In 1970, 59% of college freshmen thought that there should be capital punishment , in 2005 a random sample of 100 freshmen were asked whether there should be capital punichment or not, among these, 35 of them said no.
Question 1). Is there a significant change in the opinion of students in 2005 since 1970.
The problem that I have for question 1 is that I don't know what fractions to use for the SD. I don't know if it should be $\sqrt{0.59\cdot\ 0.61}$ or $\sqrt{0.35\cdot\ 0.65}$ in these type of problem how do I know which percentage to multiply ?
Question 2). Construct a 95% confidence interval for the proportion of all college freshmen who are against capital punishment in 2005.
For question 2 I have similar issue with picking the correct percentage, I'm not sure if SE =$\sqrt{\frac{0.35\cdot\ 0.65}{100}}$ or SE= $\sqrt{\frac{0.59\cdot\ 0.61}{100}}$ ?
Any advice? Thank you!
Let $$H_0:p=0.59$$
$$H_a:p\neq0.59$$
Under the null hypothesis we have that
$$\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\sim\mathsf N(0,1)$$
When building confidence intervals we have that a $100(1-\alpha)$% confidence interval for $p$ is given by
$$\left(\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)$$
where in our case $z_{\alpha/2}\approx1.96$