Assume that the age distribution of the Statistics 101 students has a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect.A) The sample mean's expected value is the same as the population's mean.B) The sampling distribution's standard deviation is 3 years.C) The shape of the sampling distribution is approximately normal.&nbsp;D) The standard error of the sampling distribution is equal to 0.3 years.

Question

Assume that the age distribution of the Statistics 101 students has a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect.A) The sample mean&#039;s expected value is the same as the population&#039;s mean.B) The sampling distribution&#039;s standard deviation is 3 years.C) The shape of the sampling distribution is approximately normal.&amp;nbsp;D) The standard error of the sampling distribution is equal to 0.3 years.

liannemdh · Accepted Answer

Solution :&amp;nbsp;Given that ,&amp;nbsp;mean&amp;nbsp;=μ=23&amp;nbsp;standard deviation&amp;nbsp;=σ=3&amp;nbsp;The sampling distribution has a roughly normal form (n = 100).&amp;nbsp;μx―=μ=23&amp;nbsp;and&amp;nbsp;σx―=σn=3100=0.3&amp;nbsp;The incorrect statement is,&amp;nbsp;The sampling distribution&#039;s standard deviation is 3 years.Option B) is correct.

xleb123 · Accepted Answer

A) The sample mean&#039;s expected value is the same as the population&#039;s mean.The statement is correct. According to the properties of the sampling distribution, the expected value (or mean) of the sample mean is equal to the population mean. This is expressed mathematically as:E(X¯)=μwhere E(X¯) represents the expected value of the sample mean X¯, and μ represents the population mean.B) The sampling distribution&#039;s standard deviation is 3 years.The statement is incorrect. The standard deviation of the sampling distribution, also known as the standard error, is not equal to the population standard deviation. Instead, it is given by the formula:SE(X¯)=σnwhere SE(X¯) represents the standard error of the sample mean, σ represents the population standard deviation, and n represents the sample size. In this case, the standard error would be:SE(X¯)=3100=310=0.3&amp;nbsp;yearsC) The shape of the sampling distribution is approximately normal.The statement is correct. According to the central limit theorem, when the sample size is sufficiently large (e.g., n &amp;gt; 30), the sampling distribution of the sample mean tends to follow an approximately normal distribution, regardless of the shape of the population distribution.D) The standard error of the sampling distribution is equal to 0.3 years.The statement is correct. As calculated in part B, the standard error of the sample mean is equal to 0.3 years.In summary:A) CorrectB) IncorrectC) CorrectD) CorrectTherefore, the incorrect statement is B) &#039;The sampling distribution&#039;s standard deviation is 3 years.&#039;

alenahelenash · Accepted Answer

Answer:The incorrect statement is B) The sampling distribution&#039;s standard deviation is 3 years.Explanation:Let&#039;s go through each statement and determine whether it is correct or incorrect.A) The sample mean&#039;s expected value is the same as the population&#039;s mean.The statement is correct. The expected value (or mean) of the sample mean is equal to the population mean. Mathematically, we can express this as:E(X¯)=μwhere E(X¯) represents the expected value of the sample mean X¯, and μ represents the population mean.B) The sampling distribution&#039;s standard deviation is 3 years.The statement is incorrect. The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is given by the formula:σX¯=σnwhere σX¯ represents the standard deviation of the sample mean, σ represents the population standard deviation, and n represents the sample size.Substituting the given values, we have:σX¯=3100=310Therefore, the standard deviation of the sampling distribution is 0.3 years, not 3 years.C) The shape of the sampling distribution is approximately normal.The statement is correct. According to the Central Limit Theorem, as long as the sample size is sufficiently large (typically n ≥ 30), the sampling distribution of the sample mean tends to be approximately normally distributed, regardless of the shape of the population distribution. Therefore, the shape of the sampling distribution is approximately normal.D) The standard error of the sampling distribution is equal to 0.3 years.The statement is correct. The standard error of the sampling distribution, denoted as σX¯, represents the standard deviation of the sample mean. As calculated earlier, σX¯ is equal to 0.3 years.

star233 · Accepted Answer

To solve this problem, let&#039;s go through each statement and determine whether it is correct or incorrect.A) The sample mean&#039;s expected value is the same as the population&#039;s mean.This statement is correct. The expected value (or mean) of the sample mean is equal to the population mean. In this case, the population mean is 23 years, and the sample mean&#039;s expected value will also be 23 years.B) The sampling distribution&#039;s standard deviation is 3 years.This statement is incorrect. The standard deviation of the sampling distribution of the sample mean, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the population is 3 years, but when we sample 100 students, the standard error will be 3 divided by the square root of 100, which is 0.3 years. Therefore, the correct standard deviation for the sampling distribution is 0.3 years, not 3 years.C) The shape of the sampling distribution is approximately normal.This statement is correct. According to the Central Limit Theorem, when the sample size is large enough (typically n &amp;gt; 30), the sampling distribution of the sample mean tends to be approximately normal, regardless of the shape of the population distribution. In this case, since we are sampling 100 students, which is a relatively large sample size, we can assume that the shape of the sampling distribution will be approximately normal.D) The standard error of the sampling distribution is equal to 0.3 years.This statement is correct. As mentioned earlier, the standard error of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. In this case, the population standard deviation is 3 years, and we are sampling 100 students. Therefore, the standard error is equal to 3 divided by the square root of 100, which simplifies to 0.3 years.Result:(B)&amp;nbsp;The&amp;nbsp;sampling&amp;nbsp;distribution&#039;s&amp;nbsp;standard&amp;nbsp;deviation&amp;nbsp;is&amp;nbsp;3&amp;nbsp;years.

Suppose the ages of students in Statistics 101 follow a normal distribution with a mean of 23 years and a standard deviation of 3 years

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