Rhino viruses typically cause common colds. In a test of the effectiveness of ec

Question

Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 69 of rhe 95 subjects treared with echinacea developed rhinovirus infections. In a placebo group, 83 of the 132 subjects developed rhinovirus infections. Using a 0.06 significance level, test the claim that echinacea has an effect on rhinovirus infections.

a) Determine the critical value.
b) Determine the test statistic.
c) Decide whether or not we reject or fail to reject the null hypothesis.
d) State whether or not there was sufficient evidence to support the claim.
e) Construct the confidence interval about the difference between the two proportions.
    Write the confidence interval as (LowerValue, HigherValue)

critical value =

test statistic =

[Select Decision]We reject the null hypothesisWe fail to reject the null hypothesisWe accept the alternate hypothesis

[Select Conclusion]With a 94% level of confidence, we can say echinacea has an effect on rhinovirus infections.There is not enough evidence to suggest echinacea has an effect on rhinovirus infections.

Confidence Interval about p1-p2:

Explanation / Answer

Hypothesis:
Null hypothesis: P1 = P2
Alternative hypothesis: P1 P2

a) Critical value for z statistic at 0.06 significance level = +/- 1.88
Reject H0: z > 1.88 or z < -1.88

b) p1 = 69/95 = 0.726 , n1 = 95 , p2 = 83 / 132 = 0.628 , n2 = 132

p = (p1 * n1 + p2 * n2) / (n1 + n2)
= [(0.726 * 95) + (0.628 * 132)] / (95 + 132)
= 0.669

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
SE = sqrt [ 0.669 * 0.331 * ( 1/95 + 1/132) ]
= 0.063

z = (p1 - p2) / SE
= (0.726 - 0.628)/0.063
= 1.556

we need to find p value using z = 1.556
p value = 0.119708.

c) we fail to reject the null hypothesis

d) there was not sufficient evidence to support the claim.

e)

z value at 94% CI = +/-1.88
CI = (p1 - p2) + /- z * sqrt ( p1 * (1 - p1) / n1 + p2 * ( 1 - p2) / n2)
= ( 0.726 - 0.628) + /- 1.88 * sqrt (( 0.726 * 0.274) / 95 + (0.628 * 0.372) / 132)
= (-0.018 , 0.214)

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