Lead seepage into the drinking water in Flint, Michigan has caused concerns and

Question

Lead seepage into the drinking water in Flint, Michigan has caused concerns and fears that
their water may be contaminated with high concentrations of lead. The leakage has been caused by corroding supply pipes and a failure to replace damaged pipes caused by years of neglect. The EPA has set the maximum allowable lead content to be a weighted average of 15 ppb for pipe surfaces, fittings, and fixtures. High blood lead levels are especially harmful to children and pregnant women and could cause learning disabilities and behavioral problems. (Source: https://www.epa.gov/ground-water-and-drinking-water/basic-information-about-lead-drinking-water) These fears have garnered national attention and city officials have spent millions trying to improve the quality of the drinking water for folks there. If no improvement to the public water is found, the city will be fined millions for harming their residents. Researchers have collected 252 water samples from tap water in Flint one year later and found an average lead content of 19.5 ppb with a standard deviation of 20 ppb. Does their research show evidence that the quality of their drinking water has improved with the lead content below the allowable level? Use a significance level of 1%. [Hint: Assume that the water quality has NOT improved and test to see if it has improved. If the water quality has improved, then the true average lead content in the water should be less than 15 ppb.]

a. Write out the null and alternative hypothesis. b. Check all assumptions. c. Find the test statistic and the p-value. d. Compare the p-value to the significance level. What is your conclusion? e. Find and interpret an appropriate confidence interval for the true population mean , even if you do NOT have a statistically significant result.

Explanation / Answer

a. Null HYpothesis : H0: The mean lead content in water in the area is under the maximum permissible limit of lead which is 15ppb. <= 15ppb

Alternative Hypothesis : Ha : The average lead content in water in the given area has more lead content in its water than the maximum permissible limit.   >15ppb

b. Assumptions are

(i) Data have a normal distribution (or at least is symmetric)

(ii) Data from multiple groups have the same variance.

(iv) Data collected are of independent nature.

c. Test Statistic

t = (sample mean - maximum permissible limit)/ ( standard deviation/ sqrt (sample size)

t = (19.5 - 15)/ (20/ sqrt (252) = 4.5 / 1.26 = 3.571

tcriticalfor dF = 252-1 = 251 and alpha = 0.01 , we can take it as normal distribution as n >100

so tcritical= Z0.01 for one tailed equals to 2.33

so t > tcriticalso we can reject the null hypothesis.

P - value = Pr( X>= 19.5; 15; 20/sqrt(252) ) = (3.57) = 0.0004

so P - value is under significance level s we can reject the null hypothesis.

We can conclude that there is no decrease in average lead concentration and it is more than the maximum permissible lead limit.

e. appropriate confidence interval for true population mean

As test statistics are significant so we will use sample mean as population mean.

so 99% COnfidence interval = sample Mean +- Z0.01 * (s/sqrt(n)

= 19.5 +- 2.33 * (20/sqrt(252)) = (16.565, 22.435) here we can see that the confidence interval doesn't contain value of maximum permissible limit and it is all above it so we can say that mean lead concentration in the given area is above permissible limit.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.