Using the <em>normal distribution and the central limit theorem</em>, 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.
<h3>Normal Probability Distribution</h3>In a normal distribution with mean and standard deviation
, the z-score of a measure X is given by:
- 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
.
In this problem:
- The mean is of
.
- The standard deviation is of
.
Item a:
The probability is the <u>p-value of Z when X = 270 subtracted by the p-value of Z when X = 240</u>, hence:
X = 270:
has a p-value of 0.6217.
X = 240:
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:
X = 270:
By the Central Limit Theorem
has a p-value of 0.7910.
X = 240:
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 <em>normal distribution and the central limit theorem</em>, you can check brainly.com/question/24663213