A sample of 100 cars driving on a freeway during a morning commute was drawn, and the number of occupants in each car was recorded. The results were as follows:

Question

a. Find the sample mean number of occupants.

b. Find the sample standard deviation of the number of occupants.

c. Find the sample median number of occupants.

d. Compute the first and third quartiles of the number of occupants.

e. What proportion of cars had more than the mean number of occupants?

f. For what proportion of cars was the number of occupants more than one standard deviation greater than the mean?

g. For what proportion of cars was the number of occupants within one standard deviation of the mean?

Answer 1

(a) Let denote the 100 numbers of occupants.





(b) The sample variance is





The standard deviation is

Alternatively, the sample variance can be computed as





(c) The sample median is the average of the 50th and 51st value when arranged in order. Both these values are equal to 1, so the median is 1.

(d) The first quartile is the average of the 25th and 26th value when arranged in order. Both these values are equal to 1, so the first quartile is 1. The third quartile is the average of the 75th and 76th value when arranged in order. Both these values are equal to 2, so the first quartile is 2.

(e) Of the 100 cars, 15 + 10 + 3 + 2 = 30 had more than the mean of 1.52 occupants, so the proportion is 30/100 = 0.3.

(f) The quantity that is one standard deviation greater than the mean is 1.52 + 0.9372 = 2.5472. Of the 100 cars, 10 + 3 + 2 = 15 had more than 2.8652 children, so the proportion is 15/100 = 0.15.

(g) The region within one standard deviation of the mean is 1.52 ± 0.9372 = (0.5828, 2.4572). Of the 100 cars, 70 + 15 = 85 are in this range, so the proportion is 85/100 = 0.85.