Respuesta :
Answer:
a) 86.73
b) 90.24
c) 10.56% scoring more than 90
d) 75.8
e) 50% probability that a randomly selected student will score more than 80.
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.
In this problem, we have that:
[tex]\mu = 80, \sigma = 8[/tex]
a. Find the 80th percentile.
This is the value of X when Z has a pvalue of 0.8. So X when Z = 0.841.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.841 = \frac{X - 80}{8}[/tex]
[tex]X - 80 = 8*0.841[/tex]
[tex]X = 86.73[/tex]
b. Find the cutoff for the A grade if the top 10% get an A.
This is the value of X when Z has a pvalue of 0.9. So X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 80}{8}[/tex]
[tex]X - 80 = 8*1.28[/tex]
[tex]X = 90.24[/tex]
c. Find the percentage scoring more than 90.
This is 1 subtracted by the pvalue of Z when X = 90. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{90 - 80}{8}[/tex]
[tex]Z = 1.25[/tex]
[tex]Z = 1.25[/tex] has a pvalue of 0.8944.
1 - 0.8944 = 0.1056
10.56% scoring more than 90
d. Find the score that separates the bottom 30% from the top 70%.
This is the value of X when Z has a pvalue of 0.3. So X when Z = -0.525.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.525 = \frac{X - 80}{8}[/tex]
[tex]X - 80 = 8*(-0.525)[/tex]
[tex]X = 75.8[/tex]
e. Find the probability that a randomly selected student will score more than 80.
This is 1 subtracted by the pvalue of Z when X = 80.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{80 - 80}{8}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5.
1 - 0.5 = 0.5
50% probability that a randomly selected student will score more than 80.
