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(a):

The point of placebo is for a child not to know if he actually had taken the treatment or not; thus, by giving both injection and spray the child doesn't know which one is the child taking (receiving).

Because of the placebo effect

Given:

\(\)\(\)\(\begin{array}{l}{{\hat p}_1} = 3.9\% = 0.039\\{{\hat p}_2} = 8.6\% = 0.086\\{n_1} = {n_2} = 4000\end{array}\)

Let us assume:\(\alpha = 0.05\)

Claim: superior

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\\{H_a}:{p_1} < {p_2}\end{array}\)

The sample proportion is the number of successes divided by the sample size:

\(\begin{array}{l}{x_1} = {{\hat p}_1}{n_1} = 0.039(4000) = 156\\{x_1} = {{\hat p}_1}{n_1} = 0.086(4000) = 344\\{{\hat p}_p} = \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}} = \frac{{156 + 344}}{{8000}} = \frac{{500}}{{8000}} \approx 0.0625\end{array}\)

Determine the value of the test statistic:

\(z = \frac{{{{\hat p}_1} - {{\hat p}_2}}}{{\sqrt {{{\hat p}_p}\left( {1 - {{\hat p}_p}} \right)} \sqrt {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} }} = \frac{{0.039 - 0.086}}{{\sqrt {0.0625(1 - 0.0625)} \sqrt {\frac{1}{{4000}} + \frac{1}{{4000}}} }} \approx - 8.68\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming th hypothesis is true. Determine the P-value using table the normal probability table in the appendix

\(P = P(Z < - 8.68) \approx 0\)

If the P-value is smaller than the significance level, then reject the null hypothesis:

\(P < 0.05 \Rightarrow Reject {H_0}\)

There is sufficient evidence to support the claim that the vaccine appears to be superior to the other

1. Because of the placebo effect.
2. There is sufficient evidence to support the claim that the vaccine appears to be superior to the other.