 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 

1.2

. A sample of 

95

 households is drawn. 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.

To find the probability that the sample mean number of TV sets in a U.S. household is greater than 2, we can use the Central Limit Theorem. Here's the step-by-step process:

1. **State the known parameters:**
   - Population mean (\(\mu\)) = 2.24
   - Population standard deviation (\(\sigma\)) = 1.2
   - Sample size (\(n\)) = 95
   - Sample mean (\(\bar{x}\)) = 2

2. **Calculate the standard error of the mean (SE):**
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{1.2}{\sqrt{95}} \approx 0.1231
   \]

3. **Find the Z-score for the sample mean:**
   \[
   Z = \frac{\bar{x} - \mu}{SE} = \frac{2 - 2.24}{0.1231} \approx -1.95
   \]

4. **Determine the probability corresponding to the Z-score:**
   - Using the standard normal distribution table, we find the cumulative probability for \(Z = -1.95\).
   - The cumulative probability \(P(Z < -1.95)\) is approximately 0.0256.

5. **Calculate the probability of the sample mean being greater than 2:**
   \[
   P(\bar{x} > 2) = 1 - P(Z < -1.95) = 1 - 0.0256 = 0.9744
   \]

Therefore, the probability that the sample mean number of TV sets is greater than 2 is approximately **0.9744**.

Would you like more details or have any questions?

Here are some questions you might want to ask:
1. What is the Central Limit Theorem and how is it used in statistics?
2. How do you interpret a Z-score in the context of probability?
3. What is the importance of the standard error of the mean in sampling distributions?
4. How do you use a standard normal distribution table to find probabilities?
5. Can you explain the difference between population parameters and sample statistics?

**Tip:** When dealing with probabilities and sample means, remember to use the standard error of the mean, which adjusts for sample size, ensuring accurate probability calculations based on the central limit theorem.

Calculate the probability that the sample mean number of TV sets in a US household in 2013 is greater than 2. Use the Central Limit Theorem, SE, and Z-score calculation. Population mean: 2.24, standard deviation: 1.2, sample size: 95 households.

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