Answer:
The minimum score required for an A grade is 88.
Step-by-step explanation:
When the distribution is normal, we use 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 question:
[tex]\mu = 77.5, \sigma = 8.6[/tex]
Find the minimum score required for an A grade.
This score is the 100 - 12 = 88th percentile, which is X when Z has a pvalue of 0.88. So X when Z = 1.175.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.175 = \frac{X - 77.5}{8.6}[/tex]
[tex]X - 77.5 = 8.6*1.175[/tex]
[tex]X = 87.61[/tex]
Rounding to the nearest whole number:
The minimum score required for an A grade is 88.
