Respuesta :
Using the normal distribution and the central limit theorem, it is found the power of the test is of 0.
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].
The power of a test is the probability of rejecting a true null hypothesis, which in this problem is concluding that there is significant evidence that the mean speed is above 95 m/s when in fact it isn't, which is the probability of finding a sample mean above 102.2 m.
For this problem, the parameters are:
- True mean speed of 95 m/s, hence [tex]\mu = 95[/tex]
- Standard deviation of 4 m/s, hence [tex]\sigma = 4[/tex]
- Sample of 8, hence [tex]n = 8, s = \frac{4}{\sqrt{8}}[/tex].
The probability is 1 subtracted by the p-value of Z when X = 102.2, then:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{102.2 - 95}{\frac{4}{\sqrt{8}}}[/tex]
[tex]Z = 5.1[/tex]
[tex]Z = 5.1[/tex] has a p-value of 1.
1 - 1 = 0
The power of the test is of 0.
A similar problem is given at https://brainly.com/question/24663213
