Question

The following are the means and standard deviations of some well-known standardized tests, referred to as Test A, Test B, and Test C. All three yield normal distributions.
Test Mean Standard deviation
Test A 500 100
Test B 150 8
Test C 70 15

Required:
a. A score of 450 on Test A corresponds to what score on Test B?
b. A score of 625 on Test A corresponds to what score on Test C?

Answer 1

Answer:

(a) A score of 450 on Test A corresponds to a score of 146 on Test B.

(b) A score of 625 on Test A corresponds to a score of 88.75 on Test C.

Step-by-step explanation:

We are given the means and standard deviations of some well-known standardized tests referred to as Test A, Test B, and Test C. All three yield normal distributions.

Test         Mean       Standard deviation

Test A       500                    100

Test B       150                       8

Test C        70                       15

So, $\mu_A$ = 500 and $\sigma_A$ = 100

$\mu_B$ = 150 and $\sigma_B$ = 8

$\mu_C$ = 70 and $\sigma_C$ = 15

(a) We have to find a score of 450 on Test A corresponds to what score on Test B.

For this, firstly we will find the z-score for a score on Test A and then equate with that of Test B.

      z-score of 450 on test A = $\frac{X_A-\mu_A}{\sigma_A}$

                                                = $\frac{450-500}{100}$ = -0.5

So, this z-score corresponds to the score on test B, i.e;

       z-score on test B = $\frac{X_B-\mu_B}{\sigma_B}$

                          -0.5    = $\frac{X_B-150}{8}$

                          $X_B= 150-(0.5 \times 8)$

                                 = 150 - 4 = 146

Hence, a score of 450 on Test A corresponds to a score of 146 on Test B.

(b) We have to find a score of 625 on Test A corresponds to what score on Test C.

For this, firstly we will find the z-score for a score on Test A and then equate with that of Test C.

      z-score of 625 on test A = $\frac{X_A-\mu_A}{\sigma_A}$

                                                = $\frac{625-500}{100}$ = 1.25

So, this z-score corresponds to the score on test C, i.e;

       z-score on test C = $\frac{X_C-\mu_C}{\sigma_C}$

                          1.25    = $\frac{X_C-70}{15}$

                          $X_C= 70+(1.25 \times 15)$

                                 = 70 + 18.75 = 88.75

Hence, a score of 625 on Test A corresponds to a score of 88.75 on Test C.