In: Statistics and Probability
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 | 45 | 57 | 13 | 136 |
Unlimited | 19 | 45 | 83 | 17 | 164 |
Column Total | 40 | 90 | 140 | 30 | 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: 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.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) 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 < α, we do not 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.
(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.
Let X denote the time given for the test .
and Y denote the grade obtained in the test.
Clearly X has 2 levels and Y has 4 levels.
So here we need to test whether X and Y are independent or not.
Thus the null and the alternative hypothesis are
The level of significance for the test is 0.01.
The construction of the test statistics is as follows
So the expected frequencies are as follows
Time | A | B | C | F | Row Total |
1 h | 18.13 | 40.80 | 63.47 | 13.60 | 136 |
Unlimited | 21.87 | 49.2 | 76.53 | 16.4 | 164 |
Column Total | 40 | 90 | 140 | 30 | 300 |
So the observed value of test statistic is
So the p-value is
Since the P-value is ≥ α, we do not reject the null hypothesis.
So we conclude that there is enough evidence that the Time to take a test and test score are not independent.