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 21 41 61 11 134 Unlimited 19 46 81 20 166 Column Total 40 87 142 31 300 (i) Give the value of the level of significance. State the null and alternate hypotheses. 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. 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: 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 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. 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 value of the level of significance is =1% =0.01

H0: Time to take a test and test score are independent.

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

(H0: Null Hypothesis; H1: Alternative Hypothesis)

(ii)

Sample test statistic(​​​​​​):

Observed frequencies(O):

Time	A	B	C	F	Total
1 hr	21	41	61	11	134
Unlimited	19	46	81	20	166
Total	40	87	142	31	300

Expected frequencies[E =(Row Total*Column Total/Overall Total)] :

Time	A	B	C	F	Total
1 hr	17.87	38.86	63.43	13.85	134
Unlimited	22.13	48.14	78.57	17.15	166
Total	40	87	142	31	300

Calculation of the test statistic:

Observed: O	Expected: E	(O - E)2/E
21	17.87	0.55
19	22.13	0.44
41	38.86	0.12
46	48.14	0.10
61	63.43	0.09
81	78.57	0.08
11	13.85	0.59
20	17.15	0.47
O =300	E =300	=[(O - E)2/E] =2.44

Test statistic, =2.44

(iii)

Number of rows, r =2

Number of columns, c =4

Degrees of freedom, df =(r - 1)(c - 1) =(2 - 1)(4 - 1) =3

P-value of the test statistic of =2.44 at 3 degrees of freedom is: P-value =0.49 > 0.100

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.