Question

The distribution of the commute times for the employees at a large company has mean 22.4 minutes and standard deviation 6.8 minutes. A random sample of n employees will be selected and their commute times will be recorded.

What is true about the sampling distribution of the sample mean as n increases from 2 to 10 ?


A.) The mean increases, and the variance decreases.

B.) The mean does not change, and the variance does not change.

C.) The mean does not change, and the variance increases.

D.) The mean does not change, and the variance decreases.

E.)The mean increases, and the variance increases.

Answer 1

Option D.) The mean does not change, and the variance decreases.

Step-by-step explanation:

We are given that the distribution of the commute times for the employees at a large company has mean 22.4 minutes and standard deviation 6.8 minutes.

Let X = distribution of the commute times for the employees

So, X ~ Normal($\mu=22.4 ,\sigma = 6.8$)

where, $\mu$ = population mean = 22.4 minutes

            $\sigma$ = standard deviation = 6.8 minutes

Now, A random sample of n employees will be selected and their commute times will be recorded.

• Firstly since the random samples is taken from the population data which means that the sampling distribution of the sample mean will approximately follow normal distribution as the population data follows normal distribution.

• Also, the mean of sample distribution is given by the formula;

          Sample Mean = Population Mean

                               $\bar X = \mu$

This means that distribution will have the same mean as that of population of 22.4 minutes. So, mean does not change.

• Now, coming to Variance. Variance formula for sampling distribution is given by;

                   Variance = $\frac{\sigma^{2} }{{n} }$

So, Variance of the distribution with sample size n = 2 is ;

                     $Varinace_1 =\frac{6.8^{2} }{{2} }$   = 23.12 minutes

Similarly, Variance of the distribution with sample size n = 10 is ;

                     $Varinace_2 =\frac{6.8^{2} }{{10} }$   = 4.624 inches

This means that the variance for sample size n = 2 is greater than that for sample size n = 10.

This means that the variance decrease as n increases from 2 to 10.

So, Option D reflects all the possibility and this is the correct statement that the mean does not change, and the variance decreases.