In each part, assume that the population standard deviation...

In each part, assume that the population standard deviation is known. Decide whether use of the $z$ -interval procedure to obtain a confidence interval for the population mean is reasonable. Explain your answers. a. The variable under consideration is very close to being normally distributed, and the sample size is $10 .$ b. The variable under consideration is very close to being normally distributed, and the sample size is $75 .$ c. The sample data contain outliers, and the sample size is $20 .$

In each part, assume that the population standard deviation is known. Decide whether use of the $z$ -interval procedure to obtain a confidence interval for the population mean is reasonable. Explain your answers.
a. The variable under consideration is very close to being normally distributed, and the sample size is $10 .$
b. The variable under consideration is very close to being normally distributed, and the sample size is $75 .$
c. The sample data contain outliers, and the sample size is $20 .$

Key Concepts

Z-Interval Procedure

This technique is used to construct confidence intervals for a population mean when the population standard deviation is known. It relies on the fact that the sample mean, under certain conditions, follows a normal distribution. The method uses the z-distribution quantiles to determine the margin of error and the final interval estimate.

Normality Assumption

For the z-interval procedure to be valid, the underlying population from which samples are drawn should be normally distributed or approximately so. When the population distribution significantly deviates from normality, especially in small samples, the confidence interval may not accurately reflect the true population parameter.

Central Limit Theorem (CLT)

The CLT states that, regardless of the population's distribution, the sampling distribution of the mean tends toward a normal distribution as the sample size increases. This permits the use of the z-interval procedure even when the population is not exactly normal, provided the sample size is sufficiently large.

Effect of Sample Size

The reliability of the z-interval procedure is partly dependent on the sample size. Smaller sample sizes require a closer adherence to the assumption of normality, while larger samples benefit from the Central Limit Theorem. Thus, using the z-interval in situations where the sample is small and the population is not exactly normal can be problematic.

Impact of Outliers

Outliers can distort the sample mean and increase variability, undermining the assumptions required for the z-interval procedure. The presence of outliers indicates that the data may not represent the intended structure, and caution must be exercised since these deviations can lead to misleading confidence intervals.

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