Question

Short answer

The 95% confidence interval of the percentage of green candies is equal to (11.3%, 26.7%).

As the value of 16% lies in the interval, there is sufficient evidence to support the claim of 16% green candies.

Step-by-step solution

Given information

In a sample of 100 candies, 19 are green. It is claimed that 16% of the candies are green.

Expression of the confidence interval

The confidence interval for the population proportion is shown below:

\(\hat p - E < p < \hat p + E\)

Here, E is the margin of error. It has the following formula:

\(E = {z_{\frac{\alpha }{2}}} \times \sqrt {\frac{{\hat p\hat q}}{n}} \)where

\(\hat p\)is the sample proportion of adults who have Facebook pages

\(\hat q\)is the sample proportion of adults who do not have Facebook pages

n is the sample size

\({z_{\frac{\alpha }{2}}}\) is the one-tailed critical value of z.

Sample size and the sample proportions

The sample size (n) is equal to 100.

The sample proportion of green candiesis equal to:

\(\begin{array}{c}\hat p = \frac{{19}}{{100}}\\ = 0.19\end{array}\)

The sample proportion of green candies is equal to:

\(\begin{array}{c}\hat q = 1 - \hat p\\ = 1 - 0.19\\ = 0.81\end{array}\)

Step 4:Find the margin of error

The value of\({z_{\frac{\alpha }{2}}}\)for\(\alpha = 0.05\)is equal to 1.96.

The margin of error can be computed as follows:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}} \times \sqrt {\frac{{\hat p\hat q}}{n}} \\ = 1.96 \times \sqrt {\frac{{0.19 \times 0.81}}{{100}}} \\ = 0.0769\end{array}\)

Compute the confidence interval

The confidence interval estimate of the population proportion of green candies is computed as follows:

\(\begin{array}{c}\hat p - E < p < \hat p + E\\0.19 - 0.0769 < p < 0.19 + 0.0769\\0.113 < p < 0.267\\11.3\% < p < 26.7\% \end{array}\)

The 95% confidence interval is equal to (11.3%, 26.7%).

Conclusion

Since the interval contains the value of 16%, it can be said the claim of 16% green candies is supported.