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Anton [14]
3 years ago
13

"a pollster wishes to estimate the number of left-handed scientists. how large of a sample is needed in order to be 98% confiden

t that the sample proportion will no differ from the mean by more than 4%. also suppose we know a previous study concluded that the proportion of left-handed scientists is 8%"
Mathematics
1 answer:
nadezda [96]3 years ago
4 0
The sample size should be 250.

Our margin of error is 4%, or 0.04.  We use the formula

\text{ME}=z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

To find the z-score:
Convert 98% to a decimal:  0.98
Subtract from 1:  1-0.98 = 0.02
Divide both sides by 2:  0.02/2 = 0.01
Subtract from 1:  1-0.01 = 0.99

Using a z-table (http://www.z-table.com) we see that this value has a z-score of approximately 2.33.  Using this, our margin of error and our proportion, we have:

0.04=2.33\sqrt{\frac{0.08(1-0.08)}{n}}
\\
\\0.04=2.33\sqrt{\frac{0.08(0.92)}{n}}

Divide both sides by 2.33:
\frac{0.04}{2.33}=\sqrt{\frac{0.08(0.92)}{n}}

Square both sides:
(\frac{0.04}{2.33})^2=\frac{0.08(0.92)}{n}

Multiply both sides by n:
(\frac{0.04}{2.33})^2n=0.08(0.92)

Divide both sides to isolate n:
\frac{0.08(0.92)}{(\frac{0.04}{2.33})^2}=n
\\
\\249.73=n
\\n\approx 250
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