Try the fastest way to create flashcards
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

Verified
Step 1
1 of 2

Given:

$x=32$

$n=153$

$c=90\%=0.90$

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

$\hat{p}=\dfrac{x}{n}=\dfrac{32}{153}\approx 0.2092$

For confidence level $1-\alpha=0.90$, determine $z_{\alpha/2}=z_{0.05}$ 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):

$z_{\alpha/2}=1.645$

The margin of error is then:

$E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}=1.645\cdot \sqrt{\dfrac{0.2092(1-0.2092)}{153}}\approx 0.0541$

We want to half the margin of error:

$E_N=\dfrac{0.0541}{2}=0.02705$

The sample size is then (round up!):

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

Recommended textbook solutions

Stats: Modeling the World, AP Edition

4th EditionISBN: 9780133151541David E. Bock, Paul Velleman, Richard D. De Veaux
4,112 solutions

The Practice of Statistics for the AP Exam

6th EditionISBN: 9781319113339Daren S. Starnes, Josh Tabor
2,673 solutions

Updated The Practice of Statistics for the AP Exam

6th EditionISBN: 9781319269296Daren S. Starnes, Josh Tabor
2,577 solutions

The Practice of Statistics for the AP Exam

5th EditionISBN: 9781464108730 (1 more)Daniel S. Yates, Daren S. Starnes, David Moore, Josh Tabor
2,433 solutions