Respuesta :
Answer:
a) 5.82%
b) 44.81%
Step-by-step explanation:
Problems of normally distributed samples can be 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.
(a) What percent of all applicants had scores higher than 13?
All applicants have [tex]\mu = 9.7, \sigma = 2.1[/tex]
This probability is 1 subtracted by the pvalue of Z when X = 13. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{13 - 9.7}{2.1}[/tex]
[tex]Z = 1.57[/tex]
[tex]Z = 1.57[/tex] has a pvalue of 0.9418
1 - 0.9418 = 0.0582
5.82% is the answer
(b) What percent of those who entered medical school had scores between 9 and 11?
Those who entered medical school have [tex]\mu = 10.5, \sigma = 1.6[/tex].
This probability is the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 9. So
X = 11
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{11 - 10.5}{1.6}[/tex]
[tex]Z = 0.31[/tex]
[tex]Z = 0.31[/tex] has a pvalue of 0.6217
X = 9
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{9 - 10.5}{1.6}[/tex]
[tex]Z = -0.94[/tex]
[tex]Z = -0.94[/tex] has a pvalue of 0.1736
0.6217 - 0.1736 = 0.4481 = 44.81%
