Lead is a toxic heavy metal that can be especially harmful to children. It has b

Question

Lead is a toxic heavy metal that can be especially harmful to children. It has been linked with behavior & development problems, slowed growth, and iron deficiency. Lead contamination can occur in drinking water due to lead piping or lead solder in pipes. The contamination of the water in a home can be solely due to the piping of that home or it can result from larger widespread issues and the piping belonging to the city. Cities across the U.S. frequently test for elevated levels of lead by sampling households.

While there is no safe level of lead exposure, a concentration of 15ppb in water is a typical cutoff for an “unsafe” level. Suppose a city health department is concerned about the lead levels of all homes in the city. City officials will take aggressive action if there is evidence that more than 10% of all homes in the city have an “unsafe” lead concentration. They test a sample of 271 homes and find that 45 of those homes exceeded the 15ppb lead concentration threshold.    

d.Calculate the test statistic (or z-score) for this test. Report the corresponding P-value. You may want to check that your answer makes sense based on your drawing in part (c).

e.A P-value is a probability. What is the P-value in part (d) the probability of? Explain carefully in the context of the problem. Note: I am NOT yet looking for you to give a conclusion to the test here.

f.Based on the P-value, state a conclusion to this test by discussing the strength of the evidence against the null hypothesis and in favor of the alternative hypothesis in the context of the problem.

g.Using these same data, construct a 95% confidence interval for the proportion of all homes in the city that have an “unsafe” lead concentration. . Interpret your interval in context of the problem.

h.Is the hypothesized proportion p0 = 0.10 contained in your confidence interval in part (g)? Does this make sense based on your conclusion in part (f)? Explain your reasoning in one or two complete sentences.

Explanation / Answer

we know that the z stat for proportion is given as

(phat - p)/sqrt(p*(1-p)/n)

here phat = 45/271 = 0.166
p = 0.10

putting the values we get the z stat as

z = (0.166-0.1)/sqrt(0.1*(1-0.1)/271) = 3.62

we check the corresponding z value from the table as

(1- 0.9998) = 0.0002 , hence the result is statistically signficant

p= 0.002 is the probability that our rejection of null hypothesis of "there is evidence that no more than 10% of all homes in the city have an “unsafe” lead concentration" is wrong

so as we reject the null hypothesis in favor of alternate hypothesis we conclude that

there is evidence that more than 10% of all homes in the city have an “unsafe” lead concentration"

the CI is given as

p +- z*sqrt(p*(1-p)/n)
so for 95% Ci the value of z is 1.96

put the values in the equation
0.166 +- 1.96*sqrt(0.166*(1-0.166)/271)

0.121 and 0.210 ,
this means we are 95% confident that the true value of the proportion is in the range 0.121 and 0.210

as the CI does not include the proportion 0.10 and is more than 0.10 , ranging from 0.121 to 0.210 , hence we can further restate that the proportion is greater than 0.10

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