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.

The confidence interval for Year 2 is much larger than that for the other years. What do you think may have caused this larger confidence level?

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The confidence interval for Year 2 is much larger than that for the other years. What do you think may have caused this larger confidence level?

Accepted Answer

Hello there. Today we're going to 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 research team is studying the average height of particular species of tree in a forest. They report the mean height and a 95% confidence interval for each year of their study. In the 5th year, the confidence interval for the mean height is much wider than the previous 4 years. Which of the following is most likely reason for the wider confidence interval in the 5th year? Awesome. So it appears our final answer that we're ultimately trying to solve for is we're asked to look at our multiple choice answers and determine which of these that like which of these statements is most likely the reason for the wider confidence interval in the 5th year. So with that in mind, 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 the mean tree height was lower in the 5th year, B is fewer trees were measured in the 5th year. C is the measuring equipment was more accurate in the 5th year, and D is the confidence level was decreased in the 5th year. Awesome. So, first off, in order to solve this problem, we need to recall and use the formula for a confidence interval for the mean and let us know that we're told that we have a 95% confidence interval CI for short, which is given by which we should recall and use the following equation that X bar. Plus or minus Z to the power of star. Multiplied by S divided by the square root of N, where X bar is the sample mean, S is the sample standard deviation. N is the sample size, and Z to the power of this value Z power of star is the critical value for the 95% confidence. So with that in mind, as we should recall, the width of a confidence interval depends on the sample size, the variability of the data, and the confidence level. So with that in mind, we need to analyze each option. And let's go and order from A to B to C to D. So for option A, a lower mean does not affect the width, only the center of the interval, or B, A sample so a smaller sample size increases the width of the confidence interval. For C, more accurate equipment would decrease variability, which will narrow narrow the interval, so it's narrowing the interval, and for D. Decreasing the confidence level would make the interval narrower, not wider. So with that in mind, we now need to identify the correct answer. So with that in mind, The most likely reason for a wider confidence interval is that fewer trees were measured, which means a smaller sample size was collected. So therefore, without a doubt, our final answer would have to be multiple choice answer letter B, which once again is fewer trees were measured in the 5th year. 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

Point Estimate

Sample Size and Variability

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