The problem you're dealing with asks to find the probability that the average time children watch TV is more than 4.3 hours, based on a sample of 64 students.

Here’s how to solve this problem step by step:

Step 1: Identify the given information

• Population mean ($\mu$) = 4 hours
• Population standard deviation ($\sigma$) = 1.6 hours
• Sample size ($n$) = 64
• Sample mean = 4.3 hours

Step 2: Apply the Central Limit Theorem

Since the sample size is large (64), we can apply the Central Limit Theorem. The sampling distribution of the sample mean will be approximately normally distributed with:

• Mean of the sample distribution = Population mean = 4 hours
• Standard deviation of the sample mean (standard error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{1.6}{\sqrt{64}} = \frac{1.6}{8} = 0.2$

Step 3: Calculate the z-score

The z-score formula is: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$ Substitute the values: $z = \frac{4.3 - 4}{0.2} = \frac{0.3}{0.2} = 1.5$

Step 4: Find the probability

Now, find the probability that corresponds to a z-score of 1.5. You can use a standard normal distribution table, or a calculator, which gives: $P(Z > 1.5) = 1 - P(Z \leq 1.5) = 1 - 0.9332 = 0.0668$

Conclusion:

The probability that the average time these children watch TV is more than 4.3 hours is 0.0668. Therefore, the correct answer is the second option: 0.0668.

Would you like more details on any part of the solution?

Here are some questions that can help expand your understanding:

1. How does the sample size affect the standard error of the mean?
2. What would the z-score be if the average time was 4.5 hours instead of 4.3?
3. How would the probability change if the standard deviation was 2 hours instead of 1.6 hours?
4. What happens if the sample size is much smaller, say 10, instead of 64? Can we still apply the Central Limit Theorem?
5. How would you find the probability for a two-tailed test, i.e., if you were interested in averages both less than 3.7 hours and more than 4.3 hours?

Tip: When dealing with probabilities and normal distributions, using a z-table or software tools like a scientific calculator or Excel can make your calculations faster and more accurate.