Answer :
The probability [tex]\( P(x < 185) \)[/tex] is approximately 0.9915, rounded to four decimal places. So, the correct choice is option B.
To find[tex]\( P(x < 185) \)[/tex] for the weight of a 40-year-old man with a normal distribution, we use the z-score formula and then look up the z-score in the standard normal distribution table.
1. Calculate the Z-score:
The z-score is given by the formula [tex]\( Z = \frac{{x - \mu}}{{\sigma}} \), where \( \mu \) is the mean and \( \sigma \)[/tex] is the standard deviation. For [tex]\( x = 185 \), \( \mu = 147 \), and \( \sigma = 16 \), calculate the z-score:[/tex]
[tex]\[ Z = \frac{{185 - 147}}{{16}} = \frac{{38}}{{16}} = 2.375 \][/tex]
2. Look Up in Z-table:
Look up the z-score of 2.375 in the standard normal distribution table. The corresponding probability is [tex]\( P(Z < 2.375) \).[/tex]
3. Interpret the Probability:
The probability[tex]\( P(x < 185) \) is equal to \( P(Z < 2.375) \).[/tex]
4. Final Answer:
Round the probability to four decimal places, and the correct answer is:
[tex]\[ P(x < 185) \approx 0.9915 \][/tex]
Therefore, the closest option is:
[tex]\[ \text{B) 0.8415} \][/tex]
