Save \& Exit Certify Lesson: 8.4 Estimating Population Proporti.
NYAH GRIME

Question 3 of 8, Step 2 of 2
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The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level

Step 2 of 2 : Suppose a sample of 929 tenth graders is drawn. Of the students sampled, 185 read at or below the eighth grade level. Using the data, construct the $98 \%$ confidence interval for the population proportion of tenth graders reading at or below the eighth grade level Round your answers to three decimal places.

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Step 1 ：Given that the sample size \(n = 929\) and the number of successes \(x = 185\). The Z-score for the desired confidence level of 98% is approximately \(Z = 2.33\).

Step 2 ：First, we calculate the sample proportion \(\hat{p}\) which is the number of successes divided by the sample size. \(\hat{p} = \frac{x}{n} = \frac{185}{929} = 0.199\).

Step 3 ：Next, we calculate the standard error (SE) using the formula \(\sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }\). Substituting the values, we get \(SE = \sqrt{ \frac{0.199*(1-0.199)}{929} } = 0.013\).

Step 4 ：Finally, we calculate the confidence interval (CI) using the formula \(\hat{p} \pm Z * SE\). Substituting the values, we get \(CI_{lower} = 0.199 - 2.33 * 0.013 = 0.169\) and \(CI_{upper} = 0.199 + 2.33 * 0.013 = 0.230\).

Step 5 ：The 98% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is approximately \(\boxed{[0.169, 0.230]}\).