Question

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 normal, which of the statements about M are true? Check all that apply. The standard deviation of the sampling distribution of M is equal to the standard deviation of the population divided by the square root of the sample size. 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 standard deviation of the sampling distribution of M is equal to the population standard deviation.

Answer 1

Answer: The standard deviation of the sampling distribution of M is equal to the standard deviation of the population divided by the square root of the sample size.

You can assume that the sampling distribution of M is normally distributed for any sample size.

Step-by-step explanation:

• According to the central limit theorem , if we have a population with mean $(\mu)$ and standard deviation $\sigma$ , then if we take a sufficiently large random samples from the population with replacement ,  the distribution of the sample means will be approximately normally distributed.
• When population is normally distributed , then the mean of the sampling distribution = Population mean $(\mu)$
• Standard deviation of the sampling distribution = $\dfrac{\sigma}{\sqrt{n}}$ , where $\sigma$ =  standard deviation of the population  , n=  sample size.

So, the correct statements are:

1. You can assume that the sampling distribution of M is normally distributed for any sample size.
2. The standard deviation of the sampling distribution of M is equal to the standard deviation of the population divided by the square root of the sample size.