Question

Computers Magazine has recently completed an analysis of its customer base. It has determined that 75% of the issues sold each month are subscriptions and the other 25% are sold at newsstands. It has also determined that the ages of its subscribers are normally distributed with a mean of 44.5 and a standard deviation of 7.42 years, whereas the ages of its newsstand customers are normally distributed with a mean of 36.1 and a standard deviation of 8.20 years.

(a) Computers Magazine would like to make the following statement to its advertisers: “80% of our subscribers are between the ages of . . . and . . . .” Your job is to fill in the blanks, choosing a range that is symmetric around the mean. (In other words, the mean age of subscribers should be the midpoint of the range.)

(b) What proportion of Computers Magazine newsstand customers have ages in the range you gave in (a)?

(c) What proportion of all of Computers Magazine customers have ages in the range you gave in (a)?

Answer 1

a) Z score of 1.281 represents/covers 40% observation on the right of mean. Similarly, Z score of -1.281 represents/covers 40% observation on the left of mean.

Std. deviation of the subscribers is 7.42 and average age is 44.2 years

so, range of 80% is between 44.2 - 1.281*7.42 and 44.2 + 1.281*7.42 i.e between 53.7 and 34.7

b) Newsstand customer have average age of 36.1 and std. deviation of 8.20

from (a) Lower bound is 34.7 and upper bound is 53.7

Z-score corresponding to 34.7 --> 34.7 = 36.1 - Z_Score*8.2 --> Z_Score = (36.1 - 34.7)/8.2 = 0.171

P-Value corresponding to 0.171 = 0.432 . It means 0.5 - 0.432 i.e. 0.068 or 6.8% of populations lies on the left of mean

Z-score corresponding to 53.7 --> 53.7 = 36.1 + Z_Score*8.2 --> Z_Score = (53.7 - 36.1)/8.2 = 2.146

P-Value corresponding to 2.146 = 0.016. It means 0.5 - 0.016 i.e. 0.484 or 48.8% of populations lies on the right of mean

Overall proportion between 34.7 and 53.7 among the Newsstand customer = 6.8% + 48.8% = 55.6%

c) Overall proportion between 34.7 and 53.7

75% are subscription customers and 25% are newsstand customers with 80% and 55.6% customer respectively between age range of 34.7 and 53.7 years

Overall Proportion between 34.7 and 53.7 = 0.75*0.8 + 0.25*0.556 = 0.6 + 0.139 = 0.739 or 73.9%