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.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.

Either-or probabilities are found by adding the individual probabilities.

For last Q, Start with table (.42) and insert Z score into above equation.

PsyDAG, right on thanks

To compute the probabilities and percentiles for a normal distribution, you can use the standard normal distribution table or a calculator with a built-in function to calculate normal probabilities. However, in this case, let me explain how to do it using the Z-score formula.

1. Compute P(X < 39):
To find the probability that X is less than 39, you need to calculate the Z-score first. The Z-score formula is given by: Z = (X - μ) / σ, where X is the given value, μ is the mean, and σ is the standard deviation.
In this case, X = 39, μ = 37, and σ = 5. Plug these values into the formula: Z = (39 - 37) / 5 = 0.4.
Now, you can use the Z-score table or calculator to find the probability associated with Z = 0.4.
P(X < 39) = P(Z < 0.4).

2. Compute P(X < 29 or X > 41):
To find the probability that X is less than 29 or greater than 41, you need to calculate the Z-scores for both values. Following the same Z-score formula, calculate Z1 for X = 29 and Z2 for X = 41.
Z1 = (29 - 37) / 5 = -1.6
Z2 = (41 - 37) / 5 = 0.8
Now, find the probabilities associated with Z1 and Z2 separately:
P(X < 29) = P(Z < -1.6)
P(X > 41) = P(Z > 0.8)
Finally, compute the union of these probabilities:
P(X < 29 or X > 41) = P(X < 29) + P(X > 41) - P(X < 29 and X > 41).

3. Compute P(X > 30):
To find the probability that X is greater than 30, calculate the Z-score using the formula:
Z = (X - μ) / σ.
In this case, X = 30, μ = 37, and σ = 5. Plug in these values: Z = (30 - 37) / 5 = -1.4.
Now, use the Z-score table or calculator to find the probability associated with Z = -1.4.
P(X > 30) = 1 - P(X < 30).

4. Find the 42nd percentile:
The percentile represents a specific value below which a given percentage of the data falls. In this case, you need to find the value below which 42% of the data falls.
First, find the Z-score associated with the 42nd percentile using the standard normal distribution table or calculator: Z = Z(42%) (rounded to 2 decimal places).
Then, use the Z-score formula to find the value: X = (Z * σ) + μ.

Note: If you prefer a quicker way to calculate these probabilities and percentiles, you can use a calculator or software that provides a normal distribution function.