Question

Professor Fair believes that extra time does not improve grades on exams. He randomly divided a group of 300 students into two groups and gave them all the same test. One group had exactly 1 hour in which to finish the test, and the other group could stay as long as desired. The results are shown in the following table. Test at the 0.01 level of significance that time to complete a test and test results are independent.

Time	A	B	C	F	Row Total
1 h	24	43	61	10	138
Unlimited	16	45	83	18	162
Column Total	40	88	144	28	300

(i) Give the value of the level of significance.


State the null and alternate hypotheses.

H₀: Time to take a test and test score are not independent.
H₁: Time to take a test and test score are independent. H₀: The distributions for a timed test and an unlimited test are the same.
H₁: The distributions for a timed test and an unlimited test are different.     H₀: The distributions for a timed test and an unlimited test are different.
H₁: The distributions for a timed test and an unlimited test are the same. H₀: Time to take a test and test score are independent.
H₁: Time to take a test and test score are not independent.

(ii) Find the sample test statistic. (Round your answer to two decimal places.)


(iii) Find or estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100     0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005

(iv) Conclude the test.

Since the P-value < α, we reject the null hypothesis. Since the P-value is ≥ α, we do not reject the null hypothesis.     Since the P-value < α, we do not reject the null hypothesis. Since the P-value ≥ α, we reject the null hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 1% level of significance, there is insufficient evidence to claim that time to do a test and test results are not independent. At the 1% level of significance, there is sufficient evidence to claim that time to do a test and test results are not independent.    

Answer 1

(i)

The level of signficance is 0.01.

Hypotheses are:

H0: Time to take a test and test score are independent.
H1: Time to take a test and test score are not independent.

(ii)

First we need to find the expected frequencies. Expected frequencies will be calculated as follows:

Following table shows the expeted frequencies:

Time	A	B	C	F	Row Total
1 h	18.4	40.48	66.24	12.88	138
Unlimited	21.6	47.52	77.76	15.12	162
Column Total	40	88	144	28	300

Following table shows the calculations for chi square test statistics:

O	E	(O-E)^2/E
24	18.4	1.704347826
43	40.48	0.15687747
61	66.24	0.414516908
10	12.88	0.643975155
16	21.6	1.451851852
45	47.52	0.133636364
83	77.76	0.353106996
18	15.12	0.548571429
Total		5.406884

The test statistics is:

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (2-1)*(4-1)=3

(iii)

The p-value is greater than 0.10.

Correct option:

P-value > 0.100

(iv)

Since the P-value is ≥ α, we do not reject the null hypothesis.  

(v)

At the 1% level of significance, there is insufficient evidence to claim that time to do a test and test results are not independent.