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Answer :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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:
[tex]\mu = 175, \sigma = 31[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{176 - 175}{31}[/tex]
[tex]Z = 0.03[/tex]
[tex]Z = 0.03[/tex] 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 [tex]n = 10, s = \frac{31}{\sqrt{10}} = 9.8[/tex]
This is 1 subtracted by the pvalue of Z when X = 176. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Thorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{176 - 175}{9.8}[/tex]
[tex]Z = 0.1[/tex]
[tex]Z = 0.1[/tex] 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
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{173 - 175}{9.8}[/tex]
[tex]Z = -0.2[/tex]
[tex]Z = -0.2[/tex] 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
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