INFORMATION:
We know that:
- In an earlier study, the population proportion was estimated to be 0.22
And we must calculate how large of a sample would be required in order to estimate the fraction of 10th graders reading at or below the eighth grade level at the 85% confidence level with an error of at most 0.03
STEP BY STEP EXPLANATION:
To calculate it, we need to use the following formula
[tex]E=Zc\times\sqrt{\frac{p(1-p)}{n}}[/tex]Where,
- E is the margin of error
- Zc is the value from Zc table based on the level of confidence
- p is the population proportion
- n represents the sample
From given information,
- E = 0.03
- p = 0.22
- Using the table for Zc values, we know that for this case Zc = 1.44
Now, we must replace the values in the equation and solve it for n
[tex]\begin{gathered} 0.03=1.44\times\sqrt{\frac{0.22(1-0.22)}{n}} \\ \frac{0.03}{1.44}=\frac{\sqrt{0.22(0.78)}}{\sqrt{n}} \\ \sqrt{n}\times0.03=\sqrt{0.22(0.78)}\times1.44 \\ \sqrt{n}=\frac{0.5965}{0.03} \\ \sqrt{n}=19.8833 \\ n=(19.8833)^2 \\ n=395.3456 \end{gathered}[/tex]Since n refers to students, we must round the answer to the next whole number.
Finally, the sample must be of 396 students.
ANSWER:
396
