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	45	60	15	144
Unlimited	17	46	80	13	156
Column Total	41	91	140	28	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 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 not independent.
H₁: Time to take a test and test score are independent.    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 different.
H₁: The distributions for a timed test and an unlimited test are the same.

(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 > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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 null and alternate hypotheses are:
As we need to test whether the time to complete a test and test results are independent, we need to perform a chi-square independent test.
H0: Time to take a test and test score are independent.
H1: Time to take a test and test score are not independent

II) Let's calculate the test statistic from the following formula:

Here, Oi is the observed values and Ei are the expected values for each cell

The expected value is calculated by multiplying the corresponding row and column total and divide it by 300 (Total)
The calculations/values calculated are shown in the below table:

Results
	A	B	C	F		Row Totals
1 Hour	24  (19.68)  [0.95]	45  (43.68)  [0.04]	60  (67.20)  [0.77]	15  (13.44)  [0.18]		144
Unlimited	17  (21.32)  [0.88]	46  (47.32)  [0.04]	80  (72.80)  [0.71]	13  (14.56)  [0.17]		156
Column Totals	41	91	140	28		300  (Grand Total)

The expected values are given in the parenthesis () and chi-square value for the given cell is shown in []

Chi-square statistic = Sum of individual chi-square values = 3.7321

III) P-value = 0.2918 (from the table)

P-value > 0.1000

IV) As the p-value > 0.10, we will not reject the null hypothesis

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

V) Conclusion:

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