Respuesta :
Using the normal distribution and the central limit theorem, it is found that there is a 0.1894 = 18.94% probability that the average number of pages in the sample is less than 500.
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 525, hence [tex]\mu = 525[/tex].
- The standard deviation is 200, hence [tex]\sigma = 200[/tex].
- A sample of 50 is taken, hence [tex]n = 50, s = \frac{200}{\sqrt{50}}[/tex].
The probability that the average number of pages in the sample is less than 500 is the p-value of Z when X = 500, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{500 - 525}{\frac{200}{\sqrt{50}}}[/tex]
[tex]Z = -0.88[/tex]
[tex]Z = -0.88[/tex] has a p-value of 0.1894.
0.1894 = 18.94% probability that the average number of pages in the sample is less than 500.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
