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Answer:
The minimum score required for an A grade is 89.8.
Step-by-step explanation:
We are given that a psychology professor assigns letter grades on a test according to the following scheme. A : Top 7% of scores. B : Scores below the top 7% and above the bottom 64%. C : Scores below the top 36% and above the bottom 25%. D : Scores below the top 75% and above the bottom 6%. F : Bottom 6% of scores.
Scores on the test are normally distributed with a mean of 78.4 and a standard deviation of 7.6.
<u><em>Let X = Scores on the test</em></u>
SO, X ~ Normal()
The z-score probability distribution for normal distribution is given by;
Z = ~ N(0,1)
where, = mean time = 78.4
= standard deviation = 7.6
The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.
Now, the minimum score required for an A grade so that it represents Top 7% of scores is given by;
P(X x) = 0.07 {where x is the required minimum score
P(
) = 0.07
P(Z
) = 0.07
<em>So, the critical value of x in the z table which represents the top 7% of the area is given as 1.4996, that is;</em>
= 78.4 + 11.39696 = <u>89.8 or 90</u>
Hence, the minimum score required for an A grade is 89.8.