All deal with a random variable X that follows a normal probability distribution with mean equal to 37 and a standard deviation of 5.

Compute P(X < 39)

Compute P(X < 29 or X > 41)

Compute P(X > 30)

Find the 42nd percentile for this distribution.

To compute probabilities and percentiles for a normal distribution, we can use the standard normal distribution table or use statistical software. Here's how you can find the answers to the given questions using the standard normal distribution table:

1. To compute P(X < 39), we need to find the area under the standard normal distribution curve to the left of 39. First, we calculate the z-score corresponding to 39 using the formula:
z = (X - μ) / σ
where X is the given value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:
z = (39 - 37) / 5 = 0.4

Now, we look up the area to the left of 0.4 in the standard normal distribution table. The value in the table corresponds to the probability of a standard normal random variable being less than the given z-score. From the table, we find that the area to the left of 0.4 is approximately 0.6554. Therefore, P(X < 39) ≈ 0.6554.

2. To compute P(X < 29 or X > 41), we break down the problem into two parts:
P(X < 29) - This is the area under the curve to the left of 29. So, we calculate the z-score:
z1 = (29 - 37) / 5 = -1.6
Looking up the area to the left of -1.6 in the table gives us approximately 0.0548.

P(X > 41) - This is the area under the curve to the right of 41. We calculate the z-score:
z2 = (41 - 37) / 5 = 0.8
Looking up the area to the left of 0.8 in the table gives us approximately 0.7881.

Now, we need to calculate the combined probability using the complement rule:
P(X < 29 or X > 41) = 1 - P(X ≥ 29 and X ≤ 41)
= 1 - P(X < 29) - P(X > 41)
= 1 - 0.0548 - 0.7881
= 0.1571

Therefore, P(X < 29 or X > 41) ≈ 0.1571.

3. To compute P(X > 30), we first need to find the z-score:
z = (30 - 37) / 5 = -1.4
Looking up the area to the left of -1.4 in the table gives us approximately 0.0808.

However, we need to find the probability of X being greater than 30, so we subtract the area to the left of -1.4 from 1:
P(X > 30) = 1 - P(X < 30) = 1 - 0.0808 = 0.9192

Therefore, P(X > 30) ≈ 0.9192.

4. To find the 42nd percentile for this distribution, we need to find the corresponding z-score. The percentile is the value below which a given percentage of observations fall. In this case, we want to find the value below which 42% of observations fall.

We can use the inverse normal distribution table (also known as the z-table) to find the z-score corresponding to the 42nd percentile. Looking up the area to the left of 0.42 in the table gives us a z-score of approximately -0.2103.

Now, we can use the z-score formula to find the corresponding value for X:
X = μ + (z * σ)
X = 37 + (-0.2103 * 5)
X ≈ 36.895

Therefore, the 42nd percentile for this distribution is approximately 36.895.

Please note that the values in the table are approximations, so you may see slightly different results depending on the level of precision required.