Answer :
Final answer:
The probability P(120≤x≤153) for the normal distribution with a mean of 147 and a standard deviation of 16 is found by calculating the Z-scores for 120 and 153 and finding the cumulative probabilities for these Z-scores.
Explanation:
The task is to determine P(120≤x≤153) for the weight x of a 40-year-old man where the weight follows a normal distribution with a mean of 147 pounds and a standard deviation of 16 pounds.
To solve this, we will calculate the Z-scores for both 120 and 153 and then use the standard normal distribution table or a calculator to find the probabilities for these Z-scores. The area between these Z-scores will give us the required probability.
First, the Z-score for x = 120 is calculated as
Z = (X - mean) / standard deviation = (120 - 147) / 16. Similarly, for x = 153,
Z = (153 - 147) / 16. Then we look up these Z-scores in the standard normal distribution table or use a calculator feature that provides the cumulative probability for a given Z-score.
The probability P(120≤x≤153) is the difference between the probabilities obtained for the Z-scores of 153 and 120.
