Contaminated water: In a sample of 40 water specimens taken from a construction site, 26 contained detectable levels of lead.
Part 1 of 3
(a) Construct a $99.8 \%$ confidence interval for the proportion of water specimens that contain detectable levels of lead. Round the answers to at least three decimal places.
A $99: 8 \%$ confidence interval for the proportion of water specimens that contain detectable levels of lead is $0.417
Solution
Step 1 ：Calculate the sample proportion (p̂) as the number of successes (x) divided by the sample size (n). In this case, the number of successes is the number of water specimens that contain detectable levels of lead (26), and the sample size is the total number of water specimens taken from the construction site (40). So, \(p̂ = \frac{x}{n} = \frac{26}{40} = 0.65\)
Step 2 ：Calculate the standard error (SE) as the square root of \(p̂(1 - p̂) / n\). So, \(SE = \sqrt{0.65(1 - 0.65) / 40} = 0.075\)
Step 3 ：Calculate a 90% confidence interval as \(p̂ ± Z*SE\), where Z is the Z-score that corresponds to the desired level of confidence. For a 90% confidence interval, the Z-score is 1.645. So, the 90% confidence interval is \(0.65 ± 1.645*0.075 = 0.65 ± 0.123\)
Step 4 ：Therefore, a 90% confidence interval for the proportion of water specimens that contain detectable levels of lead is \(\boxed{0.527 < p < 0.773}\)
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Solution
Step 1 ：Calculate the sample proportion (p̂) as the number of successes (x) divided by the sample size (n). In this case, the number of successes is the number of water specimens that contain detectable levels of lead (26), and the sample size is the total number of water specimens taken from the construction site (40). So, \(p̂ = \frac{x}{n} = \frac{26}{40} = 0.65\)
Step 2 ：Calculate the standard error (SE) as the square root of \(p̂(1 - p̂) / n\). So, \(SE = \sqrt{0.65(1 - 0.65) / 40} = 0.075\)
Step 3 ：Calculate a 90% confidence interval as \(p̂ ± Z*SE\), where Z is the Z-score that corresponds to the desired level of confidence. For a 90% confidence interval, the Z-score is 1.645. So, the 90% confidence interval is \(0.65 ± 1.645*0.075 = 0.65 ± 0.123\)
Step 4 ：Therefore, a 90% confidence interval for the proportion of water specimens that contain detectable levels of lead is \(\boxed{0.527 < p < 0.773}\)
