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Question

Wildlife biologists inspect 153 deer taken by hunters and find 32 of them carrying ticks that test positive for Lyme disease. If the scientists want to cut the margin of error in half, how many deer must they inspect?

Solution

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Given:

x=32x=32

n=153n=153

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

The sample proportion is the number of successes divided by the sample size:

p^=xn=321530.2092\hat{p}=\dfrac{x}{n}=\dfrac{32}{153}\approx 0.2092

For confidence level 1α=0.901-\alpha=0.90, determine zα/2=z0.05z_{\alpha/2}=z_{0.05} using table Z in appendix F (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=1.6450.2092(10.2092)1530.0541E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}=1.645\cdot \sqrt{\dfrac{0.2092(1-0.2092)}{153}}\approx 0.0541

We want to half the margin of error:

EN=0.05412=0.02705E_N=\dfrac{0.0541}{2}=0.02705

The sample size is then (round up!):

n=[zα/2]2p^(1p^)E2=1.6452(0.2092)(10.2092)0.027052612n=\dfrac{[z_{\alpha/2}]^2\hat{p}(1-\hat{p})}{E^2}=\dfrac{1.645^2 (0.2092)(1-0.2092)}{0.02705^2}\approx 612

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