Question

A new vaccination is being used in a laboratory experiment to investigate whether it is effective. There are 288 subjects in the study. Is there sufficient evidence to determine if vaccination and disease status are related?

Vaccination Status	Diseased	Not Diseased	Total
Vaccinated	81	45	126
Not Vaccinated	54	108	162
Total	135	153	288

Step 1 of 8: State the null and alternative hypothesis.

Step 2 of 8: Find the expected value for the number of subjects who are vaccinated and are diseased. Round your answer to one decimal place.

Step 3 of 8: Find the expected value for the number of subjects who are vaccinated and are not diseased. Round your answer to one decimal place.

Step 4 of 8: Find the value of the test statistic. Round your answer to three decimal places.

Step 5 of 8: Find the degrees of freedom associated with the test statistic for this problem.

Step 6 of 8: Find the critical value of the test at the 0.025 level of significance. Round your answer to three decimal places.

Step 7 of 8: Make the decision to reject or fail to reject the null hypothesis at the 0.025 level of significance.

Step 8 of 8: State the conclusion of the hypothesis test at the 0.025 level of significance.

Answer 1

Step 1: here the hypothesis is given as

H0: Vaccination status and diseased status are independent.

H1: Vaccination status and diseased status are NOT independent.

Step 2:

to find the expected value for the number of subjects who are vaccinated and are diseased.

now total number of vaccinated subjects are 126 and total number of diseased subjects are 135.

so the expected value is   

Thus the expected value for the number of subjects who are vaccinated and are diseased are 59.1.

Step 3:

to find the expected value for the number of subjects who are vaccinated and are not diseased.

now total number of vaccinated subjects are 126 and total number of not diseased subjects are 153.

so the expected value is   

Thus the expected value for the number of subjects who are vaccinated and are not diseased are 66.9.

Step 4:

the chi-square test statistics is given as

where    is the observed frequency

is the expected frequency

similar to step 2 and step 3 we have to find the expected frequency as shown below

	Diseased	Not Diseased	Total
Vaccinated			126
Not Vaccinated			162
Total	135	153	288

thus the expected frquency table is

	Diseased	Not Diseased
Vaccinated	59.0625	66.9375
Not Vaccinated	75.9375	86.0625

so the chi square statistics is

thus the test statistics is 27.267.

Step 5:

Degree of freedom (df) = (r-1)*(c-1)

where r is number of rows and c is number of columns

here r = 2 and c = 2 (i.e 2 row and 2 columns)

thus df = (2-1)*(2-1)= 1 * 1 = 1

hence df = 1.

Step 6:

the critical values for the test at 0.025 level of significance will be found using the chi square table given below

here df is 1 and l.o.s is 0.025

so, the corresponding value in the table is 5.024

thus the critical value is 5.024.

Step 7:

here if Chi square test statistics is Greater than Critical value then reject H0.

So here    27.267 > 5.024

Hence we reject the null hypothesis at 0.025 level of significance.

Step 8:

here we accpet H1, that is vaccination and disease are not independent.

hence, we conclude that there is sufficient evidence that the vaccination and disease are related.

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Thank you :)