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Sagot :
Using the normal distribution and the central limit theorem, it is found that there is a:
a) 0.1813 = 18.13% probability that a person selected at random will score between 240 and 270 on the test.
b) 0.4501 = 45.01% probability that the mean score for the sample will be between 240 and 270.
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 = 250[/tex].
- The standard deviation is of [tex]\sigma = 65[/tex].
Item a:
The probability is the p-value of Z when X = 270 subtracted by the p-value of Z when X = 240, hence:
X = 270:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{270 - 250}{65}[/tex]
[tex]Z = 0.31[/tex]
[tex]Z = 0.31[/tex] has a p-value of 0.6217.
X = 240:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{240 - 250}{65}[/tex]
[tex]Z = -0.15[/tex]
[tex]Z = -0.15[/tex] has a p-value of 0.4404.
0.6217 - 0.4404 = 0.1813.
0.1813 = 18.13% probability that a person selected at random will score between 240 and 270 on the test.
Item b:
We have a sample of 7, hence:
[tex]n = 7, s = \frac{65}{\sqrt{7}} = 24.57[/tex]
X = 270:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{270 - 250}{24.57}[/tex]
[tex]Z = 0.81[/tex]
[tex]Z = 0.81[/tex] has a p-value of 0.7910.
X = 240:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{240 - 250}{24.57}[/tex]
[tex]Z = -0.41[/tex]
[tex]Z = -0.41[/tex] has a p-value of 0.3409.
0.7910 - 0.3409 = 0.4501.
0.4501 = 45.01% probability that the mean score for the sample will be between 240 and 270.
To learn more about the normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213
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