Question

Let x be a random variable that represents the weights in pounds (lb) of adult males who are in their 40s. x has a normal distribution with mean =165 lbs and standard deviation =9 lbs. An adult male in his 40s who weighs above 170.85 is considered overweight. a) What is the probability that one adult male in his 40s is overweight? (Round your answer to four decimal places.) b) What is the probability that 10 adult males in their 40s have a mean weight over 170.85 lbs? (Round standard deviation x to two decimal places and your answer to four decimal places.) c) What is the probability that 32 adult males in their 40s have a mean weight over 170.85 lbs? (Round standard deviation x to two decimal places and your answer to four decimal places. This sampling distribution problem requires the sample size to be greater than or equal to 30.)

Answer 1

For a) the probability of one adult male being overweight is approximately 0.7422. For b) the probability that 10 adult males have a mean weight over 170.85 lbs is approximately 0.9742. For c) the probability that 32 adult males have a mean weight over 170.85 lbs is approximately 0.9996.

Step-by-step explanation:

a) To find the probability that one adult male in his 40s is overweight, we need to calculate the z-score for the weight of 170.85 lbs using the formula z = (x - μ) / σ. Substituting the given values, we get z = (170.85 - 165) / 9 = 0.65. Using a standard normal distribution table or calculator, we can find the probability corresponding to this z-score, which is approximately 0.7422.

b) To find the probability that 10 adult males in their 40s have a mean weight over 170.85 lbs, we need to find the standard deviation of the sample mean, which is given by σx = σ / sqrt(n), where n is the sample size. Substituting the values, we get σx = 9 / sqrt(10) = 2.846. We can then find the z-score for the weight of 170.85 lbs using the formula z = (x - μx) / σx. Substituting the values, we get z = (170.85 - 165) / 2.846 = 1.9651. Using a standard normal distribution table or calculator, we can find the probability corresponding to this z-score, which is approximately 0.9742.

c) To find the probability that 32 adult males in their 40s have a mean weight over 170.85 lbs, we need to find the standard deviation of the sample mean, which is given by σx = σ / sqrt(n), where n is the sample size. Substituting the values, we get σx = 9 / sqrt(32) = 1.5912. We can then find the z-score for the weight of 170.85 lbs using the formula z = (x - μx) / σx. Substituting the values, we get z = (170.85 - 165) / 1.5912 = 3.4859. Using a standard normal distribution table or calculator, we can find the probability corresponding to this z-score, which is approximately 0.9996.