Answer to Question #348086 in Statistics and Probability for Bianca

Question

Two hundred graduating students are selected and it is found that 114 will be awarded the first class honors degrees. calculate the 95% confidence interval for the proportion of the graduating students who will receive first class honor degrees

Answer 1

The critical value for $\alpha = 0.05$ is $z_c = z_{1-\alpha/2} = 1.96.$

$$\hat{p}=\dfrac{X}{N}=\dfrac{114}{200}=0.57$$

The corresponding confidence interval is computed as shown below:

$$CI(Proportion)=(\hat{p}-z_c\times \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},$$ $$\hat{p}+z_c\times \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})$$ $$=(0.57-1.96\times \sqrt{\dfrac{0.57(1-0.57)}{200}},$$ $$0.57+1.96\times \sqrt{\dfrac{0.57(1-0.57)}{40}})$$ $$=(0.5014, 0.6386)$$

Therefore, based on the data provided, the 95% confidence interval for the population proportion is $0.5014 < p < 0.6386,$ which indicates that we are 95% confident that the true population proportion $p$ is contained by the interval $(0.5014, 0.6386).$