Respuesta :
Answer:
The limits for a D-score is (56 to 64) (nearest whole numbers)
Step-by-step 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!!!
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.
Step-by-step 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.
