Estimating a Population Mean: σ Known (Section 8.1, Solutions)

CHAPTER 8 Section 8.1 -Estimating a Population Mean: σ Known 1) A simple random sample of size n is drawn from a population whose population standard deviation, σ, is known to be 3.8. The sample mean, x-bar, is determined to be 59.2. a) Compute the 90% confidence interval about µ if the sample size, n, is 45. 90%CI → zc = 1.645 n = 45  = 3.8 x = 59.2 x  zc *  n 3.8 59.2 ± 1.645 * 45 59.2 ± .9318 → (58.27, 60.13) 58.27< <60.13 For calculator feature use STAT, arrow to TESTS, and select 7:ZInterval, select Stats enter the required information, and CALCULATE b) Compute the 90% confidence interval about µ if the sample size, n, is 55. How does increasing the sample size affect the margin of error E? How does it affect the length of the interval? 90%CI → zc = 1.645 n = 55  = 3.8 x  zc *  n 59.2 ± .8429 → (58.357, 60.043) 3.8 59.2 ± 1.645 * x = 59.2 55 This is a shorter interval than the one in part (a). It’s more precise. At the same confidence level, a larger sample size produces a more precise (shorter) interval For calculator feature use STAT, arrow to TESTS, and select 7:ZInterval, select Stats enter the required information, and CALCULATE c) Compute the 98% confidence interval about µ if the sample size, n, is 45. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the size of the margin of error, E? How does it affect the length of the interval? 98%CI → zc = 2.33 (Higher confidence level) n = 45  = 3.8 x  zc *  n 59.2 ± 1.3199 → (57.88, 60.52) 3.8 59.2 ± 2.33 * x = 59.2 45 This is longer (less precise) than the interval produced in part (a). A higher confidence level produces a longer interval. With a longer interval we are now more confident that the interval contains the population mean. Use calculator feature to check answer: Z-Interval d) Can we compute a confidence interval about µ based on the information given if the sample size is n = 15? No. Why? The sample size is small. If the sample size is 15, what must be true regarding the population from which the sample was drawn? When we have a small sample size, the population should be normally distributed in order to be able to use this method to construct a confidence interval. Confidence Intervals Interpretations Refer to the first problem from the previous page. Let’s assume thousands of students took this test and we are trying to estimate the mean score (mu) of all students who took the test. In order to estimate mu we select a simple random sample of 45 students who took the test and find that the mean score of those 45 students is x-bar = 59.2. Let’s also assume that the standard deviation of the population (all thousands of test scores) is known to be σ = 3.8. You must be careful about what you say about confidence intervals. Remember that a confidence interval is an ESTIMATE for the UNKNOWN parameter mu, which is the mean score of all students who took the test. We are trying to estimate mu by selecting a sample. Our sample yields an x-bar of 59.2. This is OUR POINT ESTIMATE, and we provide a range of PLAUSIBLE values for mu by constructing a 90% confidence interval around x-bar = 59.2. Our interval estimate happens to be (58.27, 60.13) Remember that different samples will produce different x bars, and hence, different intervals. The confidence level (90%) refers to the accuracy of the method used in constructing the interval. We are expecting about 90% of random samples to yield an interval that captures the true mean, which means that about 10% of the intervals constructed will fail to capture the true value of the parameter mu. We will never know whether the interval constructed contains the mean mu or not, but we are in this case 90% confident that it does. We should say: • I am 90% confident that the interval from 58.27 to 60.13 contains the mean score of all the students who took the test. • It’s also OK to say something less formal: “I’m 90% confident that the mean score of all students who took the test is between 58.27 and 60.13. • About 90% of the intervals constructed with this method will capture the mean score of all students who took the test. • If we were to select 100 random samples of size 45 and construct a confidence interval about each of the sample means, approximately 90 of the 100 intervals would contain the population mean mu. 1) Don’t suggest that the parameter varies. A statement like “There is a 90% chance that the true mean score of all students who took the test is between 58.27 and 60.13” is just wrong. This sounds as though you think the population mean wanders around and sometimes happens to fall between 58.27 and 60.13. When you interpret a confidence interval, make it clear that you know that the population parameter mu is fixed and that it is the interval that varies from sample to sample. 2) Don’t say “90% of the test scores are between 58.27 and 60.13”. The confidence interval is about the MEAN scores not about the INDIVIDUAL scores. 3) Don’s say, “The mean of all test scores is between 58.27 and 60.13, 90% of the time” That implies that the true mean varies, when in fact it is the confidence interval that would have been different had we gotten a different sample. 4) Don’t be certain about the parameter. Saying “The mean score of all students who took the test is between 58.27 and 60.13 asserts that the population mean cannot be outside that interval. Of course, we can’t be absolutely certain of that. (Just 90% confident on this case) It’s necessary to add the statement “With 90% confidence, we can say that.....”