TV sets: According to the Nielsen Company, the mean number of TV sets in a U.S. household was 2.24. Assume the standard deviation is 1.2 . A sample of 95 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

To find the probability that the sample mean number of TV sets is greater than 2, we first need to calculate the standard error (SE) of the sample mean. The standard error can be calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{1.2}{\sqrt{95}} \approx 0.1234 \] Next, we need to determine the z-score for a sample mean of 2. The z-score can be calculated using the formula: \[ z = \frac{\bar{x} - \mu}{SE} = \frac{2 - 2.24}{0.1234} \approx -1.95 \] Now, we can look up the z-score of -1.95 in the standard normal distribution table, or use a calculator. The area to the left of z = -1.95 is approximately 0.0256. Therefore, the area to the right (which is the probability that the sample mean is greater than 2) is: \[ P(X > 2) = 1 - 0.0256 \approx 0.9744 \] So, the probability that the sample mean number of TV sets is greater than 2 is approximately **0.9744** (rounded to four decimal places).