360 Chapter 8 Confidence Intervals Objective 5 Distinguish between confidence and probability Distinguish Between Confidence and Probability In Example 8.6, a 99% confidence interval for the population mean weight �� was computed to be 20.04 < �� < 20.46. It is tempting to say that the probability is 99% that �� is between 20.04 and 20.46. This, however, is not correct. The term probability refers to random events, which can come out differently when experiments are repeated. The numbers 20.04 and 20.46 are fixed, not random. The population mean is also fixed. The population mean weight is either between 20.04 and 20.46 or it is not. There is no randomness involved. Therefore, we say that we have 99% confidence (not probability) that the population mean is in this interval. On the other hand, let’s say that we are discussing a method used to construct a 99% confidence interval. The method will succeed in covering the population mean 99% of the time, and fail the other 1% of the time. In this case, whether the population mean is covered or not is a random event, because it can vary from experiment to experiment. Therefore it is correct to say that a method for constructing a 99% confidence interval has probability 99% of covering the population mean. EXAMPLE 8.8 Interpreting a confidence level A hospital administrator plans to draw a simple random sample of 100 records of patients who were admitted for cardiac bypass surgery. She will compute the sample mean number of days spent in the hospital, and construct a 95% confidence interval for the population mean, using an appropriate method. She claims that the probability is 0.95 that the confidence interval will cover the population mean. Is she right? Solution Yes, she is right. The probability that a 95% confidence interval constructed by an appropriate method will cover the true value is 0.95. EXAMPLE 8.9 Interpreting a confidence interval Refer to Example 8.8. After drawing the sample, the hospital administrator constructs the 95% confidence interval, and it turns out to be 7.1 < �� < 7.5. The administrator claims that the probability is 0.95 that the population mean is between 7.1 and 7.5. Is she right? Solution No, she is not right. Once a specific confidence interval has been constructed, there is no more probability. She should say that she is 95% confident that the population mean is between 7.1 and 7.5. Check Your Understanding 15. A scientist plans to construct a 95% confidence interval for the mean length of steel rods that are manufactured by a certain process. She will draw a simple random sample of rods and compute the confidence interval using the methods described in this section. She says, ‘‘The probability is 95% that the population mean length will be covered by the confidence interval.’’ Is she right? Explain. Yes 16. The scientist in Exercise 15 constructs the 95% confidence interval for the mean length in centimeters, and it turns out to be 25.1 < �� < 27.2. She says, ‘‘The probability is 95% that the population mean length is between 25.1 and 27.2 centimeters.’’ Is she right? Explain. No Answers are on page 366.
navidi_monk_elementary_statistics_2e_ch7-9
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