Question

In a study designed to test the effectiveness of echinacea for treating upper respiratory tract infections
in children, 337 children were treated with echinacea and 370 other children were given a placebo.
The number of days of peak severity of symptoms for the echinacea group had a mean of 6.0 days
and a standard deviation of 2.3 days. The numbers of days of peak severity of symptoms for the
placebo group had a mean of 6.1 days and a standard deviation of 2.4 days.

(a) Construct the 95% confidence interval for the mean number of days of peak severity of symp-
toms for those who receive echinacea treatment. Use tα/2 = 1.969. Interpret the confidence

interval.

(b) Construct the 95% confidence interval for the mean number of days of peak severity of symp-
toms for those who are given a placebo. Use tα/2 = 1.969. Interpret the confidence interval.

(c) Compare the two confidence intervals. What do the results suggest about the effectiveness of
echinacea? Justify your answer based on the confidence intervals.

Answer 1

Answer:

Given,

Sample n1 = 337, n2 = 370

x1 = 6 , x2 = 6.1

standard deviation s1 = 2.3 , s2 = 2.4

a)

Degree of freedom = n1 - 1

= 337-1

= 336

At 95% CI, t(alpha/2) = 1.969

Interval = x1 +/- t*s1/sqrt(n1)

substitute the given values

= 6 +/- 1.969*2.3/sqrt(337)

= 6 +/- 0.2467

= (5.7533 , 6.2467)

b)

Now here t = 1.969

Interval = x1 +/- t*s1/sqrt(n1)

substitute the given values

= 6.1 +/- 1.969*2.4/sqrt(370)

= 6.1 +/- 0.2457

= (5.8543 , 6.3457)

c)

Here both the intervals are nearly equal. So echinacea treatment doesn't appear to be effective.