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Suppose that the weight, X, in pounds, of a 40-year-old man is a normal random variable with a mean of 157 pounds and a standard deviation of 20 pounds.

Determine P(X < 165). Round your answer to four decimal places. Provide your answer below:

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:

  1. 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]

  2. 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.

  3. 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%.

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