A researcher wishes to estimate, with 99% confidence, the population proportion of adults who are confident with their country's banking system. His estimate must be accurate within 2%
of the population proportion.
(a) No preliminary estimate is available. Find the minimum sample size needed.
(b) Find the minimum sample size needed, using a prior study that found that 25% of the respondents said they are confident with their country's banking system.
Solution
The formula for the minimum sample size required to estimate the proportion is
$$ n =p*(1-p)\bigg(\frac{z}{E}\bigg)^2 $$
where $p$ is the proportion of success, $z$ is the $Z_{\alpha/2}$ and $E$ is the margin of error.
a. No preliminary estimate is available. Thus we assume that $p = 0.5$.
Given that margin of error $E =0.02$. The confidence coefficient is $1-\alpha = 0.99$. Thus $\alpha= 0.01$.
The critical value of $Z$ is $Z_{\alpha/2} = 2.58$.
The minimum sample size required to estimate the proportion is
$$ \begin{aligned} n&= p(1-p)\bigg(\frac{z}{E}\bigg)^2\\ &= 0.5(1-0.5)\bigg(\frac{2.58}{0.02}\bigg)^2\\ &=4160.25\\ &\approx 4161. \end{aligned} $$
Thus, the sample of size $n=4161$ will ensure that the $99$% confidence interval for the proportion will have a margin of error $0.02$.
b. A prior study found that 25% of the respondents said they are confident with their country's banking system. Thus $p=0.25$.
Given that margin of error $E =0.02$. The confidence coefficient is $1-\alpha = 0.99$. That is $\alpha = 0.01$.
The critical value of $Z$ is $Z_{\alpha/2} = 2.58$.
The minimum sample size required to estimate the proportion is
$$ \begin{aligned} n&= p(1-p)\bigg(\frac{z}{E}\bigg)^2\\ &= 0.25(1-0.25)\bigg(\frac{2.58}{0.02}\bigg)^2\\ &=3120.1875\\ &\approx 3121. \end{aligned} $$
Thus, the sample of size $n=3121$ will ensure that the $99$% confidence interval for the proportion will have a margin of error $0.02$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
