Using the normal distribution, it is found that the probability is 0.16.
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.
In this problem, the mean and the standard deviation are given by, respectively, [tex]\mu = 70, \sigma = 3[/tex].
The proportion of students between 45 and 67 inches is the p-value of Z when X = 67 subtracted by the p-value of Z when X = 45, hence:
X = 67:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{67 - 70}{3}[/tex]
Z = -1
Z = -1 has a p-value of 0.16.
X = 45:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{45 - 70}{3}[/tex]
Z = -8.3
Z = -8.3 has a p-value of 0.
0.16 - 0 = 0.16
The probability is 0.16.
More can be learned about the normal distribution at https://brainly.com/question/24663213
