Question

9.14 (a) A random sample of size 144 is taken from the local population of grade-school children. Each child estimates the number of hours per week spent watching TV. At this point, what can be said about the sampling distribution?

(b) Assume that a standard deviation, σ, of 8 hours describes the TV estimates for the local population of schoolchildren. At this point, what can be said about the sampling distribution?

(c) Assume that a mean, μ, of 21 hours describes the TV estimates for the local population of schoolchildren. Now what can be said about the sampling distribution?

(d) Roughly speaking, the sample means in the sampling distribution should deviate, on average, about ___ hours from the mean of the sampling distribution and from the mean of the population.

(e) About 95 percent of the sample means in this sampling distribution should be between ___ hours and ___ hours.

Answer 1

The Central limit theorem states that if random samples of n observations are drawn from a population with finite μ and standard deviation σ, then, when n is large, the sampling distribution of the sample mean is approximately normally distributed with mean μ and standard deviation σ/√n.

(a) A random sample of size 144 is taken from the local population of grade-school children. Each child estimates the number of hours per week spent watching TV. At this point, what can be said about the sampling distribution?

At this point, according to central limit theorem, stated above, we can say that 144 children as a random sample is enough to produce a normal curve sampling distribution.

(b) Assume that a standard deviation, σ, of 8 hours describes the TV esti-mates for the local population of schoolchildren. At this point, what can be said about the sampling distribution?

At this point, according to central limit theorem, stated above, we can say that the sampling distribution of the sample mean is approximately normally distributed with standard deviation σ/√n=8/√144 = 0.67.

(c) Assume that a mean, μ, of 21 hours describes the TV estimates for the local population of schoolchildren. Now what can be said about the sam-pling distribution?

At this point, according to central limit theorem, stated above, we can say that the sampling distribution of the sample mean is approximately normally distributed with mean μ=21 and standard deviation σ/√n=8/√144 = 0.67.

(d) Roughly speaking, the sample means in the sampling distribution should deviate, on average, about 0.67 hours from the mean of the sampling distribution and from the mean of the population.

(e) According to Empirical rule, about 955 of the observations lie within 2 standard deviations from mean.

About 95 percent of the sample means in this sampling distribution should be between 19.66 hours and 22.34 hours.