Question

7. Number of Televisions Based on data obtained from AC Nielsen, the mean number of televisions in a household in the United States is 2.24. Assume that the population standard deviation number of television sets in the United States is 1.38. (a) Do you believe the shape of the distribution of number of television sets follows a normal distribution? Why or why not? (b) A random sample of 40 households results in a total of 102 television sets, What is the mean number of televisions in these 40 households? (c) What is the probability of obtaining the sample mean obtained in part (b) if the population mean is 2.24? Does the statistic from part (b) contradict the results reported by AC Nielsen?

Answer 1

(a)

The shape of the Normal distribution is symmetrical bell shape with 99.7% of the data falls within 3 standard deviations from the mean. This means, a value less than 3 standard deviations from the mean is possible with a non-zero probability.

$\mu - 3 \sigma$ = 2.14 - 3 * 1.38 = -2

So, number of television is -2 has a non-zero probability, which is not possible as the number of televisions in a household cannot be a negative number.

This, the shape of the distribution of televisions in a household does not follow normal distribution.

(b)

Mean number of televisions = 102 / 40 = 2.55

(c)

By Central limit theorem, the sample mean will follow normal distribution with mean $\mu$ = 2.24 and = 0.218

1.38/V40 σ

Probability of obtaining the sample mean of 2.55 = P($\bar{X}$ = 2.55)

= P(Z = (2.55 - 2.24) / 0.218)

= P(Z = 1.42)

= 0.1456

Since the obtained probability is not low (not below 0.05), the statistic in part (b) is possible if the population mean is 2.24 and thus the statistic does not contradict the results reported by AC Nielsen.