Question

A psychology professor assigns letter grades on a test according to the following scheme. A: Top 6%6% of scores B: Scores below the top 6%6% and above the bottom 55U% C: Scores below the top 45E% and above the bottom 24$% D: Scores below the top 76v% and above the bottom 6%6% F: Bottom 6%6% of scores

Answer 1

Question Continuation

F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 70.9 and a standard deviation of 9.8.

Find the numerical limits for a D grade.

Round your answers to the nearest whole number, if necessary.

Answer:

The numerical limits for a D grade is scores between 56 and 64.

Explanation:

Given.

D ranges between scores below the top 76% and above the bottom 6%

Mean, u = 70.9

Standard Deviation, σ = 9.8

To solve this, we'll calculate the numerical limits for (1) below the top 76% and (2) above the bottom 6%.

Calculating (1)

Using z = (x-u)/σ

Where p-value = 76% = 0.76

u = 70.9 and σ = 9.8

If the p-value = 0.76;

z-value = 1 - 0.76 = 0.24

From the z-table,

We have p(0.24) = -0.705.

Substitute these values in the above equation.

This gives.

-0.705 = (x - 70.9)/9.85 --- Solve for x

x - 70.9 = 9.85 * -0.705

x - 70.9 = -6.94425

x = 70.9 - 6.94425

x = 63.95575

x = 64 ---- Approximated

Calculating (2)

Using z = (x-u)/σ

Where p-value = 6% = 0.06

u = 70.9 and σ = 9.8

From the z-table,

We have p(0.06) = -1.555

Substitute these values in the above equation.

This gives.

-1.555 = (x - 70.9)/9.85 --- Solve for x

x - 70.9 = 9.85 * -1.555

x - 70.9 = -15.31675

x = 70.9 - 15.31675

x = 55.58325

x = 56 ---- Approximated

Hence, the calculated numerical limits for a D grade is scores approximately between 56 and 64.

Answer 2

The limits for a D-score is (56 to 64) (nearest whole numbers)

Explanation:

This is a binomial distribution problem with

Mean = μ = 70.9

Standard deviation = σ = 9.8

We will be using z-scores.

The z-score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ

The limits for a D-score: Scores below the top 76% and above the bottom 6%

Scores below the top 76% refer to the bottom 24% of the score.

Let the required limits be x' and x" & their z-scores be z' and z"

P(x' < x < x") = P(z' < z < z")

Representing this limits with inequalities.

Scores below the top 76% refer to the bottom 24% of the scores. The limit is P(x < x') = 0.24

Scores above the top 6%. The limit is P(x < x") = 0.06

Using the z-tables,

P(z < z') = 0.24

Gives a z-score of z' = -0.706

z = (x - μ)/σ

z' = (x' - μ)/σ

-0.706 = (x' - 70.9)/9.8

x' = (9.8)(-0.706) + 70.9 = 63.9812

P(z < z") = 0.06

z" = -1.555

z = (x - μ)/σ

z" = (x" - μ)/σ

-1.555 = (x" - 70.9)/9.8

x' = (9.8)(-1.555) + 70.9 = 55.661

The limits for a D-score is (55.661 to 63.9812)

Hope this Helps!!!