150 Chapter 8: Confidence Interval EstimationCHAPTER 88.1X Zn= 851.9686483.04 86.968.2X Zn= 1252.582436114.68 135.328.3If all possible samples of the same size nare taken, 95% of them include the true population average monthly sales of the product within the interval developed. Thus we are 95 percent confident that this sample is one that does correctly estimate the true average amount.8.4Since the results of only one sample are used to indicate whether something has gone wrong in the production process, the manufacturer can never know with 100% certainty that the specific interval obtained from the sample includes the true population mean. In order to have every possible interval estimate of the true mean, the entire population (sample size Nwould have to be selected.8.5To the extent that the sampling distribution of sample means is approximately normal, it is true that approximately 95% of all possible sample means taken from samples of that same size will fall within 1.96 times the standard error away from the true population mean. But the population mean is not known with certainty. Since the manufacturer estimated the mean would fall between 10.99408 and 11.00192 inches based on a single sample, it is not necessarily true that 95% of all sample means will fall within those same bounds.8.6Approximately 5% of the intervals will not include the true population mean somewhere in the interval. Since the true population mean is not known, we do not know for certain whether it is in the one interval we have developed, between 10.99408 and 11.00192 inches.8.7(a) X Zn0.9952.580.02500.98771.0023(b) Since the value of 1.0 is included in the interval, there is no reason to believe that theaverage is different from 1.0 gallon.(c) No. Since is known and n= 50, from the central limit theorem, we may assume that the sampling distribution of X is approximately normal.(d) An individual can with an observed value of 0.98 gallon of paint yields a Z-value of – 1.0. The confidence interval represents bounds on the estimate of the average of a sample of 50 cans of paint, not an individual value.(e) The reduced confidence level narrows the width of the confidence interval.(a) X Zn0.9951.960.02500.98951.0005(b) Since the value of 1.0 is still included in the interval, there is no reason to believe that the average is different from 1.0 gallon.

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151 Chapter 8: Confidence Interval Estimation8.8(a)(b) No. The manufacturer cannot support a claim that the bulbs last an average 400 hours. Based on the data from the sample, a mean of 400 hours would represent a distance of 4 standard deviations above the sample mean of 350 hours.(c) No. Since is known and n= 64, from the central limit theorem, we may assume that the sampling distribution of X is approximately normal.(d) An individual value of 320 is only 0.30 standard deviations below the sample mean of 350. The confidence interval represents bounds on the estimate of a sample of 64, not an individual value.(e) The confidence interval is narrower based on a process standard deviation of 80