Questions: Understanding the sampling distribution of M You are interested in estimating the mean of a population. You plan to take a random sample from the population and use the sample's mean as an estimate of the population mean. Assuming that the population from which you select your sample is not normal, which of the statements about M are true? Check all that apply. - The expected value of M is equal to the value of the population mean. - You can only assume that the sampling distribution of M is normally distributed for sufficiently large sample sizes. - You can assume that the sampling distribution of M is normally distributed for any sample size. - The sampling distribution of the z -score of M is normal for any sample size.

Understanding the sampling distribution of M

You are interested in estimating the mean of a population. You plan to take a random sample from the population and use the sample's mean as an estimate of the population mean.

Assuming that the population from which you select your sample is not normal, which of the statements about M are true? Check all that apply.
- The expected value of M is equal to the value of the population mean.
- You can only assume that the sampling distribution of M is normally distributed for sufficiently large sample sizes.
- You can assume that the sampling distribution of M is normally distributed for any sample size.
- The sampling distribution of the z -score of M is normal for any sample size.

Transcript text: 4. Understanding the sampling distribution of M You are interested in estimating the mean of a population. You plan to take a random sample from the population and use the sample's mean as ar estimate of the population mean. Assuming that the population from which you select your sample is not normal, which of the statements about M are true? Check all that apply. The expected value of $M$ is equal to the value of the population mean. You can only assume that the sampling distribution of $M$ is normally distributed for sufficiently large sample sizes. You can assume that the sampling distribution of M is normally distributed for any sample size. The sampling distribution of the z -score of M is normal for any sample size.

Solution

Solution Steps

To address the question, we need to apply the Central Limit Theorem (CLT) and properties of sampling distributions. The CLT states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is sufficiently large, regardless of the population's distribution. The expected value of the sample mean is equal to the population mean. The z-score of the sample mean will be normally distributed for any sample size if the population is normal, but for non-normal populations, this holds true only for large sample sizes.

Step 1: Understanding the Expected Value

The expected value of the sample mean $ M $ is equal to the population mean $ \mu $. This can be expressed mathematically as: \[ E[M] = \mu \] This statement is true.

Step 2: Applying the Central Limit Theorem

According to the Central Limit Theorem, the sampling distribution of the sample mean $ M $ approaches a normal distribution as the sample size $ n $ increases, specifically when $ n $ is sufficiently large. This means that for non-normal populations, we can only assume that the sampling distribution of $ M $ is normally distributed for large $ n $. Thus, the statement is: \[ \text{For sufficiently large } n, \quad M \sim N(\mu, \frac{\sigma^2}{n}) \] This statement is also true.

Step 3: Evaluating Other Statements

The statement that the sampling distribution of $ M $ is normally distributed for any sample size is false, as it only holds for large sample sizes when the population is not normal. Similarly, the statement regarding the z-score of $ M $ being normal for any sample size is also false unless the population itself is normal.

Final Answer

The true statements are:

The expected value of $ M $ is equal to the value of the population mean.
You can only assume that the sampling distribution of $ M $ is normally distributed for sufficiently large sample sizes.

Thus, the answer is: \[ \boxed{\text{True statements: 1 and 2}} \]

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