Question

Reports suggest that the average credit card debt for recent college graduates is $3000. FedLoan Servicing believes the average debt of graduates is much less than this, so it conducts a study of 50 randomly selected graduates and finds that the average debt is $2975, and the population standard deviation is $1000. Let’s conduct the test based on a Type I error of α=0.05.

Answer 1

Solution:

Given: the average credit card debt for recent college graduates is $3000.

FedLoan Servicing believes the average debt of graduates is much less than this.

That is we have to test if mean

Sample size = n = 50

Sample mean =

Population standard deviation =

Level of significance = α=0.05

Step 1) State H₀ and H₁:

Vs

Step 2) Test statistic:

Step 3) Find z critical value:

Level of significance = α=0.05

This is left tailed test( Since H1 is < type),

Thus look in z table for Area = 0.0500 or its closest area and find z value

Area 0.0500 is in between 0.0495 and 0.0505 and both the area are at same distance from 0.0500

Thus we look for both area and find both z values

Thus Area 0.0495 corresponds to -1.65 and 0.0505 corresponds to -1.64

Thus average of both z values is : ( -1.64+ - 1.65) / 2 = -1.645

Thus z critical value  = -1.645

Step 4) Decision Rule:
Reject null hypothesis ,if z  test statistic value < z critical value = -1.645 , otherwise we fail to reject H0.

Since z  test statistic value = > z critical value = -1.645 , we fail to reject H0.

Step 5) Conclusion:

At 0.05 level of significance, we do not have sufficient evidence to conclude that the average debt of graduates is much less than $3000.