Question

Chocolate and Happiness Use the results from part (b) of Cumulative Review Exercise 2 to construct a 99% confidence interval estimate of the percentage of women who say that chocolate makes them happier. Write a brief statement interpreting the result.

Short answer

99% confidence interval for the percentage of women who feel chocolate makes them happier is equal to (82.8%,87.2%).

There is 99% confidence that the actual percentage of women who feel chocolate makes them happier will lie between 82.8% and 87.2%.

Step-by-step solution

Given information

It is given that a survey was sponsored by a chocolate company.

Out of 1708 women who were surveyed, 85% of them said that chocolate made them happier.

Formula of the confidence interval for the single proportion

The formula to construct the confidence interval for the population proportion of women who said that chocolate made them happier is written below:

\(CI = \hat p \pm E\)

Where,

\(\hat p\)be the sample proportion of women who said that chocolate made them happier;

Eis the margin of error and has the given formula:

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

Sample proportions and sample size

Here,

\[\begin{aligned}{c}\hat p = 85\% \\ = 0.85\end{aligned}\]

And,

\(\begin{aligned}{c}\hat q = 1 - \hat p\\ = 1 - 0.85\\ = 0.15\end{aligned}\)

The sample size,n is equal to 1708.

Level of significance

The confidence level is given to be equal to 99%.

Thus, the level of significance is computed as shown:

\(\begin{aligned}{c}{\rm{Confidence}}\;{\rm{Level}} = 100\left( {1 - \alpha } \right)\\100\left( {1 - \alpha } \right) = 99\\1 - \alpha = 0.99\\\alpha = 0.01\end{aligned}\)

Critical value

From the standard normal table, the two-tailed critical value of z at 0.01 level of significance is equal to 2.5758.

Confidence interval

The 99% confidence interval is computed as follows:

\(\begin{aligned}{c}CI = \hat p \pm E\\ = 0.85 \pm {z_{\frac{{0.01}}{2}}}\sqrt {\frac{{0.85\left( {0.15} \right)}}{{1708}}} \\ = 0.85 \pm \left( {2.5758} \right)\left( {0.0086} \right)\\ = \left( {0.828,0.872} \right)\end{aligned}\)

Therefore, the 99% confidence interval for the percentage of women who feel chocolate makes them happier is equal to (82.8%,87.2%).

Interpretation

We are 99% of the time confident that the actual percentage of women who feel chocolate makes them happier will lie between 82.8% and 87.2%.