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zzz [600]
3 years ago
14

A survey of shoppers is planned to determine what percentage use credit cards. prior surveys suggest​ 63% of shoppers use credit

cards. how many randomly selected shoppers must we survey in order to estimate the proportion of shoppers who use credit cards to within​ 4% with​ 95% confidence?
Mathematics
1 answer:
Alona [7]3 years ago
3 0
We would need a sample size of 560.

We first calculate the z-score associated. with this level of confidence:
Convert 95% to a decimal:  95% = 95/100 = 0.95
Subtract from 1:  1-0.95 = 0.05
Divide by 2:  0.05/2 = 0.025
Subtract from 1:  1-0.025 = 0.975

Using a z-table (http://www.z-table.com) we see that this is associated with a z-score of 1.96.

The margin of error, ME, is given by:
ME=z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

We want ME to be 4%; 4% = 4/100 = 0.04.  Substituting this into our equation, as well as our proportion and z-score,
0.04=1.96\sqrt{\frac{0.63(1-0.63)}{n}}
\\
\\\text{Dividing both sides by 1.96,}
\\
\\\frac{0.04}{1.96}=\sqrt{\frac{0.63(0.37)}{n}}
\\
\\\text{Squaring both sides,}
\\
\\(\frac{0.04}{1.96})^2=\frac{0.63(0.37)}{n}
\\
\\\text{Multiplying both sides by n,}
\\
\\n(\frac{0.04}{1.96})^2=0.63(0.37)
\\
\\n(\frac{0.04}{1.96})^2=0.2331
\\
\\\text{Isolating n,}
\\
\\n=\frac{0.2331}{(\frac{0.04}{1.96})^2}=559.67\approx560
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