Question

Suppose that your statistics professor tells you that the scores on a midterm exam were approximately normally distributed with a mean of 78 and a standard deviation of 7 . The top \(15 \%\) of all scores have been designated A's. Your score is \(89 .\) Did you receive an A? Explain.

Short answer

With the calculated Z-score of 1.57 for the grade and 1.04 being the minimum Z-score to qualify for the top 15%, the score of 89 is in the top 15% of the scores. So, yes, the score 89 is an A.

Step-by-step solution

Define the problem in terms of Z-scores

Let's convert our problem into a Z-score problem. Z-score is a measure that indicates how many standard deviations an element is from the mean. This can be calculated using the following formula: \( Z = (X - μ) / σ \)where:- X is the raw score- μ is the population mean- σ is the standard deviation of the population

Calculate the Z-score for the given grade

Now put the given values into the Z-score formula to figure out the Z-score for the given mark of 89.\( Z = (89 - 78) / 7 = 1.57 \)

Find the Z-score of the top 15%

Using a standard Z-table, or a calculator equipped with a normal distribution function, find the Z-score that corresponds to the area that represents the top 15% of all values. This is represented by 0.85 (or 85%) on the cumulative area (since the table represents the area from the left). The corresponding Z-score is approximately 1.04.

Compare the Z-scores

Compare the two Z-scores calculated in the previous steps. If the Z-score of the grade is greater than or equal to the Z-score representing the top 15%, then the grade is in the top 15% and therefore, is an A.