# How to test if a sample proportion matches a population proportion, should I use the z-test or the t-test?

For (a), $H_0 = 0.75$ and $H_a \ne 0.75$.

For (b), how can I get an answer with a $z$-value? Why can't I use a $t$-value to answer this question? Is the idea: $p \pm Z (p(1-p)/N)^{0.5}$?

• How would you calculate the sample standard deviation for a t-statistic (and while that's possible, if non-obvious -- why would the result have a t-distribution)? Note that "infected" and "not-infected" are two states, so the sample observations are not drawn from a normal distribution. Please show your actual reasoning or an actual attempt at the question. Why do you think it's a Z -- are you working from some answers? (If so I suggest you put the answers away, and try to solve these without ever looking at the answers, by using the information you have been supplied in the subject) May 17, 2016 at 0:46

Proportions are a little different in that the standard deviation of the hypothetical curve with mean $p$ can be calculated by $\sqrt{\frac{p(1-p)}{n}}$. (This has to do with the Central Limit Theorem.) The z-score of your hypothesis test is calculated by $\frac{\hat p - p}{\sqrt{\frac{p(1-p)}{n}}} = z$. You can use the $z$ value you calculate to find a P-value and reject or fail to reject $H_0$
• @sss please show your working. I get $z=-1.756$ using Zach's formula. My guess is you used the sample proportion in the denominator; in Zach's formula $\hat{p}$ is the sample proportion and $p$ is the hypothesized proportion. Note there are no $\hat{\ }$'s over the $p$'s in the denominator May 17, 2016 at 1:07
• Remember, $p$ is the population parameter (the thing you're hypothesizing; 0.75.) $\hat p$ is the sample you took of your population (0.716) May 17, 2016 at 1:13
• If you're talking about a confidence interval (a range for $p$), that is different than a hypothesis test. The OP asked if there is evidence, so all you need to do is a hypothesis test. May 17, 2016 at 1:55