Question

3.In one of the largest and most famous public health experiments ever conducted, in 1954 a randomized controlled trial was run to see whether a vaccine developed by a doctor named Jonas Salk was effective in preventing paralytic polio. A total of 201, 974 children, chosen to be representative of those who might be susceptible to the disease, were randomized to two groups: 100,745 children where injected with a harmless saline solution and the other 101,229 children were injected with Salk's vaccine. (a) What was the point of giving saline solution to the children who didn't get the vaccine? (b) The results of the trial were as follows: 33 of the 101,229 children who got the vaccine later developed paralytic polio, whereas 115 of the other 100,745 children developed paralytic polio. Perform an appropriate statistical hypothesis test for these data to draw conclusions about the effectiveness of the Salk vaccine. (c)Describe in context what the Type I and Type Il errors are in for this scenario (do not perform any computations). What would you argue is the more detrimental error to commit? (d) Assess the assumptions needed to perform your statistical hypothesis test in (b). few sentences commenting on the validity of your conclusions in (b) in terms of these assumptions and sample size.

Answer 1

a)The saline solution is given to make a comparison between the two groups as to whether the vaccine treatment is successful in reducing polio than the group with no vaccine. It is called the control group. It helps in differentiating between the effects of treatment and non-treatment.

b)we are testing

Ho: p1 P2 VS S H1 : P1 : P1 <1 P2

where p1=with vaccine and p2=without vaccine

test statistic=$p_1-p_2/\sqrt{pq(1/n_1+1/n_2)}$, where $p=p_1n_1+p_2n_2/n_1+n_2$ =0.00073

test statistic= -6.82

c)Type 1 error is the error of rejecting the null when the null is true, while type 2 error is the error of accepting the null when the null is false

Here type 1 error would be rejecting that the proportions are equal when infact the proportions are equal.

type 2 error would be accepting that the proportions are equal when infact the proportions are not.

Type 2 error would be the more detrimental error as we are discarding the significance of a vaccine treatment. It concludes that the vaccine is not succesful in reducing polio when infact it is successful in reducing. So based on this situation here type 2 error would be more detrimental. We know type 1 error is more harmful than type 2 according to the statistical universality, but here based on the situation discarding a true alternative is more detrimrental as eradication of the disease is our main concern.

d)The assumptions of the hypothesis test are-the sample sizes should be more than 30, the two data sets must be independent of each other, data points should be equally likely, the data should be normally distributed but for more than 30 samples it is okay, sample sizes should be more or less equal.

our data set satisfies all the assumptions

we have got a test statistic score of -6.82

to reject the null hypothesis our test statistic should be less than the critical value of -1.645 at 5% level of significance.

our test statistic is lower than the critical value thus we reject the null hypothesis and accept that the proportion 1 of vaccinated trial is lesser than the proportion 2 non-vaccinated trial thus concluding that the vaccine developed is effective.