Questions: HW 73 : Sampling Distributions and CLT Question 4 of 5 (1 point) Question Attempt: 1 of Unlimited Elevator ride: Engineers are designing a large elevator that will accommodate 44 people. The maximum weight the elevator can hold safely is 8228 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have mean 186 pounds and standard deviation 60 pounds, and the weights of adult U.S. women have mean 157 pounds and standard deviation 69 pounds. Use the TI-84 Plus calculator. Part 1 of 3 (a) If 44 people are on the elevator, and their total weight is 8228 pounds, what is their average weight? The average weight is 187 pounds. Part 2 of 3 (b) If a random sample of 44 adult men ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places. The probability that the maximum safe weight will be exceeded is 0.456 Part 3 of 3 (c) If a random sample of 44 adult women ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places.

HW 73 : Sampling Distributions and CLT Question 4 of 5 (1 point) Question Attempt: 1 of Unlimited Elevator ride: Engineers are designing a large elevator that will accommodate 44 people. The maximum weight the elevator can hold safely is 8228 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have mean 186 pounds and standard deviation 60 pounds, and the weights of adult U.S. women have mean 157 pounds and standard deviation 69 pounds. Use the TI-84 Plus calculator. Part 1 of 3 (a) If 44 people are on the elevator, and their total weight is 8228 pounds, what is their average weight? The average weight is 187 pounds. Part 2 of 3 (b) If a random sample of 44 adult men ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places. The probability that the maximum safe weight will be exceeded is 0.456 Part 3 of 3 (c) If a random sample of 44 adult women ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places.
Transcript text: HW 73 : Sampling Distributions and CLT Question 4 of 5 (1 point) | Question Attempt: 1 of Unlimited Elevator ride: Engineers are designing a large elevator that will accommodate 44 people. The maximum weight the elevator can hold safely is 8228 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have mean 186 pounds and standard deviation 60 pounds, and the weights of adult U.S. women have mean 157 pounds and standard deviation 69 pounds. Use the TI-84 Plus calculator. Part 1 of 3 (a) If 44 people are on the elevator, and their total weight is 8228 pounds, what is their average weight? The average weight is 187 pounds. Part 2 of 3 (b) If a random sample of 44 adult men ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places. The probability that the maximum safe weight will be exceeded is 0.456 Part 3 of 3 (c) If a random sample of 44 adult women ride the elevator, what is the probability that the maximum safe weight will be exceeded? Round the answer to at least four decimal places.
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Solution

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Solution Steps

Step 1: Average Weight Calculation

To find the average weight of 44 people on the elevator, we divide the maximum safe weight by the number of people:

\[ \text{Average Weight} = \frac{8228 \text{ pounds}}{44} = 187.0 \text{ pounds} \]

Step 2: Probability for Men

We need to calculate the probability that the total weight of a random sample of 44 adult men exceeds the maximum safe weight of 8228 pounds. The mean weight of adult men is \( \mu = 186 \) pounds, and the standard deviation is \( \sigma = 60 \) pounds.

Using the Central Limit Theorem, we find the standard error (SE):

\[ \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{60}{\sqrt{44}} \approx 9.036 \]

Next, we calculate the Z-score for the average weight that exceeds the maximum safe weight:

\[ Z = \frac{\text{Observed Mean} - \mu}{\text{SE}} = \frac{187.0 - 186}{9.036} \approx 0.1106 \]

Using the Z-score, we find the probability:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(0.1106) \approx 0.456 \]

Step 3: Probability for Women

Similarly, we calculate the probability for a random sample of 44 adult women. The mean weight of adult women is \( \mu = 157 \) pounds, and the standard deviation is \( \sigma = 69 \) pounds.

Calculating the standard error (SE):

\[ \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{69}{\sqrt{44}} \approx 10.409 \]

Next, we calculate the Z-score for the average weight that exceeds the maximum safe weight:

\[ Z = \frac{187.0 - 157}{10.409} \approx 2.884 \]

Using the Z-score, we find the probability:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(2.884) \approx 0.002 \]

Final Answer

  • The average weight is \( \boxed{187.0} \) pounds.
  • The probability that the maximum safe weight will be exceeded for men is \( \boxed{0.456} \).
  • The probability that the maximum safe weight will be exceeded for women is \( \boxed{0.002} \).
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