The Safe Drinking Water Act, which was passed in 1974, allows ...

Question

The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.

What can the water department do to decrease the size of the confidence intervals, regardless of the amount of variance in cyanide levels?

<IMAGE>

What can the water department do to decrease the size of the confidence intervals, regardless of the amount of variance in cyanide levels?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A university laboratory is tasked with reporting the average nitrate concentration in campus drinking fountains. The 95% confidence intervals for the mean nitrate concentration are currently too wide for administrative comfort. Assuming the variance in nitrate levels cannot be changed, what should the laboratory do to obtain narrower confidence intervals? Awesome. So it appears for this particular prompt we're asked to figure out, statement wise, what should the laboratory do in order to obtain a narrower confidence intervals value. So we're trying to figure out how The laboratory or what exactly the laboratory needs to do in order to obtain narrower confidence intervals. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is increase the number of samples taken from the fountains. B is use a lower quality testing method. C is report the median instead of the mean, and D is increase the confidence level to 99%. Awesome. So first off, let us recall that the width of a confidence interval for the mean is given by the following equation, the width W is equal to 2 multiplied by Z to the power of multiplied by sigma divided by the square root of N. Where once again W is our width, Z to the power of star is the critical value for the desired confidence level. Sigma is the population standard deviation. The variance is fixed in that particular case. And N is the sample size. So in order to make the interval narrower, as we should recall, We can either decrease the Z to the power of star value, the lower confidence level, or increase N, the sample size, since sigma cannot be changed. So with that in mind, let's look over each of our multiple choice answers individually in order to see which one is the correct choice and why the other answers are incorrect. So, starting with multiple choice answer letter A. When we increase the number of samples taken from the fountains, so once again increase the number of samples taken from the fountains, this statement is indeed correct, and it's correct because increasing N sample size decreases the standard error, which is sigma divided by the square root of N, which therefore as a result, makes the confidence interval narrower. So this is technically our correct final answer. So multiple choice answer letter A is our final answer, but let's continue to look over our other multiple choice answers to see why B, C, and D are all incorrect. So, For B, use a lower quality testing method. This answer is incorrect because, as we should recall, a lower quality method could increase measurement error, potentially increasing the variance and make the interval wider, not narrower. So that's why multiple choice answer letter B is not correct. As for multiple choice answer letter C, report the median instead of the mean. This answer is incorrect because, as we should recall, reporting the median does not affect the width of the confidence interval for the mean, and the confidence interval for the median is a different statistical measure, and that is why multiple choice answer letter C is incorrect. As for multiple choice, D, increase the confidence level to 99%. This statement is incorrect because, as we should recall, increasing the confidence level increases the Z to the power of star value, which therefore, as a result, makes the interval wider, not narrower. So that means once again, without a doubt that our final answer has to be multiple choice answer letter A, increase the number of samples taken from the fountains, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Sample Size

Variance

Watch next