Questions: An elevator has a placard stating that the maximum capacity is 3500 lb-25 passengers. So, 25 adult male passengers can have a mean weight of up to 3500 / 25=140 pounds. Assume that weights of males are normally distributed with a mean of 185 lb and a standard deviation of 40 lb. a. Find the probability that 1 randomly selected adult male has a weight greater than 140 lb. b. Find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb. c. What do you conclude about the safety of this elevator? a. The probability that 1 randomly selected adult male has a weight greater than 140 lb is 0.8697. (Round to four decimal places as needed.) b. The probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb is (Round to four decimal places as needed.)

An elevator has a placard stating that the maximum capacity is 3500 lb-25 passengers. So, 25 adult male passengers can have a mean weight of up to 3500 / 25=140 pounds. Assume that weights of males are normally distributed with a mean of 185 lb and a standard deviation of 40 lb.
a. Find the probability that 1 randomly selected adult male has a weight greater than 140 lb.
b. Find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb.
c. What do you conclude about the safety of this elevator?
a. The probability that 1 randomly selected adult male has a weight greater than 140 lb is 0.8697.
(Round to four decimal places as needed.)
b. The probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb is (Round to four decimal places as needed.)
Transcript text: An elevator has a placard stating that the maximum capacity is $3500 \mathrm{lb}-25$ passengers. So, 25 adult male passengers can have a mean weight of up to $3500 / 25=140$ pounds. Assume that weights of males are normally distributed with a mean of 185 lb and a standard deviation of 40 lb . a. Find the probability that 1 randomly selected adult male has a weight greater than 140 lb . b. Find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb . c. What do you conclude about the safety of this elevator? a. The probability that 1 randomly selected adult male has a weight greater than 140 lb is 0.8697 . (Round to four decimal places as needed.) b. The probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb is (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Z-score for 140 lb

To find the probability that a randomly selected adult male weighs more than 140 lb, we first calculate the Z-score using the formula:

z=Xμσ=14018540=1.125 z = \frac{X - \mu}{\sigma} = \frac{140 - 185}{40} = -1.125

Step 2: Calculate the Probability for One Adult Male

Next, we find the probability that a randomly selected male weighs more than 140 lb. This is given by:

P(X>140)=1P(X140)=1Φ(1.125)0.1303 P(X > 140) = 1 - P(X \leq 140) = 1 - \Phi(-1.125) \approx 0.1303

Thus, the probability that 1 randomly selected male weighs more than 140 lb is approximately 0.13030.1303.

Step 3: Calculate the Probability for a Sample of 25 Adult Males

Now, we calculate the probability that the mean weight of a sample of 25 randomly selected adult males is greater than 140 lb. The standard error of the mean is given by:

σXˉ=σn=4025=8 \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{40}{\sqrt{25}} = 8

We then calculate the Z-score for the sample mean:

z=1401858=5.625 z = \frac{140 - 185}{8} = -5.625

The probability is then calculated as:

P(Xˉ>140)=1P(Xˉ140)=1Φ(5.625)1.0 P(\bar{X} > 140) = 1 - P(\bar{X} \leq 140) = 1 - \Phi(-5.625) \approx 1.0

Thus, the probability that a sample of 25 males has a mean weight greater than 140 lb is approximately 1.01.0.

Step 4: Conclusion About the Safety of the Elevator

Given that the probability that a sample of 25 adult males has a mean weight greater than 140 lb is 1.01.0, we conclude that the elevator is likely safe for 25 adult males.

Final Answer

  • The probability that 1 randomly selected male weighs more than 140 lb is approximately 0.13030.1303.
  • The probability that a sample of 25 males has a mean weight greater than 140 lb is approximately 1.01.0.
  • The elevator is likely safe for 25 adult males.

P(X>140)0.1303,P(Xˉ>140)1.0 \boxed{P(X > 140) \approx 0.1303, \quad P(\bar{X} > 140) \approx 1.0}

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