AP Statistics Hypothesis Testing Chapter 9. Intro to Significance Tests

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1 Intro to Significance Tests Name Hr For the following pairs, indicate whether they are legitimate hypotheses and why For each situation, state the null and alternate hypothesis. (Define your parameter.) 5. Researchers have postulated that, due to the math excitement in Mrs. Sapp s classroom, her students have a higher mean systolic blood pressure than other students at MHS. Suppose that the mean systolic blood pressure in students at MHS is known to be A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well. 7. In 2010, the proportion of US students who graduated from high school in 4 years was 93%. Do this year s data show a different proportion? 8. Data show that the mean household income in Manhattan, KS is $37,000 per year 1. A market research firm questions shoppers at Manhattan Town Center Mall. The researchers suspect that the mean household income of mall shoppers is higher than that of the general population. 9. The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is too small or too large, the motor will not work properly. The manufacturer measures the distance in a sample of the motors to determine whether the mean diameter has moved away from the target. 1

2 10. The Mentor reports that 81% of MHS students favor mathematics classes over science classes. You suspect that the true proportion is actually lower. For each of the following P-values, determine whether you would reject or fail to reject the null hypothesis when Type I and II Errors 15. A recent study compared the crime rates between Chicago and Los Angeles. The null hypothesis is that the crime rates are the same, and the alternative is that Chicago has a higher crime rate. Identify each statement as representing a Type I or Type II error: The rates are the same between Chicago and Los Angeles, but the researchers say that Chicago has a higher crime rate. Chicago has a higher crime rate, but the researchers say the rates of crime are the same When you set the level of significance ( ) for a test, what are you controlling for? a) Type I error b) Type II error c) The critical value d) The p-value e) Overpopulation of sloths 17. When conducting a formal hypothesis test, there are different errors that may be made, depending on your decision. One decision is to reject the null hypothesis. If you falsely reject the null, what type of error have you made? a) Type I error b) Type II error c) Critical error d) Standard error e) Not buying Mrs. Sapp enough chocolate error

3 18. A researcher plans to conduct a significance test at the significance level. She designs her study to have power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will make a Type II error for the particular alternative value of the parameter at which she computed the power is: a) 0.01 b) 0.10 c) 0.89 d) 0.90 e) Food inspectors for the USDA test samples of food products to see if they are safe. This can be thought of as a hypothesis test where the food is safe the food is not safe For you as a consumer, which type of error would be the worst for the food inspector to make, Type I or Type II? Explain. 20. Josh and Duncan each want to test if the mean score on the ACT of students in Kansas 23, as claimed in the Kansas Times. Josh takes a random sample of 200 adults and uses. Duncan takes a random sample of 200 adults and uses. Suppose that the newspaper s average is actually right. a) Is it possible for Josh and Duncan to make a Type I error? If so, who is more likely to do so? b) Is it possible for Josh and Duncan to make a Type II error? If so, who is more likely to do so? 21. Now suppose that the newspapers average was actually wrong. a) Is it possible for Josh and Duncan to make a Type I error? If so, who is more likely to do so? b) Is it possible for Josh and Duncan to make a Type II error? If so, who is more likely to do so?

4 22. A Hi-Def TV manufacturer claims that only 10% of its TVs will need service during the first three years. A consumer agency has heard complaints and doubts the manufacturer s claim. They obtain a random sample of 100 purchasers and ask each whether the TV purchased needed repair during the first three years after purchase. Let be the sample proportion of responses indicating a repair was necessary. Let indicate the true proportion of TVs by this manufacturer needing repair within the first three years. Obviously the agency does not want to claim false advertising unless the sample evidence strongly suggests that. The appropriate hypotheses are: a) In the context of this problem, describe a Type I and Type II error, and discus the consequences of each. b) Would you recommend a test procedure that uses or? Explain. 23. In order to avoid making Type I errors, why not just always use very large critical values? 24. A P-value measures: a) The probability the null hypothesis is true. b) The probability the null hypothesis is false. c) The probability of seeing a result at least as far from as the one observed, given that the null hypothesis is true. d) The probability of seeing a result at least as far from as the one observed, given that the null hypothesis is false.

5 25. You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test: If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% level against the alternative is a) Explain what power = 0.23 means. b) You could get higher power against the same alternative with the same α by changing the number of measurements you make. Should you make more or less measurements to increase power? c) If you decide to use in place of, with no other changes to the test, will the power increase or decrease? Explain. d) If you shift your interest to the alternative, with no other changes, will the power increase or decrease? Justify your answer. One Sample z-test for Means (σ known) 26. Explain in plain language why a significance test that is significant at the 1% level ( ) must always be significant at the 5% level ( ). If a test is significant at the 5% level, what can you say about its significance at the 1% level?

6 27. To determine whether the mean nicotine content of a certain brand of cigarettes is greater than the advertised value of 1.4 mg, a health advisory group tests versus. The calculated value of the test statistic is. a) Is the result significant at the 5% level? Why or why not? b) Is the result significant at the 1% level? Why or why not? c) What decision would you make about in part (a)? Part (b)? Explain. 28. Suppose that Gambino s, a local pizza parlor, claims that the average delivery time is 23.5 minutes with a standard deviation (σ) of 2.5 minutes. However, every time you order a pizza it seems to take more than 23.5 minutes. Suppose that we take a simple random sample and 40 delivery times are obtained. The average of these times is minutes. Is there sufficient evidence to suggest that the average delivery time is greater than 23.5 minutes? Use. 29. A laboratory is asked to evaluate the claim that the concentration of the active ingredient in a specimen is 0.86%. The mean of three random repeated analyses of the specimen is %, and the process is known to be normally distributed with standard deviation of σ = The lab chooses the 1% level of significance. Is there sufficient evidence to say that the level of active ingredient is different from 0.86%?

7 30. A random sample of 25 values of monthly rent of two bedroom apartments was selected from a recent edition of the Manhattan Mercury. The distribution of two bedroom apartments is approximately normal. The mean of the sample was $575, and you may assume that the population standard deviation was $165. a) Are the assumptions met for conducting a hypothesis test? b) A group of students thinks the true mean rent for this type of apartment is. Is there statistical evidence to reject their claim? c) A newspaper reporter reports that the average rent for two bedroom apartments last year was $500. Is there statistical evidence of a change in mean rent from last year to this year? d) Construct a 95% confidence interval for the true average rent for Manhattan two bedroom apartments. Could you have made a decision in parts (b) and (c) using only the confidence interval? Explain. 31. Suppose an advertising firm reported in 2013 that Chocoholics Chocolate Chip Cookies were selling at a mean rate of $1323 per week with a known standard deviation (σ) of $275. Suppose a random sample of 30 weeks in 2011 showed that the cookies were selling at the average rate of $1258. Does this indicate that the sales of the cookies are different from the earlier figure?

8 32. Bottles of a popular cola are supposed to contain 300 ml of cola. There is some variation from bottle to bottle because the filling machinery is not perfect. (Suppose the process is normally distributed with σ = 3 ml.) An inspector who suspects that the bottler is under filling randomly measures the contents of six bottles. Is this convincing evidence that the mean content of cola bottles is less than the advertised 300 ml? One Sample t-test for Means (σ unknown) 33. An SRS of 100 postal employees found that the average time these employees had worked for the postal service was x = 7 years with standard deviation s = 2 years. Assume the distribution of the time the population of employees have worked for the postal service is approximately normal with mean μ. Are these data evidence that μ has changed from the value of 7.5 years of 20 years ago? To determine this we test the hypotheses H : 7.5, H : 7.5, using the one-sample t test. o a The P-value for the one-sample t test is: A) larger than B) between 0.10 and C) between 0.05 and D) below a) Find and interpret a 95% confidence interval for the mean time μ the population of postal service employees have spent with the postal service. (Assume assumptions are met.) b) Can you make a conclusion based only on this confidence interval? Explain.

9 34. If a good estimate of the population standard deviation is 3.0 C, then approximately what minimum sample size would you need in a simple random sample if you wanted a 95% confidence interval for the mean to have a margin of error of 0.5 C or less? 35. Researchers at Kansas State University conducted a study that included the collection of a large amount of data on the water temperature of streams in northeast Kansas. The study found that the mean temperature for a random sample of 178 streams that were under a closed canopy of vegetation was C with a sample standard deviation of 2.83 C. a) If you perform a hypothesis test for the mean stream temperature based on the t-distribution, then, in practice, what conditions must be satisfied? I. The sample must be random and the population must be approximately normal. II. The sample must be random, but because the sample size is large (178), the population does not have to be approximately normal. III. The sample doesn t need to be random and the population doesn t have to be approximately normal because the sample size is so large (178). IV. The type of canopy must have been randomly assigned to the streams. V. The sample must be random and np 10 and n(1-p) 10. b) Test the hypothesis that the true mean stream temperature under a closed canopy of vegetation is more than 15.5 using a 0.05 level of significance. (Include a full writeup: Name the test; (you have already checked conditions), provide null and alternative hypotheses, compute the test statistic and P-value, and write a conclusion in context.)

10 36. Mrs. Sapp conducted a random survey of 40 MHS teens. The teens were found to sleep an average of 6.2 hours per school night with a standard deviation of 1 hour. The distribution had no outliers but is slightly skewed toward the larger values. a) An earlier research report suggested that MHS teens average about 6.5 hours of sleep per night. Is there statistically significant evidence at the 1% level of significance that the mean number of hours of sleep at MHS per night isn t 6.5? (Include a full writeup: Name the test and check conditions, provide null and alternative hypotheses, compute the test statistic and P-value and make a sketch, and write a conclusion in context.) b) If the researchers claimed that the number of hours of sleep for teens was less than 6.5 hours, then how would the P-value change? Is this stronger or weaker evidence than that in part (a)? Explain. c) Would the evidence in part (a) be stronger or weaker if the sample size were increased, assuming everything else remained the same? Explain.

11 One Sample z-test for Proportions 37. MHS makes a change that should improve student s satisfaction with the parking situation. Before the change, 37% of the school s students approved of the parking that was provided. After the change, Mr. Hoyt surveys an SRS of 100 of the over 1800 students at the school. In all, 83 students say that they approve of the new parking arrangement. Mr. Hoyt cites this as evidence that the change was effective. a) Perform a test of Mr. Hoyt s claim at the α = 0.05 significance level. Be sure to check assumptions! b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. c) The test has a power of 0.75 to detect that p =.37. Explain what this means. d) Identify two ways to increase the power in part (c).

12 38. Suppose a Gallop Poll is taken in order to measure the proportion of US adults who approve of attempts to clone humans. Gallop finds that 2% of the US adults surveyed approve. What sample size is necessary to be 95% confident within +/ of the true proportion of adults who approve of attempts to clone humans? 39. We hear that newborn babies are more likely to be boys than girls. Is this true? A random sample of 25,468 firstborn children included 13,173 boys. Boys do make up more than half of the sample, but of course we don t expect a perfect split in a random sample. a) To what population can the results of the study be generalized: all children or all firstborn children? Justify your answer. b) Do these data give convincing evidence that boys are more common than girls in the population? Carry out a significance test to help answer this question. (Use ) 40. Dillard s took a random sample of 100 orders for women s shoes and found that 23 of the orders were for shoe size 8. Construct a 95% confidence interval for the proportion of Dillard s customers who wear size 8 shoes. (Assume the assumptions are met.) A report in USA today said that 29% of US women wear size 8 shoes. Does the data above indicate that the proportion of Dillard s orders for size 8 shoes is different from the population of women wearing size 8 shoes in the general population? Explain.