Respuesta :
Answer:
a) 25% of the students exam scores fall below 55.6.
b) The minimum score for an A is 84.68.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set 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]
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 p-value, we get the probability that the value of the measure is greater than X.
Mean of 61 and a standard deviation of 8.
This means that [tex]\mu = 61, \sigma = 8[/tex]
(a) What score do 25% of the students exam scores fall below?
Below the 25th percentile, which is X when Z has a p-value of 0.25, that is, X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 61}{8}[/tex]
[tex]X - 61 = -0.675*8[/tex]
[tex]X = 55.6[/tex]
25% of the students exam scores fall below 55.6.
(b) Suppose the professor decides to grade on a curve. If the professor wants 0.15% of the students to get an A, what is the minimum score for an A?
This is the 100 - 0.15 = 99.85th percentile, which is X when Z has a p-value of 0.9985. So X when Z = 2.96.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.96 = \frac{X - 61}{8}[/tex]
[tex]X - 61 = 2.96*8[/tex]
[tex]X = 84.68[/tex]
The minimum score for an A is 84.68.
