Question 4 (3 points). A pharmaceutical company has produced a flu vaccine. In a

Question

Question 4 (3 points). A pharmaceutical company has produced a flu vaccine. In a test of its effectiveness, 1,000 people were randomly selected; of these, 500 were injected with the vaccine, and the other 500 went untreated. The number of people in each group who contacted the flu during the next three months is summarized in the table below NUMBER OF PEOPLE CONDITION Developed the flu Did not develop the flu Treated with Vaccine 65 435 Untreated 125 375 Does this study provide significant evidence that the vaccine is effective in preventing the flu? In other words, do proportionally fewer people develop the flu when treated with vaccine? Conduct a complete and appropriate hypothesis test using = .05, the p-value approach, and 5-step procedure discussed in class Step 1: State the null and alternative hypotheses Step 2: State the decision rule for rejecting the null hypothesis Step 3: State the p-value of the hypothesis test Step 4: Evaluate the null hypothesis Step 5: State the practical conclusion(s) to be drawn from the hypothesis test, in the context of the problem, in plain English 4b. Construct a 95% confidence interval estimate of the difference in the proportion of people who do and do not develop the flu when treated with vaccine. Make sure that your interval estimate is in the correct format, as discussed in class. Interpret the practical meaning of the resulting interval estimate, in plain English (i.e., in the context of the problem)

Explanation / Answer

Q4a.

Given table data is as below

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calculation formula for E table matrix

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expected frequecies calculated by applying E - table matrix formulae

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calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above

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set up null vs alternative as

null, Ho: no evidence that study provide significant evidence that vaccine is effective in preventing flu

alternative, H1: have evidence that study provide significant evidence that vaccine is effective in preventing flu

level of significance, = 0.05

from standard normal table, chi square value at right tailed, ^2 /2 =3.8415

since our test is right tailed,reject Ho when ^2 o > 3.8415

we use test statistic ^2 o = (Oi-Ei)^2/Ei

from the table , ^2 o = 23.3918

critical value

the value of |^2 | at los 0.05 with d.f (r-1)(c-1)= ( 2 -1 ) * ( 2 - 1 ) = 1 * 1 = 1 is 3.8415

we got | ^2| =23.3918 & | ^2 | =3.8415

make decision

hence value of | ^2 o | > | ^2 | and here we reject Ho

^2 p_value =0

ANSWERS

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null, Ho: no evidence that study provide significant evidence that vaccine is effective in preventing flu

alternative, H1: have evidence that study provide significant evidence that vaccine is effective in preventing flu

test statistic: 23.3918

critical value: 3.8415

p-value:0

decision: reject Ho

have evidence that study provide significant evidence that vaccine is effective in preventing flu

Q4b.

TRADITIONAL METHOD
given that,
sample one, x1 =65, n1 =500, p1= x1/n1=0.13
sample two, x2 =435, n2 =500, p2= x2/n2=0.87
I.
standard error = sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where
p1, p2 = proportion of both sample observation
n1, n2 = sample size
standard error = sqrt( (0.13*0.87/500) +(0.87 * 0.13/500))
=0.0213
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
margin of error = 1.96 * 0.0213
=0.0417
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.13-0.87) ±0.0417]
= [ -0.7817 , -0.6983]
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DIRECT METHOD
given that,
sample one, x1 =65, n1 =500, p1= x1/n1=0.13
sample two, x2 =435, n2 =500, p2= x2/n2=0.87
CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.13-0.87) ± 1.96 * 0.0213]
= [ -0.7817 , -0.6983 ]
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interpretations:
1) we are 95% sure that the interval [ -0.7817 , -0.6983] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the difference between
true population mean P1-P2

we have significant evidence that the vaccine is eflèctive in preventing the flu

Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 65 95 -30 900 9.4737 125 95 30 900 9.4737 435 405 30 900 2.2222 375 405 -30 900 2.2222 ^2 o = 23.3918

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