To find the probability that a randomly selected male's systolic blood pressure (denoted as $X$) is between 116 and 120, we can use the Z-score formula for a normal distribution.

The formula for the Z-score is:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the value of interest,
• $\mu = 135$ is the mean,
• $\sigma = 8$ is the standard deviation.

Step 1: Calculate the Z-scores for 116 and 120.

1. For $X = 116$:

$Z_1 = \frac{116 - 135}{8} = \frac{-19}{8} = -2.375$

2. For $X = 120$:

$Z_2 = \frac{120 - 135}{8} = \frac{-15}{8} = -1.875$

Step 2: Use the Z-scores to find the corresponding probabilities from the standard normal distribution table.

• For $Z_1 = -2.375$, the cumulative probability (from standard normal tables) is approximately 0.0087.
• For $Z_2 = -1.875$, the cumulative probability is approximately 0.0304.

Step 3: Find the probability between 116 and 120.

The probability is the difference between the cumulative probabilities for $Z_2$ and $Z_1$:

$P(116 < X < 120) = P(Z_2) - P(Z_1) = 0.0304 - 0.0087 = 0.0217$

Thus, the probability that a randomly selected male's systolic blood pressure is between 116 and 120 is 0.022, to the nearest thousandth.

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Would you like more details on this? Here are 5 related questions for practice:

1. What is the probability that a male's systolic blood pressure will be less than 120?
2. What is the probability that a male's systolic blood pressure will be greater than 140?
3. How would the probability change if the mean systolic blood pressure were 130 instead of 135?
4. What systolic blood pressure value corresponds to the 90th percentile of this distribution?
5. What is the probability that a male's systolic blood pressure will be between 120 and 140?

Tip: When using Z-scores, always check if your values align with the standard normal table, as small errors in reading tables can lead to incorrect probabilities.