Respuesta :
Answer:
Grades between 61 and 68 are the numerical limits for a D grade.
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 = 76.9, \sigma = 9.6[/tex]
D: Scores below the top 82% and above the bottom 5%
So scores between 5th and the 100-82 = 18th percentile.
5th percentile:
value of X when Z has a pvalue of 0.05. So X when Z = -1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 76.9}{9.6}[/tex]
[tex]X - 76.9 = -1.645*9.6[/tex]
[tex]X = 61[/tex]
18th percentile:
value of X when Z has a pvalue of 0.18. So X when Z = -0.925.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.925 = \frac{X - 76.9}{9.6}[/tex]
[tex]X - 76.9 = -0.925*9.6[/tex]
[tex]X = 68[/tex]
Grades between 61 and 68 are the numerical limits for a D grade.
