3 points each) (a) State the Type I Error for the test above: (b) State the Type II Error for the test above: Page 6

7 11. Hypothesis Testing Assume that a simple random sample has been selected from a normally distributed population. Find and state the: I Null and alternative hypothesis II List all given variables used III Perform the calculations and list the calculator program used, and all quantities/variables from that program. IV Draw a picture of the distribution, showing your test statistic or P -value, and make your comparison. V State the final conclusion using the original claim. (8 points each) (a) Heavy children: Are children heavier now than they were in the past? The National Health and Nutrition Examination Survey (NHANES) taken between 1999 and 2002 reported that the mean weight of six-year-old girls in the United States was 49.3 pounds. Another NHANES survey, published in 2008, reported that a sample of 193 six-year-old girls weighed between 2003 and 2006 had an average weight of 51.5 pounds. Assume the population standard deviation is σ = 15 pounds. Can you conclude that the mean weight of six-year-old girls is higher in 2006 than in 2002? Use the α = 0.01 level of significance. Page 7

8 (b) Google it: The marketing research company comscore, Inc. reported that 65% of online searches in March 2010 used Google as the search engine. A network administrator wants to determine whether the percentage of searches that use Google is different in his company. He samples 400 searches and nds that 295 of them use Google. Can he conclude that the percentage of searches that use Google in his company differs from 65%? Use the α = 0.04 level of significance. Page 8

9 (c) Playing Times of Popular Songs Listed below are the playing times (in seconds) of songs that were popular at the time of this writing. (The songs are by Timberlake, Furtado, Daughtry, Stefani, Fergie, Akon, Ludacris, Beyonce, Nickelback, Rihanna, Fray, Lavigne, Pink, Mims, Mumidee, and Omarion.) Use a α = 0.05 significance level to test the claim that the songs are from a population with a standard deviation less than one minute A battery manufacturer claims that a recently developed calculator battery has a mean life longer than 100 hours. Assuming that a hypothesis test has been conducted, and that the conclusion is to reject the null hypothesis, state the final conclusion using the original claim. (2 points) 13. EXTRA CREDIT You can increase the width of a confidence interval by... (circle one) (4 points) A. increasing the confidence level or increasing the sample size B. increasing the confidence level or decreasing the sample size C. decreasing the confidence level or increasing the sample size D. decreasing the confidence level or decreasing the sample size Page 9