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Chapter 53: Problem 2

Calculate and interpret standard errors. Two samples, \(A\) and \(B\), gave the following descriptive statistics (measured in the same units): Sample \(A\), mean \(=16.2\), standard deviation \(=12.7\), number of data values \(=12\); Sample B, mean \(=\) 13.2, standard deviation 14.4, number of data values \(=20\). Which has the lower standard error in absolute terms and in proportion to the sample mean? (Express answers to three significant figures.)

Short Answer

Expert verified
To get the short answer, calculate the values for standard errors and proportional standard errors for both samples based on the above steps. Compare these values to determine which sample has the lower standard error in absolute terms and in proportion to the sample mean.

Step by step solution

01

Calculate the standard error for Sample A

To calculate the standard error for Sample A, use the formula which is given by: \(SE = \frac{SD}{\sqrt{n}}\), where \(SE\) is the standard error, \(SD\) is the standard deviation and \(n\) is the number of data values. Substituting these values in, the standard error for Set A is: \(SE_A = \frac{12.7}{\sqrt{12}}\).
02

Calculate the standard error for Sample B

Similarly, to calculate the standard error for Sample B, substitute the values \(SD = 14.4\) and \(n = 20\) into the formula. Thus the standard error for Set B is: \(SE_B = \frac{14.4}{\sqrt{20}}\).
03

Compare standard errors and interpret

Once you have calculated the two standard errors, compare them in absolute and relative (to the sample mean) terms. Whichever sample has the smaller standard error in absolute terms is the sample with the more precise estimate of the population mean. Similarly, the sample with the lower standard error relative to its mean has the more precise estimate in proportional terms.
04

Calculate the proportional standard error for both samples

The proportional standard error is calculated by dividing the standard error by the mean of the sample. For sample A and B, the proportional errors would be calculated as \(PSE_A = \frac{SE_A}{16.2}\) and \(PSE_B = \frac{SE_B}{13.2}\) respectively. Compare these two to find out which sample has a lower standard error in proportion to the sample mean.

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Most popular questions from this chapter

Compute a mean value correctly. A researcher finds that the mean vitamin concentration in three replicate samples designated \(A, B\) and \(C\) is \(3.0,2.5\) and \(2.0 \mathrm{mg}\), respectively. He computes the mean vitamin concentration as \(2.5 \mathrm{mg}\), but forgets that the sample sizes were 24,37 and 6, respectively. What is the true mean vitamin concentration? (Answer to three significant figures.)
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