Respuesta :
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 Â Â Â 500 Â Â Â Â Â Â Â Â Â Â 100
Test B Â Â Â 150 Â Â Â Â Â Â Â Â Â Â Â 8
Test C Â Â Â Â 70 Â Â Â Â Â Â Â Â Â Â Â 15
So, [tex]\mu_A[/tex] = 500 and [tex]\sigma_A[/tex] = 100
[tex]\mu_B[/tex] = 150 and [tex]\sigma_B[/tex] = 8
[tex]\mu_C[/tex] = 70 and [tex]\sigma_C[/tex] = 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 = [tex]\frac{X_A-\mu_A}{\sigma_A}[/tex]
                        = [tex]\frac{450-500}{100}[/tex] = -0.5
So, this z-score corresponds to the score on test B, i.e;
    z-score on test B = [tex]\frac{X_B-\mu_B}{\sigma_B}[/tex]
             -0.5   = [tex]\frac{X_B-150}{8}[/tex]
             [tex]X_B= 150-(0.5 \times 8)[/tex]
                 = 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 = [tex]\frac{X_A-\mu_A}{\sigma_A}[/tex]
                        = [tex]\frac{625-500}{100}[/tex] = 1.25
So, this z-score corresponds to the score on test C, i.e;
    z-score on test C = [tex]\frac{X_C-\mu_C}{\sigma_C}[/tex]
             1.25   = [tex]\frac{X_C-70}{15}[/tex]
             [tex]X_C= 70+(1.25 \times 15)[/tex]
                 = 70 + 18.75 = 88.75
Hence, a score of 625 on Test A corresponds to a score of 88.75 on Test C.
