Question

The following graph shows the curve for a normally distributed variable. Superimposed are the curves for the sampling distributions of the sample mean for two different sample sizes.

a. Explain why all three curves are centered at the same place.

b. Which curve corresponds to the larger sample size? Explain your answer.

c. Why is the spread of each curve different?

d. Which of the two sampling-distribution curves corresponds to the sample size that will tend to produce less sampling error? Explain your answer.

c. Why are the two sampling-distribution curves normal curves?

Short answer

Part (a) The mean of the sample means is always equal to $\mu$ regardless of sample size.

Part (b) Curve $B$corresponds to the larger sample size.

Part (c) ${\sigma }_{\overline{x}}$differs depending on the sample size.

Part (d) Because the curve $B$has the lesser standard deviation of the two sampling distributions ($A$and $B$).

Part (e) because of the population variable.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

$\text{population mean and standard deviation:}{\mu }_{\overline{x}}=\mu \text{and}{\sigma }_{\overline{x}}=\sigma /\sqrt{n}\text{.}$

Part (a) Step 3: Explanation

We know that if a population variable has a mean of $\mu$and a standard deviation of $\sigma$, then the sample mean has a mean of $\mu$and a standard deviation of $\frac{\sigma }{\sqrt{n}}$, where $n$is the sample size.

As a result, the mean of sample means is equal to the population mean $\mu$and is independent of sample size, i.e., the mean of sample means is always equal to $\mu$regardless of sample size.

All three curves are located at the same spot because the normal curves are centered at the mean $\mu$

Part (b) Step 1: Explanation

The higher sample size is shown by curve $B$

Because the square root of sample size $n$ is inversely related to the standard deviation of sample mean ${\sigma }_{\overline{x}}$ That is, the larger the sample size, the lower the standard deviation of the sample mean, and hence the smaller the spread of the normal curve. Curve $B$ has the lower spread in this case. As a result, curve $B$ represents the greater sample size.

Part (c) Step 1: Explanation

We know that the standard deviation of the sample mean is proportional to the sample size ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

As a result, ${\sigma }_{\overline{x}}$differs depending on the sample size.

As a result, the spread of each normal curve varies depending on the standard deviation of sample means${\sigma }_{\overline{x}}$

Part (d) Step 1: Explanation

Because curve $B$ has the lesser standard deviation of the two sampling distributions ($A$ and $B$), it has less sampling error. We know that the smaller the standard deviation of sample means, the smaller the sampling error tends to be.

Part (e) Step 1: Explanation

Because of the population variable, sample mean follows normal distribution regardless of sample size for a normally distributed population variable.