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 24% of the respondents said they are confident with their country's banking system.

c. Compare the results from parts (a) and (b).

#### Solution

The formula to estimate the 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 estimates is available. Assume that $p =0.5$.

Given that margin of error $E =0.02$. The confidence coefficient is $0.99$.

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. Given that $p = 0.24$.

Given that margin of error $E =0.025$. The confidence coefficient is $0.99$.

The critical value of $Z$ is `$Z_{\alpha/2} = 2.58$`

.

The minimum sample size required to estimate the proportion is

` $$ \begin{aligned} n&= p(1-p)\bigg(\frac{z}{E}\bigg)^2\\ &= 0.24(1-0.24)\bigg(\frac{2.58}{0.025}\bigg)^2\\ &=1942.6038\\ &\approx 1943. \end{aligned} $$ `

Thus, the sample of size $n=1943$ will ensure that the $99$% confidence interval for the proportion will have a margin of error $0.025$.

c. Having an estimate of the population proportion reduces the minimum sample size needed.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators