In a survey of 1070 Ann Arbor residents, 59% supported a ban on bicycles on downtown sidewalks in certain areas with high pedestrian traffic. A city administrator wants to determine if a majority of all Ann Arbor residents support such a ban. She decides to conduct the hypotheses $H_0:\; p = 0.50$ versus $H_a:\; p > 0.50$ at a 5% significance level.

Calculate the appropriate test statistic value for testing the majority hypothesis.

Is it just $$z = \frac{0.59-0.50}{\sqrt{0.50(1-0.50)\big/1070}} = 5.89?$$

I'm confused, because it seems that the z value is very high. Any ideas?


Your z value would have to be about that, yes.

Recall that the margin of error on a sample of 1000 with a binomial is only a little over 3%. At 9%, you have almost 3 times that, - which means it must be nearly 6 s.d.'s, as you saw.

You have a nice large sample, so as long as it's a random sample (or is otherwise constructed to have similar properties to one), and the other assumptions you need to hold do in fact hold, then yes, everything there looks correct.


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