Question

Consider two populations of coins, one of 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?

• Yes, the conditions for normality have been met because the distributions of age for the two populations are approximately normal.

  A

• Yes, the conditions for normality have been met because the sample sizes taken from both populations are large enough.

  B

• 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.

  C

• 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.

  D

• 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.

• E

For a weekly town council meeting in a certain town, the distribution of the duration of the meeting is approximately normal with mean 53 minutes and standard deviation 2.5 minutes. For a weekly arts council meeting in the same town, the distribution of the duration of the meeting is approximately normal with mean 56 minutes and standard deviation 5.1 minutes. Let x¯1x¯1 represent the average duration, in minutes, of 10 randomly selected town council meetings, and let x¯2x¯2 represent the average duration, in minutes, of 10 randomly selected arts council meetings.

Which of the following is the best reason why the sampling distribution of x¯1−x¯2x¯1−x¯2 can be modeled by a normal distribution?

• The sample sizes are equal.

  A

• Both sample sizes are large enough to satisfy the normality condition.

  B

• The population distributions are approximately normal.

  C

• The population standard deviations are assumed equal.

  D

• The sample sizes are less than 10% of the corresponding population sizes.

• E

Answer 1

(1) According to the central limit theorem, sample sizes should be greater than or equal to 30 for the theorem to hold, which says that the distribution of sample means approximate to a normal distribution if the sample size is sufficiently large (>= 30). Here, the size of the random sample of pennies is 25, indicating that it is not sufficiently large. Hence, our answer is - 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.

(2) In the question here, it is already mentioned that both the distributions of the duration of the town council meeting and the arts council meeting are approximately normal. Hence, the sampling distribution of the difference of their sample means will also be approximately normal. Thus, our answer is - The population distributions are approximately normal.