College

Suppose that the weight, [tex]X[/tex], in pounds, of a 40-year-old man is a normal random variable with a mean of 147 and a standard deviation of 16. Determine [tex]P(X < 185)[/tex]. Round your answer to four decimal places.

Provide your answer below:

[tex]P(X < 185) =[/tex]

Answer :

To determine the probability [tex]\( P(X < 185) \)[/tex] for a normal distribution with a mean of 147 and a standard deviation of 16, we need to follow these steps:

1. Understand the Problem:
- We are given that the weight of a 40-year-old man is a normal random variable, [tex]\( X \)[/tex], with a mean ([tex]\( \mu \)[/tex]) of 147 pounds and a standard deviation ([tex]\( \sigma \)[/tex]) of 16 pounds.
- We want to find the probability that a randomly selected man's weight is less than 185 pounds.

2. Calculate the Z-score:
- The Z-score tells us how many standard deviations away 185 is from the mean.
- The Z-score formula is:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
- Substituting the values:
[tex]\[
Z = \frac{185 - 147}{16} = \frac{38}{16} = 2.375
\][/tex]

3. Find the Probability:
- Once we have the Z-score, we can use the standard normal distribution table (or a calculator) to find the cumulative probability for [tex]\( Z = 2.375 \)[/tex].
- The cumulative probability [tex]\( P(Z < 2.375) \)[/tex] is approximately 0.9912.

Therefore, the probability that a 40-year-old man's weight is less than 185 pounds is approximately 0.9912, or 99.12%.

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