Question

help with sketching rejection region corresponding to level of significance?

Answer 1

Hello! I'm trying to figure out how to sketch the rejection region corresponding to level of significance of 0.05.
Here is the problem:
"A pollster states that a majority of a city's voters will vote for a new stadium. A random sample of 400 voters found that 53% would vote for the new stadium. Sketch the region corresponding to level of significance 0.05 and include any critical value(s) in this sketch."

Answer 2

I have found, so far that n=400, σ= 9.98, α =0.05, p=0.53, q =0.47.. where to from here? I can't seem to figure out the degree of freedom and I think that's the next step.

Answer 3

You're setting up a hypothesis test for a proportion, which suggests a binomial distribution (or normal, if the sample is considered large enough). Degrees of freedom are typically associated with a chi square distribution.

First, you have to set up the null and alternative hypotheses. The pollster is claiming that \(\textbf{a majority}\) of voters will vote for the new stadium, which means the claim is that the proportion of voters \(p\) is greater than \(0.5\) (assuming anything greater than 50% is considered a majority). To test the claim, the pollster would sample the voters' opinions to find that \(53\%=0.53\) would vote yes.

So, the null hypothesis is that \(p_0>0.5\), which means the alternative hypothesis is that \(p_0\le0.5\). The sample proportion is given by \(\hat{p}=0.53\).

The rejection region is the part of the distribution that disagrees with the null hypothesis, which here means that the test statistic \(Z\) is less than the critical value \(Z_\alpha\). This critical value is associated with the significance level. It's the value \(k\) such that \(P(Z<k)=\alpha\), which in this case means \(Z_\alpha\approx-1.645\) since \(P(Z<-1.645)=0.05\). The \(\textbf{rejection region}\) itself is \(Z<Z_\alpha\).

Next, you would determine the value of \(Z\) and compare it to \(Z_\alpha\), where
\[Z=\frac{n\hat{p}-np_0}{\sqrt{np_0(1-p_0)}}=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]and come to some conclusion about your hypotheses.