Answer :
To determine the probability that a 40-year-old man weighs less than 165 pounds, given that the weight [tex]X[/tex] is a normal random variable with a mean [tex]\mu = 157[/tex] pounds and a standard deviation [tex]\sigma = 20[/tex] pounds, we need to calculate [tex]P(X < 165)[/tex].
Here's a step-by-step approach:
Identify the Z-score:
- The Z-score is a way of standardizing a value from a normal distribution. It tells us how many standard deviations a data point is from the mean.
- The formula for the Z-score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex] - In this case, [tex]X = 165[/tex], [tex]\mu = 157[/tex], and [tex]\sigma = 20[/tex].
- Substitute the values into the formula:
[tex]Z = \frac{165 - 157}{20} = \frac{8}{20} = 0.4[/tex]
Use the Standard Normal Distribution Table:
- A Z-score of 0.4 corresponds to a probability, which gives us the probability that a standard normal random variable is less than 0.4.
- Look up the Z-score of 0.4 in the standard normal distribution table or use a calculator to find [tex]P(Z < 0.4)[/tex].
- The probability associated with a Z-score of 0.4 is approximately 0.6554.
Conclusion:
- Therefore, [tex]P(X < 165) \approx 0.6554[/tex].
So, the probability that the weight of a 40-year-old man is less than 165 pounds is approximately 0.6554 or 65.54%.
