Math 1530 Practice Problems for Final Exam Fall 2010 Means Chapter 7 The Sampling Distribution of the Sample Mean 1 One barge from Inland Waterways Inc can carry a load of 5516 8 lb Records of past trips show that the weight of cans that it carries have a mean of 82 lb and a standard deviation of 16 lb For samples of size 64 find the mean and standard deviation of the distribution of the sample mean 2 The mean height for a population is 65 inches and the standard deviation is 3 inches Let denote the mean height for a simple random sample of people from the population True or False The standard deviation of the distribution of for samples of size 30 is greater than the standard deviation of the distribution of for samples of size 20 3 Scores on an aptitude test are normally distributed with a mean of 280 and a standard deviation of 30 Determine the percentage of samples of size 4 that have a mean score within 21 points of the population mean 280 4 For the population of one town the number of siblings is a random variable whose relative frequency histogram is very right skewed Let denote the mean number of siblings per person for a random sample of size 30 For samples of size 30 which of the following statements concerning the sampling distribution of the sample mean is true a is approximately normally distributed b The distribution of is very right skewed c is normally distributed d None of the above is true 5 Answer question 4 for samples of size 3 where size 3 represents the mean of a random sample of Chapter 8 Confidence Interval for a Population Mean 6 Find the value of critical t value that corresponds to a confidence level of 98 given a sample size of n 23 7 Thirty students were selected randomly from a large group of students taking a certain calculus test The mean score for the students in the sample was 86 the standard deviation was s 8 9 Construct a 99 confidence interval for the mean score of all students taking the test Tell the critical value of t and give the MOE Interpret your answer in terms of the original data 8 Mary wishes to estimate the mean height of women aged 18 24 She picks a sample of 100 women aged between 18 and 24 and constructs a 99 confidence interval for the population mean If she were to repeat this procedure 200 times in total she would get 200 different confidence intervals How many of these intervals would you expect to contain the population mean Explain your thinking 9 Based on a sample of size 45 a 95 confidence interval for the mean score of all students is from 66 to 72 Find the margin of error 10 A sample mean is used to estimate a population mean To obtain a margin of error of 1 5 units at a confidence level of 95 a sample size of 120 is needed Would a larger or smaller sample size be needed if the researcher wished to a increase the confidence level while keeping the same margin of error b decrease the margin of error while keeping the same confidence level 11 For a t distribution with df 15 find t0 025 12 For a t distribution with df 6 fine the two t values that divide the area under the curve into a middle 99 and two outside areas each of which is 0 005 This is the critical t value for a 99 CI and its negative 13 The scores on a school aptitude test are normally distributed The following SRS of 6 scores is selected 80 0 81 3 87 1 71 8 86 3 79 4 Determine a 90 CI for the mean score of all students who took the test Give the critical t value and the MOE Interpret your results in terms of the original data 14 Rank the following t and z values in order from smallest to largest a z 05 b t 05 c t 05 than that for b d and e d t 025 e t 01 The df for b d and e is the same The df for c is greater Chapter 9 Hypothesis Tests for One Population Mean 15 A health insurer has determined that the reasonable and customary fee for a certain medical procedure is 1200 They suspect that the average fee charged by one particular clinic for this procedure is higher than 1200 The insurer wants you to perform a hypothesis test to determine whether their suspicion is correct a State an appropriate null hypothesis and alternative hypothesis b Will the hypothesis test be a right tailed left tailed or two tailed test 16 You with to test the following hypothesis Ho 40 Ha 40 Would you be inclined to reject the null hypothesis if the sample mean turned out to be much smaller than 40 Explain 17 In 1990 the average math SAT score for students at one school was 473 Five years later a teacher wants to perform a hypothesis test to determine whether the average math SAT score has changed from the 1990 mean of 473 The hypotheses are Ho 473 Ha 473 Explain the meaning of a Type I error 18 In the past the mean lifetime of a certain type of flashlight battery has been 8 4 hours The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has increased as a result The hypotheses are Ho 8 4 hours Ha 8 4 hours Suppose that the results of sampling lead to rejection of the null hypothesis If the mean lifetime has not in fact increased classify this conclusion as a Type I error a Type II error or a correct decision 19 A researcher claims that the amounts of acetaminophen in a certain brand of cold tablet have a mean different than the 600 mg claimed by the manufacturer Test this claim at a 2 level of significance The mean acetaminophen content for a random sample of n 35 tablets is 604 8 mg Assume that the standard deviation is 4 7 mg a Assuming no outliers what type of test should you use b State the null and alternative hypotheses c Do the test using the p value approach You may always use your calculator to find the p value d Interpret your result in terms of the original values 20 A test of sobriety involves measuring the subject s motor skills The scores on such a test are normally distributed Twenty randomly selected sober subjects take the test and produce a mean score of 37 4 with a standard deviation of 3 7 At the 0 01 level of significance test the claim that the true mean score for all sober subjects is equal to 35 Repeat steps a d of the previous problem in working this one