Question

1 In a graduate teacher college, a survey was conducted to determine the proportion of students who want to

major in Mathematics. If 378 out of 900 students said Yes, with 95% confidence, what interpretation can we

make regarding the probability that all students in the teacher graduate college want to major in Mathematics.

Answer 1

The sample proportion is computed as follows, based on the sample size $n = 900$ and the number of favorable cases $X = 378:$

$$\hat{p}=\dfrac{X}{n}=\dfrac{378}{900}=0.42$$

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:

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

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

$$=(0.42-1.96\sqrt{\dfrac{0.42(1-0.42)}{900}},$$

$$0.42+1.96\sqrt{\dfrac{0.42(1-0.42)}{900}})$$

$$=(0.388,0.452)$$

Therefore, based on the data provided, the 95% confidence interval for the population proportion is $0.388 < p < 0.452,$ which indicates that we are 95% confident that the true population proportion $p$ is contained by the interval $(0.388, 0.452).$