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) x = 88.75 sx = 8.897431411 n = 30 a. what are the requirements for using the methods for the mean to construct a confidence interval estimate of a population mean? a. the sample is a simple random sample and the population is normally distributed or n ≤ 30 b. the sample is a simple random sample and the population is normally distributed or n > 30. c. the sample is a simple random sample, the population is normally distributed, and n > 30 d. the sample is a simple random sample, the population is normally distributed, and n ≤ 30 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

Explanation:

For part (a), when constructing a confidence interval for the population mean using the t - distribution (which is the case here as the population standard deviation is unknown), the requirements are that the sample is a simple random sample and either the population is normally distributed or \(n>30\) (by the Central Limit Theorem).
For part (b), the term “robust” in statistics means that the method is not overly affected by violations of its assumptions. In the context of confidence intervals for the mean, it means the methods work well with non - normal distributions as long as the departures from normality are not too extreme. A somewhat symmetric distribution, unimodal (only one mode), and no outliers are characteristics of a distribution that is “not too extreme” in its non - normality.

Answer:

a. B. The sample is a simple random sample and the population is normally distributed or \(n > 30\)
b. 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