5 You are on the staff of a congresswoman She wants to poll

5. You are on the staff of a congresswoman. She wants to poll her constituency on a controversial tax increase. To get a good sense as to how the citizens of her district (which has over 1 million registered voters) feel, you arrange for a poll of 8250 people in the district. Of those surveyed, 4205 say they are in favor of the tax increase. What hypothesis test should you set up? Given that politicians are notoriously cautious of voting for tax increases, what level of significance will you use in your test? Do the results of the survey represent statistically significant evidence a majority of the district favors the tax increase? What do you recommend to the congresswoman? Show all work, especially calculator functions.

Solution

We setup a hypothesis test if the proportion is greater than 0.5.

We choose 0.01 level of significance, as they are cautious.

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   <=   0.5
Ha:   p   >   0.5
As we see, the hypothesized po =   0.5      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.50969697      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.005504819      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    1.761542024      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.039073353      

significance level =    0.01      

As P > 0.01 we   FAIL TO REJECT THE NULL HYPOTHESIS.  

Thus, there is no significant evidence that a mjority will vote for the tax increase at 0.01 level. [CONCLUSION]

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You can use 1-propZTest if you have a ti-84 to get the right tailed test here.