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Sagot :
Using the normal probability distribution and the central limit theorem, it is found that the probabilities are given by:
a) 0.0119 = 1.19%.
b) 0.121 = 12.1%.
c) 0.9257 = 92.57%.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, which is the percentile of X.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem:
- The mean is of [tex]\mu = 45000[/tex].
- The standard deviation is of [tex]\sigma = 12720[/tex].
- A sample of 144 is taken, hence [tex]n = 144, s = \frac{12720}{\sqrt{144}} = 1060[/tex].
Item a:
The probability is the p-value of Z when X = 42600, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{42600 - 45000}{1060}[/tex]
[tex]Z = -2.26[/tex]
[tex]Z = -2.26[/tex] has a p-value of 0.0119.
The probability is of 0.0119 = 1.19%.
Item b:
The probability is the 1 subtracted by the p-value of Z when X = 46240, hence:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{46240 - 45000}{1060}[/tex]
[tex]Z = 1.17[/tex]
[tex]Z = 1.17[/tex] has a p-value of 0.879.
1 - 0.879 = 0.121.
The probability is of 0.121 = 12.1%.
Item c:
The probability is the p-value of Z when X = 46980 subtracted by the p-value of Z when X = 43190, hence:
X = 46980:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{46980 - 45000}{1060}[/tex]
[tex]Z = 1.87[/tex]
[tex]Z = 1.87[/tex] has a p-value of 0.9693.
X = 43190:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{43190 - 45000}{1060}[/tex]
[tex]Z = -1.71[/tex]
[tex]Z = -1.71[/tex] has a p-value of 0.0436.
0.9693 - 0.0436 = 0.9257.
The probability is of 0.9257 = 92.57%.
To learn more about the normal probability distribution and the central limit theorem, you can check https://brainly.com/question/24663213
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