Using the <em>normal distribution and the central limit theorem</em>, we have that:
a) A normal model with mean 0.3 and standard deviation of 0.0458 should be used.
b) There is a 0.2327 = 23.27% probability that more than one third of this sample wear contacts.
<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, for a <u>proportion p in a sample of size n</u>, the sampling distribution of sample proportion is approximately normal with mean
and standard deviation
, as long as
and
.
In this problem:
- 30% of students at a university wear contact lenses, hence p = 0.3.
- We randomly pick 100 students, hence n = 100.
Item a:
Hence a normal model is appropriated.
The mean and the standard deviation are given as follows:
Item b:
The probability is <u>1 subtracted by the p-value of Z when X = 1/3 = 0.3333</u>, hence:
By the Central Limit Theorem
has a p-value of 0.7673.
1 - 0.7673 = 0.2327.
0.2327 = 23.27% probability that more than one third of this sample wear contacts.
To learn more about the <em>normal distribution and the central limit theorem</em>, you can take a look at brainly.com/question/24663213