Question

Consider two populations of coins, one of the pennies and one of quarters. A random sample of 25 pennies was selected, and the mean age of the sample was 32 years. A random sample of 35 quarters was taken, and the mean age of the sample was 19 years.
For the sampling distribution of the difference in sample means, have the conditions for normality been met?

a. Yes, the conditions for normality have been met because the distributions of age for the two populations are approximately normal.
b. Yes, the conditions for normality have been met because the sample sizes taken from both populations are large enough.
c. No, the conditions for normality have not been met because neither sample size is large enough and no information is given about the distributions of the populations.
d. No, the conditions for normality have not been met because the sample size for the pennies is not large enough and no information is given about the distributions of the populations.
e. No, the conditions for normality have not been met because the sample size for the quarters is not large enough and no information is given about the distributions of the populations.

Answer 1

Answer:

d. No, the conditions for normality have not been met because the sample size for the pennies is not large enough and no information is given about the distributions of the populations.

Step-by-step explanation:

In this example, we learn that the sample taken of pennies was of 25, while the sample of quarters was of 35. According to the central limit theorem, a sample must be equal to or larger than 30 for the theorem to hold. The theorem states that when random variables are added, they tend towards a normal distribution even if the original variables themselves are not normally distributed. In this question, the sample of pennies is not large enough for the theorem to hold. On top of this, we have no information about the distribution of the populations. Therefore, the conditions for normality have not been met.