Question

7. 116 ptsl The Salk polio vaccine trial was carried out in 1954. The table below shows the incidence of polio accine and half who did rn Polio Victim? Yes No 142 199,858 Placebo Sal k Vaccine 56 199,944 (a) [6 pts] Calculate a 95% confidence interval for the difference in the proportion of children who got polio between the placebo and vaccine groups (in this order). Show your calculations. Interpret the interval. (b) [8 ptsl Carry out a statistical test to assess whether or not rates of polio differ between the placebo and vaccine groups. (Start with stating the hypotheses, then find the test statistic, then calculate the p-value, finish with an appropriate conclusion based on your findings.) (c) 12 pts] Are all the necessary conditions of the procedures you used in (a) and (b) satisfied? Explain.

Answer 1

$\hat{p}_1$ = 142/200000 = 0.00071 and 1 - $\hat{p}_1$ = 0.99929

$\hat{p}_2$ = 0.00028 and 1 - $\hat{p}_2$ = 0.99972

The Zcritical (2 tail) for $\alpha$ = 0.05, is 1.96

The Confidence Interval is given by (
P1
- $\hat{p_2}$ ) $\pm$ ME, where

(
P1
- $\hat{p_2}$ ) = 0.00071 – 0.00028 = 0.00043

0.00071 0.99929 0.00028 0.99972 -Zcritical* = 1.96* 200000+ 7l 200000 0.00014

The Lower Limit = 0.00043 - 0.00014 = - 0.00029

The Upper Limit = 0.00043 + 0.00014 = 0.00057

The 95% Confidence Interval is 0.00029 < p1 - p2 < 0.00057

We are 95% confident that the population difference in proportion between those who got polio after vaccination and placebo lies with the limits from 0.00029 to 0.00057.

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We take Alpha = 0.05 as the default level

The Hypothesis:

H0: p1 = p2 : The proportion of children who get polio after vaccination is equal to the the proportion who get polio after being given a placebo.

Ha: p1 $\neq$ p2 : The proportion of children who get polio after vaccination is different from the the proportion who get polio after being given a placebo.

The Test Statistic:

pl - p2 p( 1-מ()-+ 0.00071一0.00028 = 6.06 、 /0.0005* 0.9995 * ) 20000 ( 20000 )

The p Value: The p value (2 Tail) for Z = 6.06, is; p value = 0.000

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the proportion of children who get polio after vaccination is different from the the proportion who get polio after being given a placebo.

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(c) Yes, All the conditions are satisfied.

(1) n * $\hat{p}_1$ , n * (1 - $\hat{p}_1$ ), n * $\hat{p}_2$ and n * (1 - $\hat{p}_2$ ) are all greater than 10

(2) The sample can be assumed to be a simple random samples.

(3) The samples can be assumed to be independent of each other.

(4) The sample size is less than 5% of the total population.

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