Which of the following statements about the sampling distribution of the sample mean is incorrect? (a) The standard deviation of the sampling distribu

To solve the question and provide the correct statement, we will analyze each option and determine which one is incorrect.
(a) The standard deviation of the sampling distribution will decrease as the sample size increases.
$\mathbf{\text{This statement is correct.}}$ As the sample size increases, the standard deviation of the sampling distribution decreases. This phenomenon is known as the Central Limit Theorem. The standard deviation of the sampling distribution is given by $\frac{\sigma }{\sqrt{n}}$, where $\sigma$ represents the population standard deviation and $n$ is the sample size.
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
$\mathbf{\text{This statement is correct.}}$ The standard deviation of the sampling distribution provides a measure of how much the sample mean varies among different samples drawn from the same population. It quantifies the spread or variability of the sample means.
(c) The sample mean is an unbiased estimator of the population mean.
$\mathbf{\text{This statement is correct.}}$ The sample mean is an unbiased estimator of the population mean. This means that, on average, the sample mean will be equal to the population mean. The unbiasedness property is a desirable characteristic of estimators.
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
$\mathbf{\text{This statement is correct.}}$ The sampling distribution illustrates how the sample mean will vary across multiple samples drawn from the same population. It provides information about the distribution of sample means and helps us understand the behavior of the estimator.
(e) The sampling distribution shows how the sample was distributed around the sample mean.
$\mathbf{\text{This statement is incorrect.}}$ The sampling distribution does not show how the sample was distributed around the sample mean. Instead, it shows the distribution of sample means. It provides information about the variability of sample means and the shape of the distribution they follow.
Therefore, the $\mathbf{\text{incorrect statement}}$ is (e) 'The sampling distribution shows how the sample was distributed around the sample mean.'

(a) The standard deviation of the sampling distribution will $\mathbf{\text{not}}$ decrease as the sample size increases.
The standard deviation of the sampling distribution is given by the formula ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$, where ${\sigma }_{\overline{x}}$ represents the standard deviation of the sampling distribution, $\sigma$ represents the population standard deviation, and $n$ represents the sample size. As the sample size $n$ increases, the denominator $\sqrt{n}$ becomes larger, resulting in a $\mathbf{\text{decrease}}$ in the standard deviation of the sampling distribution. Therefore, statement (a) is $\mathbf{\text{incorrect}}$.
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
This statement is $\mathbf{\text{correct}}$. The standard deviation of the sampling distribution, ${\sigma }_{\overline{x}}$, measures how much the sample means from different samples would vary around the population mean. It provides a measure of the precision of the sample mean as an estimator of the population mean.
(c) The sample mean is an unbiased estimator of the population mean.
This statement is $\mathbf{\text{correct}}$. The sample mean is an unbiased estimator of the population mean, which means that on average, the sample mean is equal to the population mean. This property holds regardless of the sample size.
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
This statement is $\mathbf{\text{correct}}$. The sampling distribution of the sample mean illustrates how the sample means from different samples would vary if we were to repeat the sampling process multiple times. It provides information about the distribution of the sample means and allows us to make inferences about the population mean.
(e) The sampling distribution shows how the sample was distributed around the sample mean.
This statement is $\mathbf{\text{incorrect}}$. The sampling distribution does not show how the sample was distributed around the sample mean. Instead, it shows how the sample means from different samples would be distributed around the population mean. The sampling distribution focuses on the behavior of the sample means, not the individual observations within each sample.
Therefore, the $\mathbf{\text{incorrect}}$ statement is (a) The standard deviation of the sampling distribution will decrease as the sample size increases.

Step 1:
(a) The standard deviation of the sampling distribution will decrease as the sample size increases.
This statement is correct. According to the central limit theorem, as the sample size ($n$) increases, the sampling distribution of the sample mean approaches a normal distribution with a smaller standard deviation. The standard deviation of the sampling distribution, often denoted as ${\sigma }_{\overline{x}}$, is inversely proportional to the square root of the sample size ($n$). So as $n$ increases, ${\sigma }_{\overline{x}}$ decreases.
Step 2:
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
This statement is correct. The standard deviation of the sampling distribution, ${\sigma }_{\overline{x}}$, represents the spread or variability of the sample means when multiple samples are taken from the same population. It provides a measure of how much the sample mean is expected to vary across different samples.
Step 3:
(c) The sample mean is an unbiased estimator of the population mean.
This statement is correct. The sample mean, denoted as $\overline{x}$, is an unbiased estimator of the population mean, denoted as $\mu$. This means that, on average, the sample mean will equal the population mean. Unbiasedness ensures that the sample mean is not systematically overestimating or underestimating the population mean.
Step 4:
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
This statement is correct. The sampling distribution illustrates how the sample mean will vary if multiple samples are taken from the same population. It provides information about the distribution of sample means and allows us to make inferences about the population mean based on the behavior of the sample mean.
Step 5:
(e) The sampling distribution shows how the sample was distributed around the sample mean.
This statement is incorrect. The sampling distribution does not show how the sample was distributed around the sample mean. Instead, it shows how the sample means are distributed around the population mean. The sampling distribution is concerned with the behavior and variability of the sample means, not the individual values within a single sample.
In summary, the incorrect statement is (e).