The NBC hit comedy Friends was Tivo\'s most popular show during the week of Apri
ID: 2359800 • Letter: T
Question
The NBC hit comedy Friends was Tivo's most popular show during the week of April 18-24, 2004. According to the Nielsen ratings, 29.7% of Tivo owners either recorded Friends or watched it live. Suppose you select a random sample of 50 Tivo owners. a. What is the probability that more than half the people in the sample watched or recorded Friends? b. What is the probability that less than 25% of the people in the sample watched or recorded Friends? c. If a random sample of size 500 is taken, how does this change your answers to a and b?Explanation / Answer
The NBC hit comedy Friends was TiVo's most popular show during the week of April 18-24, 2004. According to the Nielsen ratings, 29.7% of TiVo owners in the United States either recorded Friends or watched it live ("Prime - Time Nielsen Ratings," USA Today, April 28, 2004, p. 3D)
Suppose you select a random sample of 50 TiVo owners.
Let X be the number of TiVo owners who watched or recorded out of the 50 selected. Then X follows the binomial distribution with parameters n = 50 and p = 29.7/100 = 0.297
The mean of X is np = 50*0.297 = 14.85 and the variance is np(1-p) = 50*0.297*(1-0.297)=10.440. The standard deviation is sqrt(10.440)=3.231.
Since n>30, the distribution of X can be approximated by the normal distribution with mean 14.95 and standard deviation 3.231.
Z = (X-14.95)/3.231 follows the standard normal distribution.
a)What is the probability that more than half the people in the sample watched or recorded Friends?
half of 50 is 25
P(X>25) is required.
The Z value for X = 25 is (25-14.95)/3.231 = 3.141
P(Z>3.141) = 1-P(Z<3.141) = 1-0.999159=0.000841
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b)What is the probability that less than 25% of the people in the sample watched or recorded Friends?
25% of 50 is 12.5
P(X<12.5) is required.
The Z value for X = 12.5 is (12.5-14.95)/3.231 = -0.727
P(Z<-0.727) = 0.234
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c)If a random sample of size 500 is taken, how does this change your answers to (a) and (b)?
mean of X is np = 500*0.297 =149.5
Standard deviation of x is sqrt(500*0.297*(1-0.297)) = 10.217
part(a) becomes
half of 50 is 25
P(X>25) is required.
The Z value for X = 25 is (25-149.5)/10.217 = -12.087
P(Z>-12.087) = 1-P(Z<-12.087) = 1-0 = 1
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Part(b) becomes
25% of 50 is 12.5
P(X<12.5) is required.
The Z value for X = 12.5 is (12.5-149.5)/10.217 = -13.311
P(Z<-13.311) = 0