Question

chapter 5 practice, part 2
name

1. using the central limit theorem, determine whether the sampling distribution is approximately

normal or not. state your reasoning.
a. the distribution of the population is unknown. the sample size is 5.
b. the distribution of the population is right - skewed. the sample size is 35.
c. the distribution of the population is approximately normal. the sample size is 8.

Explanation:

Step1: Recall the Central Limit Theorem conditions

The Central Limit Theorem (CLT) states that if the sample size \(n\) is large (\(n\geq30\)), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal, regardless of the population distribution. If the population is normal, the sampling distribution of \(\bar{X}\) is normal for any sample size \(n\).

Step2: Analyze each case

• Case a:

The population distribution is unknown and \(n = 5\). Since \(n<30\) and the population distribution is unknown, the sampling distribution of the sample mean is not approximately normal.

• Case b:

The population is right - skewed (not normal), but \(n=35\geq30\). By the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal.

• Case c:

The population is normal. For a normal population, the sampling distribution of the sample mean \(\bar{X}\) is normal for any sample size \(n\). Here \(n = 8\), so the sampling distribution of the sample mean is approximately normal.

Answer:

a. Not normal. Reason: \(n = 5<30\) and population distribution is unknown.
b. Normal. Reason: \(n=35\geq30\) (Central Limit Theorem applies for \(n\geq30\) even if population is skewed).
c. Normal. Reason: Population is normal (sampling distribution of \(\bar{X}\) is normal for any \(n\) when population is normal).