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Chapter 6: Problem 108

In Exercise \(6.107,\) we see that plastic microparticles are contaminating the world's shorelines and that much of the pollution appears to come from fibers from washing polyester clothes. The same study referenced in Exercise 6.107 also took samples from ocean beaches. Five samples were taken from each of 18 different shorelines worldwide, for a total of 90 samples of size \(250 \mathrm{~mL}\). The mean number of plastic microparticles found per \(250 \mathrm{~mL}\) of sediment was 18.3 with a standard deviation of 8.2 . (a) Find and interpret a \(99 \%\) confidence interval for the mean number of polyester microfibers per \(250 \mathrm{~mL}\) of beach sediment. (b) What is the margin of error? (c) If we want a margin of error of only ±1 with \(99 \%\) confidence, what sample size is needed?

Short Answer

Expert verified
The 99% confidence interval will give a range of values within which the true population mean number of polyester microfibers per 250 mL of sediment is likely to fall. The margin of error shows the range of values around our sample mean that our population mean could be. Based on this, an increased sample size will be necessary to achieve a smaller margin of error of ±1, while maintaining the same level of confidence at 99%.

Step by step solution

01

Find and interpret the 99% confidence interval

This can be computed using the formula for confidence intervals: \[CI = x̄ ± Z \frac{σ}{√n}\] Here, x̄ is the sample mean = 18.3, Z is the Z-score which corresponds to the confidence level (for 99% confidence, Z=2.57), σ is the standard deviation = 8.2, n is the sample size = 90. Upon inserting these values into the formula, we should find the confidence interval around the mean.
02

Find the margin of error

The margin of error (E) is essentially the numeric value that is added and subtracted from the mean to achieve the confidence interval. It is found in the equation: \[E = Z \frac{σ}{√n}\]
03

Find the required sample size

Given we want a margin of error (E) of only ±1 with 99% confidence, we can use the formula: \[n = (Z \frac{σ}{E})^2\] Then, depending upon whether we get a whole number or a fraction, we could decide what the minimum required sample size would be. The number of samples has to be a whole number, so if we get a fraction, we will need to round up to the next nearest whole number.

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Most popular questions from this chapter

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