Questions: Question 6 10 pts A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of x̄ is approximately normal because of the central limit theorem approximately normal because the sample size is large in comparison to the population size approximately normal because x̄ is always approximately normally distributed normal if the population is normally distributed

Question 6
10 pts

A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of x̄ is approximately normal because of the central limit theorem
approximately normal because the sample size is large in comparison to the population size
approximately normal because x̄ is always approximately normally distributed
normal if the population is normally distributed

Transcript text: Question 6 10 pts A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of $\bar{x}$ is $\qquad$ approximately normal because of the central limit theorem approximately normal because the sample size is large in comparison to the population size approximately normal because $\bar{x}$ is always approximately normally distributed normal if the population is normally distributed Previous Next

Solution

Solution Steps

Solution Approach

To determine the sampling distribution of the sample mean ($\bar{x}$), we need to consider the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large, typically $n \geq 30$. However, if the population is normally distributed, the sampling distribution of the sample mean will be normal regardless of the sample size.

Step 1: Determine Sample Size and Population Size

We have a sample size of $ n = 24 $ and a population size of $ N = 150 $.

Step 2: Check Conditions for Normality

To assess the normality of the sampling distribution of the sample mean $ \bar{x} $, we evaluate two conditions:

The sample size is not large enough for the Central Limit Theorem to apply, as $ n < 30 $.
The sample size relative to the population size is $ \frac{n}{N} = \frac{24}{150} = 0.16 $, which is greater than $ 0.1 $. This indicates that the sample size is large in comparison to the population size.

Step 3: Conclusion on Distribution

Since the sample size is not large enough for the Central Limit Theorem to apply, but it is large relative to the population size, we conclude that the sampling distribution of $ \bar{x} $ is approximately normal because the sample size is large in comparison to the population size.

Final Answer

The answer is approximately normal because the sample size is large in comparison to the population size. Thus, we can box the final answer as follows:

$\boxed{\text{approximately normal because the sample size is large in comparison to the population size}}$

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