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# The article “Leachate from Land Disposed Residential Construction Waste” (W. Weber, Y. Jang, et al., Journal of Environmental Engineering, 2002: 237-245) presents a study of contamination at landfills containing construction and demolition waste. Specimens of leachate were taken from a test site. Out of 42 specimens, 26 contained detectable levels of lead, 41 contained detectable levels of arsenic, and 32 contained detectable levels of chromium. a. Find a 90% confidence interval for the probability that a specimen will contain a detectable level of lead. b. Find a 95% confidence interval for the probability that a specimen will contain a detectable level of arsenic. c. Find a 99% confidence interval for the probability that a specimen will contain a detectable level of chromium.

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(a) Given:

$x=26$

$n=42$

$c=90\%=0.90$

We add 2 successes and 2 failures to the sample. The sample proportion is the number of successes divided by the sample size:

$\hat{p}=\dfrac{x+2}{n+4}=\dfrac{26+2}{42+4}=\dfrac{28}{46}\approx 0.6087$

For confidence level $1-\alpha=0.90$, determine $z_{\alpha/2}=z_{0.05}$ using using the normal probability table in the appendix (look up 0.05 in the table, the z-score is then the found z-score with opposite sign):

$z_{\alpha/2}=1.645$

The margin of error is then:

$E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n+4}}=1.645\times \sqrt{\dfrac{0.6087(1-0.6087)}{42+4}}\approx 0.1184$

The boundaries of the confidence interval are then:

$\hat{p}-E=0.6087-0.1184= 0.4903$

$\hat{p}+E=0.6087+0.1184= 0.7271$

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