Question

The Moon’s orbit is not circular, and its distance from the Earth varies. One day, when the Moon is at closest approach, 25 different astronomers each take one measurement of the Earth-Moon distance. The mean of these measurements comes out to be 363,105 kilometers, and their SD is 10 km. For each of the following four statements about this data, either state that it is true (in which case you need do nothing further) or false (in which case, explain why). You may assume the Gauss model, with no bias.

A. The mean of these 25 measurements is about 2 km away from 363,105 km.

B. The interval between 363,101 and 363,109 km is a 95% confidence interval for the average of the 25 measurements.

C. Each of the 25 measurements is about 10 km away from 363,105 km.

D. If a 26th measurement were made, there is a 95% chance it would be within 4 km of the actual Earth-Moon distance.

Answer 1

A)

Answer: False

Explanation:

The sample mean of the sampling distribution is always equal to the population means.

B)

Answer: True

Explanation: Since the population standard deviation is not known, the t distribution is used to construct the confidence interval.

The 95% confidence interval is obtained using the following formula,

The t critical value is obtained from the t critical value table for significance level = 0.05, and degree of freedom = n-1 = 24,

Now,

C)

Answer: False

Explanation: The standard deviation value = 10 km which means the on average a data value lies 10 km from its mean, it doesn't mean that each measurement is 10 km away from the mean.

D)

Answer: True

Explanation:

The margin of error for the 95% CI is,

The margin of error value is approximately 4 km hence we can say that the 26th measurement would be within 4 km of the actual Earth-Moon distance.