Question

In: Statistics and Probability

Professor Fair believes that extra time does not improve grades on exams. He randomly divided a...

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.    

Solutions

Expert Solution

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.


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