Question

A 2011 Gallup poll found that 76% of Americans believe that high achieving high school students should be recruited to become teachers. This poll was based on a random sample of 1002 Americans. a) Find a 90% confidence interval for the proportion of Americans who would agree with this. b) Interpret your interval in this context. c) Explain what "90% confidence" means. d) Do these data refute a pundit 's claim that 2/ 3 of Americans believe this statement? Explain.

Solution

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Step 1

1 of 5

Given:

n=1002n=1002

p^=76%=0.76\hat{p}=76\%=0.76

c=90%=0.90c=90\%=0.90

(a) For confidence level 1α=0.901-\alpha=0.90, determine zα/2=z0.05z_{\alpha/2}=z_{0.05} using the normal probability table in the appendix (look up 0.05 in the table, the z-score is then the found z-score with opposite sign):

zα/2=1.645z_{\alpha/2}=1.645

The margin of error is then:

E=zα/2p^(1p^)n=1.645×0.76(10.76)10020.0222E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}=1.645\times \sqrt{\dfrac{0.76(1-0.76)}{1002}}\approx 0.0222

The boundaries of the confidence interval are then:

p^E=0.760.0222=0.7378=73.78%\hat{p}-E=0.76-0.0222= 0.7378=73.78\%

p^+E=0.76+0.0222=0.7822=78.22%\hat{p}+E=0.76+0.0222= 0.7822=78.22\%

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