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The test for the effectiveness of Echinacea involves two groups:

In the treatment group, 40 of 45 subjects developed infections.

In the placebo group, 88 of 103 subjects developed infections.

The significance level is \(\alpha = 0.05\) .

a.

To test the effectiveness, let\({p_1},{p_2}\)be the proportion of subjects who develop infections in the treatment and subject groups, respectively.

From the claim, the null and alternative hypotheses are as follows.

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\,\\\,{H_1}:{p_1} \ne {p_2}\end{array}\)

From the given information, summarize the following:

\(\begin{array}{l}{n_1} = 45\\{x_1} = 40\,\\{n_2} = 103\,\\{x_2} = 88\end{array}\)

The sample proportions are as follows.

\(\begin{array}{c}{{\hat p}_1} = \frac{{{x_1}}}{{{n_1}}}\\ = \frac{{40}}{{45}}\\ = 0.8889\end{array}\)

and

\(\begin{array}{c}{{\hat p}_2} = \frac{{{x_2}}}{{{n_2}}}\\ = \frac{{88}}{{103}}\\ = 0.8544\end{array}\).

The sample pooled proportions are calculated as follows.

\(\begin{array}{c}\bar p = \frac{{\left( {{x_1} + {x_2}} \right)}}{{\left( {{n_1} + {n_2}} \right)}}\,\\ = \frac{{\left( {40 + 88} \right)}}{{\left( {45 + 103} \right)}}\\ = 0.8649\end{array}\)

and

\(\begin{array}{c}{\rm{\bar q}} = 1 - {\rm{\bar p}}\\ = 1 - 0.8649\\ = 0.1351\end{array}\)

To conduct a hypothesis test of two proportions, the test statistic is computed as

\(z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\left( {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} \right)} }}\).

Here, \({\rm{\bar p}}\)is the pooled sample proportion, and\({\rm{\bar q}} = 1 - {\rm{\bar p}}\)

Substitute the values. So,

\(\begin{array}{c}z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\left( {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} \right)} }}\\ = \frac{{\left( {0.8889 - 0.8544} \right) - 0}}{{\sqrt {\left( {\frac{{0.8649 \times 0.1351}}{{45}} + \frac{{0.8649 \times 0.1351}}{{103}}} \right)} }}\\ = 0.565\end{array}\).

The value of the test statistic is 0.565.

Referring to the standard normal table for the positive z-score of 0.7157, the cumulative probability of 0.57 is obtained from the cell intersection for row 0.5 and the column value of 0.07.

For the two-tailed test, the p-value is twice the area to the right of the test statistic, as shown below.

\(\begin{array}{c}2 \times P\left( {Z > 0.57} \right) = 2 \times \left( {1 - P\left( {Z < 0.57} \right)} \right)\\ = 2 \times \left( {1 - 0.7157} \right)\\ = 0.5686\end{array}\).

Thus, the p-value is 0.569.

As the p-value is greater than 0.05, the null hypothesis is failed to be rejected. Thus, there is not enough evidence to support the claim that Echinacea is effective on rhinovirus infections.

b.

The general formula for the confidence interval of the difference of proportions is as follows.

\({\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E\,\,,\,\,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\)\(\)

Here, E is the margin of error calculated as follows.

\({\rm{E}} = {z_{\frac{\alpha }{2}}} \times \sqrt {\left( {\frac{{{{\hat p}_1} \times {{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2} \times {{\hat q}_2}}}{{{n_2}}}} \right)} \)

For the two-tailed test with a significance level of 0.05, the confidence level will be 95%.

The critical value\({z_{\frac{\alpha }{2}}}\)has the cumulative area to its left as\(1 - \frac{\alpha }{2}\).

Mathematically,

\(\begin{array}{c}P\left( {Z < {z_{\frac{\alpha }{2}}}} \right) = 1 - \frac{\alpha }{2}\\P\left( {Z < {z_{0.025}}} \right) = 0.975\end{array}\)

From the standard normal table, the area of 0.975 is observed corresponding to the intersection of the row value 1.9 and the column value 0.06 as 1.96.

The margin of error is as follows.

\(\begin{array}{c}{\rm{E}} = {z_{\frac{\alpha }{2}}} \times \sqrt {\left( {\frac{{{{\hat p}_1} \times {{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2} \times {{\hat q}_2}}}{{{n_2}}}} \right)} \\ = 1.96 \times \sqrt {\left( {\frac{{0.8889 \times 0.1111}}{{45}} + \frac{{0.8544 \times 0.1456}}{{103}}} \right)} \\ = 0.1143\end{array}\).

Substitute the value of E as follows.

\(\begin{array}{c}{\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E\,\,,\,\,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\\ = \left( {\left( {0.8889 - 0.8544} \right) - 0.1143\,,\,\left( {0.8889 - 0.8544} \right) + 0.1143} \right)\\ = \left( { - 0.0798\,\,,\:0.149} \right)\end{array}\).

Thus, the 95% confidence interval for two proportions is\( - 0.0798 < \left( {{p_1} - {p_2}} \right) < 0.149\).

As 0 is included in the interval, there is not sufficient evidence to support the effectiveness of Echinacea.

c.

From the results, Echinacea does not appear to have a significant effect on reducing the infection rate of rhinovirus.