Question

A sociology professor assigns letter grades on a test according to the following scheme. A: Top 13% of scores B: Scores below the top 13% and above the bottom 59% C: Scores below the top 41% and above the bottom 23% D: Scores below the top 77% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 70.8 and a standard deviation of 8.6. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Answer 1

To find the minimum score required for an A grade, use the z-score formula with the given mean and standard deviation. Then, use a calculator or z-score table to find the score. Round the answer to the nearest whole number.

Step-by-step explanation:

To find the minimum score required for an A grade, we need to determine the score at the top 13% of the distribution.

Since the scores on the test are normally distributed with a mean of 70.8 and a standard deviation of 8.6, we can use the z-score formula.

The z-score formula is z = (x - μ) / σ, where z is the z-score, x is the score we want to find, μ is the mean, and σ is the standard deviation. We want to find the score that corresponds to the 87th percentile, which is the complement of the top 13%.

We can use a z-score table or a calculator to find the z-score that corresponds to the 87th percentile. Once we have the z-score, we can plug it into the z-score formula to find the score.

Using the TI-83, 83+, 84, 84+ Calculator, we can enter the z-score and the mean and standard deviation values to find the score. Rounding the answer to the nearest whole number gives us the minimum score required for an A grade.

Answer 2

Answer: the minimum score required for an A grade is 81

Step-by-step explanation:

Since the scores on the test are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = scores on the test.

µ = mean score

σ = standard deviation

From the information given,

µ = 70.8

σ = 8.6

The probability value for the top 13% of the scores would be (1 - 13/100) = (1 - 0.13) = 0.87

Looking at the normal distribution table, the z score corresponding to the probability value is 1.13

Therefore,

1.13 = (x - 70.8)/8.6

Cross multiplying by 8.6, it becomes

1.13 × 8.6 = x - 70.8

9.718 = x - 70.8

x = 9.718 + 70.8

x = 81 to the nearest whole number