A student scores 56 on a geography test and 285 on a mathematics test. The geography test has a mean of 80 and a standard deviation of 20. The mathematics test has a mean of 300 and a standard deviation of 10. If the data for both tests are normally distributed, on which test did the student score better relative to the other students in each class? Justify your answer

We are given that a student scores 56 on a geography test and 285 on a mathematics test. The geography test has a mean of 80 and a standard deviation of 20. The mathematics test has a mean of 300 and a standard deviation of 10.

Let X = Student scores in geography test

Y = Student scores in mathematics test

Here, Mean score of geography test, [tex]\mu_X[/tex] = 80

Standard deviation of geography test, [tex]\sigma_X[/tex] = 20

Also, Mean score of mathematics test, [tex]\mu_Y[/tex] = 300

Standard deviation of mathematics test, [tex]\sigma_Y[/tex] = 10

For comparing on which test did the student score better relative to the other students in each class we will find the z score of both the test as data for both tests are normally distributed;

Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)

Z score for geography test, [tex]Z_G[/tex] ;

[tex]Z_G[/tex] = [tex]\frac{X_G-\mu_X}{\sigma_X}[/tex] = [tex]\frac{56-80}{20}[/tex] = -1.2

Z score for mathematics test, [tex]Z_M[/tex] ;

[tex]Z_M[/tex] = [tex]\frac{X_M-\mu_Y}{\sigma_Y}[/tex] = [tex]\frac{285-300}{10}[/tex] = -1.5

From both the z scores it is clear that z score of geography test is higher than that of mathematics test.

Therefore, the student score better relative to the other students in each class in geography test .