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Answer :
Answer:
0.7019; no, it does not appear to be safe.
Step-by-step explanation:
We want to find P(X > 185). However, in a z table, we are given the area under the curve to the left of the value; this means we want to find
1 - P(X ≤ 185)
The z score for the mean of a sample is given by
[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]
For this situation, the mean, μ, is 189 and the standard deviation, σ, is 39. Our sample size, n, is 27. This gives us
z = (185-189)/(39÷√27) = -4/(39÷5.1962) = -4/7.5055 = -0.53
Using a z table, we see that the area under the curve to the left of this value is 0.2981. This means our probability is
1-0.2981 = 0.7019.
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Final answer:
The question calculates the statistical probability of an event with normal distribution, specifically if the average weight of 27 adult males in an elevator exceeds 185 lbs. The Central Limit Theorem is applied, showing the mean equals the original mean, and the new standard deviation is calculated as the original divided by the square root of the sample size. Once the Z-score is found, standard normal distribution tables identify the probability of average weight exceeding 185 lbs. If the probability is small, the elevator is typically deemed safe.
Explanation:
This question is about understanding the statistical probability of an event under the conditions of normal distribution, in this case, the weight of adult male passengers in an elevator.
To answer this question, we need to calculate the probability of the average weight of 27 males exceeding 185 lbs using the Central Limit Theorem. The normal distribution has a mean of 189 lb and a standard deviation of 39 lb. If a sample of 27 adult males is taken, by the Central Limit Theorem, this sample will roughly follow a normal distribution with mean equal to the original mean and standard deviation equals to the original standard deviation divided by sqrt(n), where n is the number of observations.
Therefore, the new standard deviation will be 39/sqrt(27), and you can find the z-score of 185 lbs within this new distribution. The Z-score formula is Z = (X-μ)/σ, where X is the observation (185 in this case), μ is the mean and σ is the standard deviation. Using standard normal distribution tables, you will find the probability of average weight exceeding 185 lbs, and if this probability is sufficiently small, the elevator can be considered safe.
Learn more about Central Limit Theorem
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