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	58	11	136
Unlimited	19	48	81	16	164
Column Total	43	91	139	27	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 independent.
H₁: Time to take a test and test score are not 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 not independent.
H₁: Time to take a test and test score are 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 reject the null hypothesis.

Since the P-value < α, we do not 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

Here we are using Excel for calculation:

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Column 1	2	43	21.5	12.5
Column 2	2	82	41	98
Column 3	2	139	69.5	264.5
Column 4	2	27	13.5	12.5
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	3726	3	1242.13	12.82	0.0161	16.694
Within Groups	387.5	4	96.875
Total	4114	7

i) State the null and alternate hypotheses.

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.    

ii) Test statistics:

F = 12.82

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

0.010 < P-value < 0.025

iv) Conclude the test.

Since the P-value is ≥ α, we do not 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.