Question

X=842, n=1200, 95% confidence

Answer 1

The sample proportion is computed as follows, based on the sample size $N = 1200$ and the number of favorable cases $X = 842:$

$$\hat{p}=\dfrac{X}{N}=\dfrac{842}{1200}\approx0.701667$$

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

The corresponding confidence interval is computed as shown below:

$$CI(Proportion)=(\hat{p}-z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},$$

$$\hat{p}+z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})$$

$$=(0.701667-1.96\sqrt{\dfrac{0.701667(1-0.701667)}{1200}},$$

$$0.701667+1.96\sqrt{\dfrac{0.701667(1-0.701667)}{1200}})$$

$$=(0.615,0.788)$$

Therefore, based on the data provided, the 95% confidence interval for the population proportion is $0.615 < p < 0.788,$ which indicates that we are 95% confident that the true population proportion $p$ is contained by the interval $(0.615, 0.788).$