Savannah earned a score of 43 on Exam A that had a mean of 36 and a standard
deviation of 5. She is about to take Exam B that has a mean of 650 and a standard
deviation of 40. How well must Savannah score on Exam B in order to do
equivalently well as she did on Exam A? Assume that scores on each exam are
normally distributed.
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
To find out how well Savannah must score on Exam B in order to do equivalently well as she did on Exam A, we need to convert her score on Exam A to a z-score and then use that z-score to find the corresponding score on Exam B.
First, let's calculate the z-score for Savannah's score on Exam A:
z = (x - μ) / σ
where x is the score on Exam A, μ is the mean of Exam A, and σ is the standard deviation of Exam A.
x = 43
μ = 36
σ = 5
z = (43 - 36) / 5
z = 7 / 5
z = 1.4
Now that we have the z-score for Savannah's score on Exam A, we can use it to find the score she must achieve on Exam B.
Using the formula for z-score:
z = (x - μ) / σ
where z is the z-score, x is the score on Exam B, μ is the mean of Exam B, and σ is the standard deviation of Exam B.
μ = 650
σ = 40
z = 1.4
Solving for x:
1.4 = (x - 650) / 40
Multiply both sides by 40:
1.4 * 40 = x - 650
56 = x - 650
Add 650 to both sides:
x = 56 + 650
x = 706
Therefore, Savannah must score at least 706 on Exam B in order to do equivalently well as she did on Exam A.
To determine how well Savannah must score on Exam B in order to perform equivalently to Exam A, we need to standardize the scores using z-scores.
The z-score formula is: z = (x - μ) / σ
Where:
- z is the standardized score
- x is the individual score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Let's calculate the z-score for Exam A:
z_A = (x_A - μ_A) / σ_A
Given:
x_A = 43 (Savannah's score on Exam A)
μ_A = 36 (mean of Exam A)
σ_A = 5 (standard deviation of Exam A)
z_A = (43 - 36) / 5
z_A = 7 / 5
z_A = 1.4
Now, using the z-score formula, we can calculate Savannah's score on Exam B:
z_B = (x_B - μ_B) / σ_B
Given:
μ_B = 650 (mean of Exam B)
σ_B = 40 (standard deviation of Exam B)
To find x_B, rearrange the formula:
(x_B - μ_B) = z_B * σ_B
x_B = z_B * σ_B + μ_B
Substituting in the known values:
x_B = 1.4 * 40 + 650
x_B = 56 + 650
x_B = 706
Therefore, Savannah must score at least 706 on Exam B in order to perform equivalently to her score of 43 on Exam A.