Questions: Suppose a large population has mean μ and standard deviation σ, and a simple random sample of size n is taken. The sampling distribution of the sample mean x̄ has mean and variance respectively equal to a) μ / n and σ^2 / n b) μ and σ / n c) μ / n and σ^2 / n. d) μ and σ^2 / n.

Suppose a large population has mean μ and standard deviation σ, and a simple random sample of size n is taken. The sampling distribution of the sample mean x̄ has mean and variance respectively equal to
a) μ / n and σ^2 / n
b) μ and σ / n
c) μ / n and σ^2 / n.
d) μ and σ^2 / n.

Transcript text: 22. Suppose a large population has mean $\mu$ and standard deviation $\sigma$, and a simple random sample of size $n$ is taken. The sampling distribution of the sample mean $\bar{x}$ has mean and variance respectively equal to a) $\mu / \mathrm{n}$ and $\sigma^{2} / \mathrm{n}$ b) $\mu$ and $\sigma / n$ c) $\mu / \mathrm{n}$ and $\sigma^{2} / \mathrm{n}$. d) $\mu$ and $\sigma^{2} / \mathrm{n}$.

Solution

Solution Steps

Step 1: Understand the properties of the sampling distribution of the sample mean

The sampling distribution of the sample mean $\bar{x}$ is derived from taking repeated samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$. The mean of the sampling distribution of $\bar{x}$ is equal to the population mean $\mu$, and the variance of the sampling distribution of $\bar{x}$ is equal to the population variance $\sigma^2$ divided by the sample size $n$.

Step 2: Identify the correct mean of the sampling distribution

The mean of the sampling distribution of $\bar{x}$ is $\mu$. This eliminates options a) and c), which incorrectly state the mean as $\mu / n$.

Step 3: Identify the correct variance of the sampling distribution

The variance of the sampling distribution of $\bar{x}$ is $\sigma^2 / n$. This eliminates option b), which incorrectly states the variance as $\sigma / n$.

Step 4: Match the correct option

The correct option is d), which states that the mean is $\mu$ and the variance is $\sigma^2 / n$.