Question

a professor has a bias towards selecting the answer "B" on multiple-choice exams. Evaluate this claim at the 95% and 99% confidence levels.

data:

1 C

2 B

3 D

4 B

5 B

6 E

7 E

8 D

9 C

10 B

11 A

12 C

13 A

14 B

15 E

16 C

17 B

18 E

19 C

20 C

21 D

22 E

23 B

24 D

25 B

1. what is H0 and H1

2 what is the result of the statistical test (Examples: "The p-value was 0.061" or "the upper bound of the population mean was 12.4g")

Answer 1

The Probablity of correct answer is B if we selct one question randomly = 0.25 [unbiased result]

Ho: p = 0.25 , professor is not biased towards answer B.

Ha : p > 0.25 , Professor is Biased towards answer B.

sample size n = 25

favourable cases (correct answer is B) = 8

p = 8 / 25 = 0.32

1) 95 %

The critical value for α=0.05 is

The corresponding confidence interval is computed as shown below:

The following information is provided: The sample size is N = 25, the number of favorable cases is X = 8, and the sample proportion is

and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p=0.25

Ha:p>0.25

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=0.808≤zc​=1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.2095, and since p=0.2095≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than p0​, at the α=0.05 significance level.

2) 99%

The critical value for α=0.01 is

. The corresponding confidence interval is computed as shown below:

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p=0.25

Ha: p>0.25

.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a right-tailed test is zc​=2.33.

The rejection region for this right-tailed test is R={z:z>2.33}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=0.808≤zc​=2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.2095, and since p=0.2095≥0.01, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than p0​, at the α=0.01 significance level.

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