To solve this problem, let's analyze each part of the question step-by-step.

Given:

• Sample size $n = 19$
• Population mean $\mu = 5$
• Population standard deviation $\sigma = 4$

Since the sample size is 19, we need to consider if it's appropriate to use the normal distribution for the sampling distribution of the sample mean $\bar{x}$. Given that the population standard deviation is known and the sample size is reasonably large (typically $n \geq 30$ is standard, but with known variance, the normal approximation can work for slightly smaller samples), it might be appropriate here.

1. Determine the standard error of the mean $\bar{x}$: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{19}} \approx 0.918$

Part (a)

Is it appropriate to use the normal distribution to find probabilities for $\bar{x}$?

Yes, it is appropriate to use the normal distribution to find probabilities for $\bar{x}$ because the Central Limit Theorem allows the sampling distribution of the sample mean to be approximately normal when the sample size is moderately large (often $n \geq 30$ is typical, but smaller sizes can work, especially with known population variance).

Part (b)

Find the probability that $\bar{x}$ will be greater than 4.

To find $P(\bar{x} > 4)$:

1. Convert 4 to a z-score using the formula: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{4 - 5}{0.918} \approx -1.09$
2. Use the standard normal distribution to find $P(z > -1.09)$.

From standard normal tables, $P(z > -1.09) \approx 0.8621$.

So, the probability that $\bar{x}$ will be greater than 4 is approximately 0.8621.

Part (c)

Find the 30th percentile of $\bar{x}$.

The 30th percentile corresponds to finding the value $k$ such that $P(\bar{x} < k) = 0.30$.

1. Find the z-score corresponding to the 30th percentile, which is approximately $z = -0.524$.
2. Convert this z-score back to the original scale using: $k = \mu + z \cdot \sigma_{\bar{x}} = 5 + (-0.524)(0.918) \approx 4.52$

So, the 30th percentile of $\bar{x}$ is approximately 4.52.

Summary of Answers:

1. (a) It is appropriate to use the normal distribution.
2. (b) The probability that $\bar{x}$ will be greater than 4 is 0.8621.
3. (c) The 30th percentile of $\bar{x}$ is 4.52.

Would you like further details on any of these steps?

Here are some related questions you might find helpful:

1. How does the Central Limit Theorem justify using the normal distribution here?
2. What if the population standard deviation was unknown? How would that change the approach?
3. Why is the sample mean distribution's standard error smaller than the population's standard deviation?
4. How would the answer change if the sample size were less than 19?
5. How would we find the 70th percentile instead of the 30th?

Tip: When using z-scores, remember that tables often provide the cumulative probability from the left. For probabilities greater than a value, you might need to subtract from 1.