You are interested in estimating the mean of a population. you plan to take a random sample from the population and use the samples mean as an estimate of population mean.
Assuming that the population from which you select your sample is normal, which of the statements about the mean are true?
a. its expected value (mean) is equal to the value of the population mean.
b. It follows a normal distribution only for sufficeiently large samples sizes.
c. It follows a normal distribution for any sample size.
d. The variance of its sampling distribution is equal to the population variance.
e. Its expected value is equal to the value of the population mean divided by the sample size.
f. The variance of its sampling distribution is equal to the variance of the population divided by the sample size.
a b c
The correct statements about the mean are:
a. Its expected value (mean) is equal to the value of the population mean. This means that, on average, the sample mean will be equal to the population mean.
b. It follows a normal distribution only for sufficiently large sample sizes. As the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution. This is known as the Central Limit Theorem.
d. The variance of its sampling distribution is equal to the population variance divided by the sample size. The standard deviation of the sampling distribution of the mean, often referred to as the standard error, is equal to the population standard deviation divided by the square root of the sample size.
So, the correct statements are a, b, and d.
To answer this question, we need to consider the properties of the sampling distribution of the sample mean when sampling from a normal population.
a. Statement a is true. The expected value (mean) of the sampling distribution of the sample mean is equal to the value of the population mean.
b. Statement b is not true. Regardless of the sample size, the sampling distribution of the sample mean is approximately normal due to the Central Limit Theorem.
c. Statement c is true. The sampling distribution of the sample mean follows a normal distribution for any sample size, as long as the population itself is normally distributed.
d. Statement d is not true. The variance of the sampling distribution of the sample mean is equal to the population variance divided by the sample size, not the population variance itself.
e. Statement e is not true. The expected value of the sampling distribution of the sample mean is equal to the population mean, not divided by the sample size.
f. Statement f is true. The variance of the sampling distribution of the sample mean is equal to the variance of the population divided by the sample size.
So, the correct statements are a, c, and f.