Answer to Question #348087 in Statistics and Probability for Bianca

Question

200 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 honors 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 $(0.5014, 0.6386).$