Question

5. A random sample of 120 people is drawn from a population and asked how much time each week they spend watching television. The population has a mean of 187 minutes with a standard deviation of 32 minutes.

(a) Find the mean and standard error of the sampling distribution of ¯x.

(b) What is the probability that this sample will have a mean that is at least 192 minutes?

(c) What proportion of all samples of size 120 will have a mean that is at least 192 minutes? (d) What is the 90’th percentile

Answer 1

a) Since the sample size is large, by Central Limit Theorem the sampling distribution of mean approaches normal distribution (as n increases)

so mean= 187 and

SD=32/10.95445

{square root of 120 is 10.95445}

b) The probability that the sample mean is at least 192 minutes is given by P[X>=192]

=0.04348216

c) Proportion of samples of size 120 that will have a mean that is at least 192 minutes is 120*0.04348216 = 5.217859

ie, 5 percent

d) 90 th percentile is 201.0387