Evan earned a score of 610 on Exam A that had a mean of 650 and a standard deviation of 40. He is about to take Exam B that has a mean of 600 and a standard deviation of 50. How well must Evan score on Exam B in order to do equivalently well as he did on Exam A? Assume that scores on each exam are normally distributed.
one s.d. below the mean on the 1st exam
one s.d. below the mean on the 2nd exam ... 600 - 50 = ?
To determine how well Evan must score on Exam B in order to perform equivalently well as he did on Exam A, we can use the concept of z-scores.
The z-score formula is given by:
z = (x - μ) / σ
where:
- x is the individual score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
First, let's calculate the z-score for Evan's score on Exam A:
z-score for Exam A = (610 - 650) / 40
= -40 / 40
= -1
This z-score represents how many standard deviations below or above the mean Evan's score on Exam A is.
To perform equivalently well on Exam B, Evan's z-score on Exam B should be the same as his z-score on Exam A (-1 in this case). We can use this information to find the corresponding score on Exam B.
Using the z-score formula, we can rearrange it to solve for x (the score on Exam B):
x = z * σ + μ
Now, let's substitute the values we know for Exam B:
z-score for Exam B = -1
μ for Exam B = 600
σ for Exam B = 50
Plugging these values into the formula:
x = -1 * 50 + 600
= -50 + 600
= 550
Therefore, Evan must score 550 on Exam B in order to perform equivalently well as he did on Exam A.
To find out how well Evan must score on Exam B in order to do equivalently well as he did on Exam A, we need to standardize his Exam A score and then use the standardized score to determine his score on Exam B.
First, let's standardize Evan's Exam A score. Standardization involves calculating the z-score, which measures the number of standard deviations a data point is from the mean.
The formula for calculating the z-score is:
z = (x - mean) / standard deviation
For Exam A:
x = 610 (Evan's score on Exam A)
mean = 650 (mean score of Exam A)
standard deviation = 40 (standard deviation of Exam A)
Substituting the values into the formula, we get:
z = (610 - 650) / 40
z = -1
Now that we have the z-score for Evan's Exam A score, we can use it to determine his Exam B score. We'll use the formula for converting a z-score to a raw score.
The formula for converting a z-score to a raw score is:
x = (z * standard deviation) + mean
For Exam B:
z = -1 (Evan's z-score on Exam A)
mean = 600 (mean score of Exam B)
standard deviation = 50 (standard deviation of Exam B)
Substituting the values into the formula, we get:
x = (-1 * 50) + 600
x = 550
Therefore, Evan must score at least 550 on Exam B in order to do equivalently well as he did on Exam A.