Question: Is caffeine addictive? Does the caffeine in coffee, tea, and cola induce an addiction similar to that induced by alcohol, tobacco, heroin, and cocaine? In an attempt to answer this question, researchers at Johns Hopkins University examined 27 caffeine drinkers and found 25 who displayed some type of withdrawal symptoms when abstaining from caffeine. [Note: The 27 caffeine drinkers volunteered for the study.] Furthermore, of 11 caffeine drinkers who were diagnosed as caffeine dependent, 8 displayed dramatic withdrawal symptoms (including impairment in normal functioning) when they consumed a caffeine-free diet in a controlled setting. The National Coffee Association claimed, however, that the study group was too small to draw conclusions. Is the sample large enough to estimate the true proportion of caffeine drinkers who are caffeine dependent to within .05 of the true value with 99% confidence? Explain.

The sample size of caffeine drinkers is 27.

The number of success be 11.

The confidence level is 99%.

The sampling error is 0.05.

The point estimate of the population proportion is obtained below:$$\begin{gathered} \hat{p}=\frac{Number of success in the sample \left(x\right)}{Sample\backslash \mathrm{kern}1pt size \left(n\right)} \\ =\frac{11}{27} \end{gathered}$$

Let the confidence level be 0.99.

$$\begin{gathered} For \left(1-\alpha \right)=0.99 \\ \alpha =0.01 \\ \frac{\alpha }{2}=0.005 \end{gathered}$$

The$Z_{\frac{\alpha }{2}}$obtained from the standard normal table is,

$$\begin{gathered} Z_{\frac{\alpha }{2}}=Z_{0.005} \\ =2.575 \end{gathered}$$

The formula for sample size is given below:

$n=\frac{{\left(Z_{\frac{\alpha }{2}}\right)}^2\left(pq\right)}{{\left(SE\right)}^2}$

Where SE is the sampling error.

The value of the pq is unknown; it can be estimated by using the sample fraction of success,from a prior sample.

Let the sample proportion$\hat{p}$is 0.407.

Here, the value pq is unknown. Which can be obtained by using the sample fraction of success$\hat{p}$,$\hat{p}$.

The product of pq is computed as

$$\begin{gathered} pq=p\left(1-p\right) \\ =\left(0.407\right)\left(1-0.407\right) \\ =\left(0.407\right)\left(0.593\right) \\ =0.241351 \end{gathered}$$

The sample size is computed as

$$\begin{gathered} n=\frac{{\left(2.575\right)}^2\left(0.407\right)\left(0.593\right)}{{\left(0.005\right)}^2} \\ =\frac{1.6003}{0.0025} \\ =640.123 \\ \approx 640 \end{gathered}$$

Hence, the required sample size is 640. So, the sample size is not large enough to estimate the true proportion of caffeine drinkers who are caffeine dependent to be within 0.05 with 99% confidence. Because the pq values are not closed to 0.5.