The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. In an earlier study, the
population proportion was estimated to be 0.23 .
How large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the confidence level with an
error of at most 0.03 ? Round your answer up to the next integer.
Answer
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Question

The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. In an earlier study, the
population proportion was estimated to be 0.23 .
How large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the

confidence level with an
error of at most 0.03 ? Round your answer up to the next integer.
Answer
How to enter your answer (opens in new window)

UpStudy AI · Accepted Answer

To estimate the sample size required to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level at a 98% confidence level with an error of at most 0.03, we can use the formula for sample size calculation for proportions:

$$ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) $$

Where:
- $$ n $$ is the sample size
- $$ Z $$ is the Z-score corresponding to the desired confidence level (98% confidence level corresponds to a Z-score of 2.33)
- $$ p $$ is the estimated population proportion (0.23)
- $$ E $$ is the margin of error (0.03)

Substitute the values into the formula and calculate the sample size.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(2.33^{2}\times 0.23\left(1-0.23\right)\right)}{0.03^{2}}$$
- step1: Remove the parentheses:
  $$\frac{2.33^{2}\times 0.23\left(1-0.23\right)}{0.03^{2}}$$
- step2: Subtract the numbers:
  $$\frac{2.33^{2}\times 0.23\times 0.77}{0.03^{2}}$$
- step3: Convert the expressions:
  $$\frac{\left(\frac{233}{100}\right)^{2}\times 0.23\times 0.77}{0.03^{2}}$$
- step4: Convert the expressions:
  $$\frac{\left(\frac{233}{100}\right)^{2}\times 0.23\times 0.77}{\left(\frac{3}{100}\right)^{2}}$$
- step5: Multiply:
  $$\frac{\frac{96145819}{100^{4}}}{\left(\frac{3}{100}\right)^{2}}$$
- step6: Evaluate the power:
  $$\frac{\frac{96145819}{100^{4}}}{\frac{9}{10000}}$$
- step7: Multiply by the reciprocal:
  $$\frac{96145819}{100^{4}}\times \frac{10000}{9}$$
- step8: Rewrite the expression:
  $$\frac{96145819}{100^{4}}\times \frac{100^{2}}{9}$$
- step9: Reduce the numbers:
  $$\frac{96145819}{100^{2}}\times \frac{1}{9}$$
- step10: Multiply the fractions:
  $$\frac{96145819}{100^{2}\times 9}$$
- step11: Multiply:
  $$\frac{96145819}{90000}$$
The sample size required to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level at a 98% confidence level with an error of at most 0.03 is approximately 1068.29.

Rounding up to the next integer, the required sample size is 1069.
The required sample size is 1069.

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