Question

The city government has collected data on the square footage of houses within the city. They found that the average square footage of homes within the city limit is 1,240 square feet while the median square footage of homes within the city limits is 1,660 square feet. The city government also found that the standard deviation of home square footage within the city limits is 198 square feet. A statistician hired by a local home-carpeting company is going to randomly select a sample of 24 houses and record the square footage of the homes using public records. Which of the following is true?

** Please tell me the answer(s) as well as how you got those answer(s).

a. The shape of the sampling distribution of the mean square footage of homes will be right skewed.

b. The shape of the sampling distribution of the mean square footage of homes will be left skewed.

c. If the statistician sampled 11 more homes within the city limits and added their data to the original sample of 24 homes then the shape of the sampling distribution of the mean square footage of all 35 homes will be approximately symmetric.

d. The sampling distribution of the mean square footage will have a smaller standard deviation when compared to the standard deviation of square footage among all homes within the city limits.

e. The sampling distribution of the mean square footage will have a standard deviation equal to or larger than the standard deviation of square footage among all homes within the city limits.

Answer 1

(a)

In case of right skewed data, mean lies right to median in the graph i.e. mean is greater than median. In the given data, mean (1240) is not greater than median (1660). So, the shape of the sampling distribution of the mean square footage will not be skewed.

So, the given statement is False.

(b)

In case of left skewed data, mean lies left to median in the graph i.e. mean is lesser than median. In the given data, mean (1240) is lesser than median (1660). So, the shape of the sampling distribution of the mean square footage will be skewed.

So, the given statement is True.

(c)

The population data shows that mean (1240) is lesser than median (1660). So, the population data is left skewed. Hence there is no dependency upon number of sample units to obtain a symmetric sample, it is always (more or less) left skewed.

So, the given statement is False.

(d)

Suppose, be the random sample obtained.

Mean square footage is given by

So, standard deviation is given by which is clearly lesser than sample standard deviation 198.

So, the given statement is True.

(e)

Based on calculation in (d) we can conclude that the given statement is False.