Question

Flu vaccine Refer to Exercise$R9.5$. Of the $1000$ adults who were given the vaccine, $43$ got the flu. Do these data provide convincing evidence to support the company’s claim?

Short answer

There is not sufficient evidence to support the claim.

Step-by-step solution

Step 1:Given information

$\text{No of adults}\left(x\right):1000$

$\text{No of peoples got flu}\left(n\right):43$

Step 2:Calculation

So finding out the hypotheses:

$H_0:p=5\%=0.05$

$H_a=p<0.05$

The sample proportion is the no. of successes divided by the sample size:

$p=\frac{x}{n}=\frac{43}{1000}=0.043$

Determine the value of test statistic:

$z=\frac{\hat{p}-p_o}{\sqrt{\frac{p_o\left(1-p_o\right)}{n}}}$

$z=\frac{0.043-0.05}{\sqrt{\frac{0.05(1-0.05)}{1000}}}$

$z\approx -1.02$

$P$- value is the probability of obtaining the value of test statistic, or a value more extreme.

$\text{For determining the}P\text{- value in table}A\text{:}$

$P=P(z<-1.02)=0.1539$

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

$P>0.05\text{, thus fails to reject the}{H}_0\text{.}$