Respuesta :
Answer:
a) The minimum score required to be admitted into this program will be 68.8.
b) The minimum score required to be admitted into this program will be 53.05.
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 = 50, \sigma = 10[/tex]
a. What will be the minimum score required to be admitted into this program?
Top 3%, so the minimum score is X when Z has a pvalue of 1-0.03 = 0.97. So X when Z = 1.88.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.88 = \frac{X - 50}{10}[/tex]
[tex]X - 50 = 10*1.88[/tex]
[tex]X = 68.8[/tex]
The minimum score required to be admitted into this program will be 68.8.
b. One state wants to allow all students with scores in the top 38 into a special advanced program. What will be the minimum score required to be admitted into this program?
Top 38%, so the minimum score is X when Z has a pvalue of 1-0.32 = 0.68. So X when Z = 0.305.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.305 = \frac{X - 50}{10}[/tex]
[tex]X - 50 = 10*0.305[/tex]
[tex]X = 53.05[/tex]
The minimum score required to be admitted into this program will be 53.05.
