Problems Practice for Test 3
1.) (6.5) The distribution of the time it takes a student to complete the fall class registration has a mean of 19 minutes, with a standard deviation of 7.6 minutes. For a random sample of 45 students, a) What is the mean and the standard error of the sampling distribution of the sample mean? b) What can you say about the shape of the sampling distribution of the sample mean? c) What is the probability that the average of this sample will be less than 17.5 minutes? d) What is the probability that the average of this sample will be within 1 minute of the population mean. 2) (6.4) According to the National Center for Health Statistics, 22.4% of adults are smokers. Suppose a random sample of 300 adults is obtained. a) What is the mean and the standard error of the sampling $ distribution of the sample proportion, p ? b) What can you say about the shape of the sampling $ distribution of p ? c) Using the Empirical Rule, almost all of the sample proportion values will be between _____ and _____. d) Is it likely or unlikely to observe a sample proportion of 0.32 if the population proportion really is 0.224?
3) (7.1) The mean monthly expense for entertainment per household was obtained from data collected by telephone interviews of 1000 randomly selected households, and the 95% confidence interval was computed as: ($86.50, $161.50) a) Does the sample mean fall within the interval? If so, find its value. b) Does the population mean fall within the interval? If so, find its value. c) What is the margin of error? d) A magazine claims that on average households spend $120 per month on entertainment. Would we reject this claim? 4) (7.2) In a poll conducted in May 2000, a news agency took a simple random sample of 1068 Americans and asked them the following question Have you ever been on TV? Of the 1068 Americans, 120 responded Yes and the rest answered No. a) Calculate the point estimate for the proportion of Americans that answered Yes to this question. b) Can a valid 92% confidence interval be constructed for p? c) Determine the standard error and margin of error for a 92% confidence interval for p? d) Give the lower and upper limits for this 92% confidence interval.
5. (7.3) A random sample of 60 eggs from an egg farm was selected and each egg was weighed. The calculated sample mean was 2.1 oz. and the sample standard deviation was 0.35. A 90% confidence interval of the mean weight of eggs was constructed. a) If a 95% confidence interval was constructed the using same sample, which confidence interval would be wider: the 90% C.I. or the 95% C.I.? b) If a 90% confidence interval was constructed using a sample size of 100 eggs with the same sample mean of 2.1 oz, which 90% confidence interval would be narrower (this one or the n=60 one)? c) If you took 500 samples of 60 eggs and created a 90% confidence interval for each sample, approximately how many of these 90% confidence intervals would contain the population mean? 6. (7.3) In a random sample of n = 50 households, the average number of TV viewing hours per week was 35.3 with a sample standard deviation of 12.7 hours. Construct a 99% confidence interval for the mean number of TV viewing hours per week for the population. a) What is the point estimate? b) Find the standard error of the sample mean. c) To obtain a 99% confidence interval of the mean number of TV viewing hours per week in the population, what do we multiply the standard error by to find the margin of error? Find that value. d) For the 99% confidence interval, what is the margin of error? e) Find the upper and lower limits for the 99% confidence interval.
7) (7.4) In a previous sample taken in May 2000, 39% of the people polled answered yes to the following question: do you happen to have in your home or garage any guns or revolvers? How large a sample do we need to estimate the population proportion of ALL people who answered yes to within 0.03 with a 95% level of confidence if a) we want to use the May 2000 sample as our guideline? b) we dont want to guess the proportion by using any previous study? c) Compare the results you obtained in a) and b). Is it something that you would expect? 8) (7.4) A doctor wants to estimate the mean serum HDL cholesterol of all 20-29 year old males. How many subjects would he need in order to estimate the mean serum HDL cholesterol of all 20-29 year old males to within 1.5 points with 95% confidence? Use a sample standard deviation of 12.5. Also, what will happen to the sample size if (i) we increase the margin of error?, or (ii) the sample standard deviation increased? 9) (Chapter 6) Suppose a random sample of 7 data values was selected from a population with mean 6 and standard deviation 0.9. The sample data values are: 5.2, 5.8, 7.9, 8.1, 4.9, 6.2, 6.8 a) What is the sample mean and sample standard deviation? b) What is the mean of the sampling distribution of x ? c) What conditions must be satisfied in order to compute a confidence interval using these values?