Question

We wish to determine if two instructors teaching the same introductory course have significantly different “DF rates” (defined as the proportion of students who receive a course grade of D or F among all students who complete the course).Each instructor teaches 60 students. Among the first instructor’s students, 21 receive a D or F. Among the second instructor’s students, 16 receive a D or F. Assume these can be treated as independent simple random samples from their respective populations.

Use this sample data to test the claim H0:(p1−p2)=0H0:(p1−p2)=0 against HA:(p1−p2)≠0HA:(p1−p2)≠0, using a significance level of 5%.

The value of the test statistic is z=

A large pharmaceutical company advertises that its newly-deleveloped drug is “more effective” at curing a certain disease than the existing drug of its leading competitor. To test this claim, we can compare the proportions of patients who are cured of the disease, depending on which drug they are using for treatment.In a simple random sample of 80 patients being treated with the newly-developed drug, 20 patients were successfully cured of the disease. In an independent simple random sample of 80 patients being treated with the existing drug, 12 patients were successfully cured of the disease.

Use the sample data to test the hypotheses H0:(p1−p2)=0H0:(p1−p2)=0 against HA:(p1−p2)>0HA:(p1−p2)>0, using a significance level of 5%.

The value of the test statistic is z =

In a certain population, it is believed that the proportion of men having red/green color blindness is higher than the proportion of women having red/green color blindness. A hypothesis test can be used to test this claim.

Independent simple random samples of 950 men and 2500 women are tested. Among the men in the sample, 81 have red/green color blindness. Among the women in the sample, 9 have red/green color blindness.The value of the test statistic is z=

Answer 1

1.We wish to determine if two instructors teaching the same introductory course have significantly different “DF rates” (defined as the proportion of students who receive a course grade of D or F among all students who complete the course).Each instructor teaches 60 students. Among the first instructor’s students, 21 receive a D or F. Among the second instructor’s students, 16 receive a D or F. Assume these can be treated as independent simple random samples from their respective populations.

Level of significance :

We are given that

  

It is a two tailed test since the alternative hypothesis is two tailed.

Test statistic :

   has a standard Normal distribution.

where

The value of the test statistic is

The critical value of Z for a 5% level of significance is 1.96,( p-value is 0.3228>0.05). Since the calculated Z value is <Critical value, we do not reject the null hypothesis.

Hence the two instructors' DF rates are same.

2.A large pharmaceutical company advertises that its newly-deleveloped drug is “more effective” at curing a certain disease than the existing drug of its leading competitor. To test this claim, we can compare the proportions of patients who are cured of the disease, depending on which drug they are using for treatment.In a simple random sample of 80 patients being treated with the newly-developed drug, 20 patients were successfully cured of the disease. In an independent simple random sample of 80 patients being treated with the existing drug, 12 patients were successfully cured of the disease.

Level of significance :

We are given that

  

It is an one tailed test since the alternative hypothesis is two tailed.

Test statistic :

   has a standard Normal distribution.

where

The value of the test statistic is

The critical value of Z for a 5% level of significance is 1.6445( the p-value is 0.05705>0.05). Since the calculated Z value is <Critical value, we do not reject the null hypothesis.

Therefore there is no evidence that its newly-deleveloped drug is “more effective” at curing a certain disease than the existing drug of its leading competitor as advertised.

3.

In a certain population, it is believed that the proportion of men having red/green color blindness is higher than the proportion of women having red/green color blindness. A hypothesis test can be used to test this claim.

Independent simple random samples of 950 men and 2500 women are tested. Among the men in the sample, 81 have red/green color blindness. Among the women in the sample, 9 have red/green color blindness.

Let the subscript 1 denote men and 2 denotes Women. We need to test the belief that the proportion of men having red/green color blindness is higher than the proportion of women having red/green color blindness. Hence the alternative is one-tailed

Level of significance :

We are given that

  

It is an one tailed test since the alternative hypothesis is two tailed.

Test statistic :

   has a standard Normal distribution.

where

The value of the test statistic is

The critical value of Z for a 5% level of significance is 1.6445( the p-value is 1.642E-41<0.05). Since the calculated Z value is >Critical value, we reject the null hypothesis.

Hence there is a strong evidence in the belief that the proportion of men having red/green color blindness is higher than the proportion of women having red/green color blindness.