In a Gallup poll, 1025 randomly selected adults were surveyed and 29% of them said that they used the internet for shopping at least a few times a year.
a. Find the point estimate of the percentage of adults who use the internet for shopping.
b. Find a 99% confidence interval estimate of the percentage of adults who use the internet for shopping.
Solution
a. Given that sample size $n = 1025$.
The sample proportion is $\hat{p}=0.29$.
So the point estimate of the percentage of adults who use the internet for shopping is $\hat{p}=0.29$.
b. For 99% confidence interval estimate of the percentage of adults who use the internet for shopping
Step 1 Specify the confidence level $(1-\alpha)$
Confidence level is $1-\alpha = 0.99$. Thus, the level of significance is $\alpha = 0.01$.
Step 2 Given information
Given that sample size $n =1025$.
The estimate of the proportion of success is $\hat{p} =0.29$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval for population proportion is
$$ \begin{aligned} & \hat{p} - E \leq p \leq \hat{p} + E. \end{aligned} $$
where
$$ \begin{aligned} E=Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \end{aligned} $$
and $Z_{\alpha/2}$ is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.
Step 4 Determine the critical value
The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.
Thus $Z_{\alpha/2} = Z_{0.005} = 2.576$.
Step 5 Compute the margin of error
The margin of error for proportions is
$$ \begin{aligned} E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}\\ & = 2.576 \sqrt{\frac{0.29*(1-0.29)}{1025}}\\ & =0.037. \end{aligned} $$
Step 6 Determine the confidence interval
$99$% confidence interval estimate for population proportion is
$$ \begin{aligned} \hat{p} - E & \leq p \leq \hat{p} + E\\ 0.29 - 0.037 & \leq p \leq 0.29 + 0.037\\ 0.2535 & \leq p \leq 0.3265. \end{aligned} $$
Thus, $99$% confidence interval estimate for population proportion $p$ is $(0.2535,0.3265)$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
