Questions: TV sets: According to the Nielsen Company, the mean number of TV sets in a U.S. household in 2013 was 2.24. Assume the standard deviation is 1.1. A sample of 80 households is drawn. Part: 0 / 5 Part 1 of 5 (a) What is the probability that the sample mean number of TV sets is greater than 2? Round your answer to at least four decimal places. The probability that the sample mean number of TV sets is greater than 2 is

Solution Steps

The standard error of the mean is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

where \(\sigma = 1.1\) is the population standard deviation and \(n = 80\) is the sample size. Thus:

\[ SE = \frac{1.1}{\sqrt{80}} \approx 0.1230 \]

The z-score is calculated using the formula:

\[ z = \frac{\bar{x} - \mu}{SE} \]

where \(\bar{x} = 2\) is the sample mean value and \(\mu = 2.24\) is the population mean. Thus:

\[ z = \frac{2 - 2.24}{0.1230} \approx -1.9515 \]

The probability that the sample mean is greater than 2 is given by:

\[ P(\bar{x} > 2) = 1 - P(Z \leq -1.9515) \]

Using the standard normal distribution, we find:

\[ P(Z \leq -1.9515) \approx 0.0255 \]

Thus:

\[ P(\bar{x} > 2) = 1 - 0.0255 = 0.9745 \]

Final Answer