Answer:
a. 48.80%
b. 46.02%
c. 57.93% probability of the sample mean weight being above the weight limit, which is a high probability, meaning that the elevator does not appear to have the correct weight limit
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
![\mu = 175, \sigma = 31](https://tex.z-dn.net/?f=%5Cmu%20%3D%20175%2C%20%5Csigma%20%3D%2031)
a. find the probability that if a person is randomly selected, his weight will be greater than 176 pounds.
This is 1 subtracted by the pvalue of Z when X = 176. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{176 - 175}{31}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B176%20-%20175%7D%7B31%7D)
![Z = 0.03](https://tex.z-dn.net/?f=Z%20%3D%200.03)
has a pvalue of 0.5120
1 - 0.5120 = 0.4880
48.80% probability that if a person is randomly selected, his weight will be greater than 176 pounds.
b. Find the probability that 10 randomly selected people will have a neam that is greater than 176 pounds.
Now ![n = 10, s = \frac{31}{\sqrt{10}} = 9.8](https://tex.z-dn.net/?f=n%20%3D%2010%2C%20s%20%3D%20%5Cfrac%7B31%7D%7B%5Csqrt%7B10%7D%7D%20%3D%209.8)
This is 1 subtracted by the pvalue of Z when X = 176. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Thorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{176 - 175}{9.8}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B176%20-%20175%7D%7B9.8%7D)
![Z = 0.1](https://tex.z-dn.net/?f=Z%20%3D%200.1)
has a pvalue of 0.5398
1 - 0.5398 = 0.4602
46.02% probability that 10 randomly selected people will have a neam that is greater than 176 pounds.
c. Does the elevator appear to have the correct weight limit?
The weight limit is 173 pounds in a sample of 10.
The probability that the mean weight is larger than this is 1 subtracted by the pvalue of Z when X = 173. So
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{173 - 175}{9.8}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B173%20-%20175%7D%7B9.8%7D)
![Z = -0.2](https://tex.z-dn.net/?f=Z%20%3D%20-0.2)
has a pvalue of 0.4207
1 - 0.4207 = 0.5793
57.93% probability of the sample mean weight being above the weight limit, which is a high probability, meaning that the elevator does not appear to have the correct weight limit