Respuesta :
Answer:
65.78% probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test.
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 = 502, \sigma = 100, n = 90, s = \frac{100}{\sqrt{90}} = 10.54[/tex]
What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
This is the pvalue of Z when X = 502+10 = 512 subtracted by the pvalue of Z when X = 502-10 = 492.
X = 512
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{512 - 502}{10.54}[/tex]
[tex]Z = 0.95[/tex]
[tex]Z = 0.95[/tex] has a pvalue of 0.8289
X = 492
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{492 - 502}{10.54}[/tex]
[tex]Z = -0.95[/tex]
[tex]Z = -0.95[/tex] has a pvalue of 0.1711
0.8289 - 0.1711 = 0.6578
65.78% probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test.
