Question

refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the old faithful geyser. the confidence level of 95% was used. complete parts (a) through (d) below. interval (85.74,91.76) ( \bar{x}=88.75 ) ( s x=0.007431411 ) ( n=36 ) b. what does it mean to say that the confidence interval methods for the mean are robust against departures from normality? a. the methods work well with distributions that are not normal, provided that departures from normality are extreme. the distribution should not be symmetric or have only one mode, and should include outliers. b. the methods only work with distributions that are not normal. c. the methods work well with distributions that are not normal, provided that departures from normality are not too extreme. the distribution should be somewhat close to symmetric, should have only one mode, and should not include outliers. d. the methods do not work well with distributions that are not normal. b. does the sample used for the accompanying data display satisfy the requirements for constructing a confidence interval estimate of the population mean ( mu ) ? explain. a. no, the sample is not a simple random sample. b. yes, the sample is a sample random sample and the sample size is 36 , which is greater than 30 . c. no, the population is not normally distributed. d. yes, the sample is a simple random sample and the population is normally distributed.

Explanation:

For part (a), the concept of robustness in confidence interval methods for the mean means that the methods can still be effective even if the data distribution is not perfectly normal, as long as the deviations from normality are not extreme. A somewhat symmetric distribution with one mode and no outliers is acceptable.
For part (b), when constructing a confidence interval for the population mean, if the population is not known to be normal, a sample size \(n> 30\) (by the Central Limit Theorem) can be used. Here, the sample is a simple random sample and \(n = 36>30\).

Answer:

a. C. The methods work well with distributions that are not normal, provided that departures from normality are not too extreme. The distribution should be somewhat close to symmetric, should have only one mode, and should not include outliers.
b. B. Yes, the sample is a sample random sample and the sample size is 36, which is greater than 30.