A random variable follows the normal probability distribution with a mean of 117 and a standard deviation of 23. Complete parts (a) through (d) below.
Click here to view page 1 of the standard normal probability table. Click here to view page 2 of the standard normal probability table.
a) What is the probability that a randomly selected value from this population is between 103 and 133?
(Round to four decimal places as needed.)
b) What is the probability that a randomly selected value from this population is between 124 and 161?
(Round to four decimal places as needed.)
c) What is the probability that a randomly selected value from this population is between 69 and 85?
(Round to four decimal places as needed.)
d) What is the probability that a randomly selected value from this population is between 109 and 165?
(Round to four decimal places as needed.)

To solve these problems, we first need to standardize the values using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

where:
- \( X \) is the given value,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation, and
- \( Z \) is the standardized value.

Then, we'll use the standard normal probability table to find the probabilities.

a) For \( X = 103 \) and \( X = 133 \):
\[ Z_1 = \frac{103 - 117}{23} = -0.609 \]
\[ Z_2 = \frac{133 - 117}{23} = 0.696 \]

Using the standard normal probability table, the probability between \( Z_1 \) and \( Z_2 \) is approximately 0.5279.

b) For \( X = 124 \) and \( X = 161 \):
\[ Z_1 = \frac{124 - 117}{23} = 0.304 \]
\[ Z_2 = \frac{161 - 117}{23} = 1.913 \]

The probability between \( Z_1 \) and \( Z_2 \) is approximately 0.4664.

c) For \( X = 69 \) and \( X = 85 \):
\[ Z_1 = \frac{69 - 117}{23} = -2.087 \]
\[ Z_2 = \frac{85 - 117}{23} = -1.391 \]

The probability between \( Z_1 \) and \( Z_2 \) is approximately 0.0822.

d) For \( X = 109 \) and \( X = 165 \):
\[ Z_1 = \frac{109 - 117}{23} = -0.348 \]
\[ Z_2 = \frac{165 - 117}{23} = 2.087 \]

The probability between \( Z_1 \) and \( Z_2 \) is approximately 0.8664.