Answer :
Answer:
a) 20.33% probability that the participant is less than 64.5 inches.
b) 33.72% probability that the participant is more than 68.25 inches
Step-by-step explanation:
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.
In this problem, we have that:
[tex]\mu = 67, \sigma = 3[/tex]
a.) Find the probability that the participant is less than 64.5 inches?
This is the pvalue of Z when X = 64.5.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{64.5 - 67}{3}[/tex]
[tex]Z = -0.83[/tex]
[tex]Z = -0.83[/tex] has a pvalue of 0.2033
20.33% probability that the participant is less than 64.5 inches.
b.) Find the probability that the participant is more than 68.25 inches?
This is 1 subtracted by the pvalue of Z when X = 68.25.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{68.25 - 67}{3}[/tex]
[tex]Z = 0.42[/tex]
[tex]Z = 0.42[/tex] has a pvalue of 0.6628
1 - 0.6628 = 0.3372
33.72% probability that the participant is more than 68.25 inches