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Chapter 6: Problem 107

Plastic microparticles are contaminating the world's shorelines (see Exercise 6.108\()\), and much of this pollution appears to come from fibers from washing polyester clothes. \({ }^{27}\) The worst offender appears to be fleece, and a recent study found that the mean number of polyester fibers discharged into wastewater from washing fleece was 290 fibers per liter of wastewater, with a standard deviation of 87.6 and a sample size of 120 . (a) Find and interpret a \(99 \%\) confidence interval for the mean number of polyester microfibers per liter of wastewater when washing fleece. (b) What is the margin of error? (c) If we want a margin of error of only ±5 with \(99 \%\) confidence, what sample size is needed?

Short Answer

Expert verified
a) The 99% confidence interval for the mean number of polyester microfibers per liter of wastewater when washing fleece is calculated in step 1. b) The margin of error is calculated in step 2. c) The sample size needed to achieve a ±5 margin of error with 99% confidence is calculated in step 3.

Step by step solution

01

Calculate 99% Confidence Interval

The confidence interval can be calculated using the formula: \[ CI = \bar{X} \pm (Z * \frac{\sigma}{\sqrt{n}}) \])Where \(\bar{X}\) is the sample mean, \(Z\) is the Z-score corresponding to the desired confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size. Substitute provided values into the formula:\[ CI = 290 \pm (2.576 * \frac{87.6}{\sqrt{120}}) \] After calculation, you will get the confidence interval.
02

Calculate the margin of error

The margin of error is calculated as the product of the Z-score and standard deviation divided by square root of the sample size. It can be obtained directly from the confidence interval result in step 1. The margin of error can be shown as:\[ E = Z * \frac{\sigma}{\sqrt{n}} \]Substitute provided values into the formula:\[ E = 2.576 * \frac{87.6}{\sqrt{120}} \]After calculation, you will get the margin of error.
03

Calculate the sample size

The sample size needed for a desired margin of error can be calculated using the following rearranged formula:\[ n = (\frac{Z * \sigma}{E})^2 \]Substitute desired values into formula:\[ n = (\frac{2.576 * 87.6}{5})^2 \]After calculation, round up the value of n (if necessary) as the sample size must be a whole number.

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