Question

There is a population of people: one-sixth of the people have seen 1 movie in the last year. One-sixth of the people have seen 6 movies in the last year. One-sixth of the people have seen 7 movies in the last year. Two-sixths (one-third) of the people have seen 9 movies in the last year. And one-sixth of the people have seen 10 movies in the last year. If a sample of 225 people is picked, what is the probability that the sample mean will be smaller than 7.5?

Answer 1

Based on the given data,

No. of movies (x)	Probability (p)	(x) (p)	(x-Mean )	(x) (x-Mean)
1	1/6	1/6	36	36
6	1/6	6/6	1	6
7	1/6	7/6	0	0
9	2/6	18/6	4	36
10	1/6	10/6	9	90
SUM = 33	SUM = 1	SUM = 42 / 6 = 7	SUM = 50	SUM = 168
MEAN (M) = 7 / 1 = 7
VAR = 168 / 33 = 5.091
SD (s) = SQRT (VAR) = 2.256

Hence, the expectation (or) mean for the given data is obtained as 7.

For n = 225,

Pr( M < 7.5)

From standard normal table,

= 0.99955

Hence, the probability that the sample mean will be smaller than 7.5 would be 0.99955.