Question

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The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85. Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?

A.) Both distributions are approximately normal with mean 65 and standard deviation 3.5.

B.) Both distributions are approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.

C.) Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.

D.) Only the distribution for size 85 is approximately normal. Both distributions have mean 65 and standard deviation 3.5.

E.) Only the distribution for size 85 is approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.

Answer 1

Option C.) Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.

Step-by-step explanation:

We are given that the distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches.

Let X = distribution of height for a certain population of women

So, X ~ Normal($\mu=65 ,\sigma = 3.5$)

where, $\mu$ = mean height = 65 inches

            $\sigma$ = standard deviation = 3.5 inches

Now, Consider two different random samples taken from the population, one of size 5 and one of size 85.

• Firstly since both the random samples are taken from the population which means that both the distributions 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 both distribution will have the same mean of 65 inches.

• Now, coming to standard deviation. Standard deviation formula for sampling distribution is given by;

                   S.D. = $\frac{\sigma}{\sqrt{n} }$

So, Standard deviation of the distribution with sample size n = 5 is ;

                     $S.D_1 =\frac{3.5}{\sqrt{5} }$   = 1.565 inches

Similarly, Standard deviation of the distribution with sample size n = 85 is ;

                     $S.D_2 =\frac{3.5}{\sqrt{85} }$   = 0.379 inches

This means that the standard deviation for sample size 5 is greater than that for sample size 85.

So, Option C reflects all the possibility and this is the correct statement that both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.