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Chapter 7: Problem 40

Suppose it is known that \(60 \%\) of employees at a company use a Flexible Spending Account (FSA) benefit. a. If a random sample of 200 employees is selected, do we expect that exactly \(60 \%\) of the sample uses an FSA? Why or why not? b. Find the standard error for samples of size 200 drawn from this population. What adjustments could be made to the sampling method to produce a sample proportion that is more precise?

Short Answer

Expert verified
a) No, we don't expect that exactly \(60\%\) of the sample will use a FSA. This is because in random sampling, the sample proportion can vary due to sampling variability. b) The standard error for this sample is calculated by the formula \(\sqrt{(p*(1-p)/n)}\), resulting in a specific value. To improve sample proportion precision, we could increase the sample size, ensure random selection of the sample and ensure that the sample is representative of the overall population.

Step by step solution

01

Understand Proportions

Start by understanding the concept of proportions. A population proportion is the fraction or percentage of a group possessing a certain characteristic (in this case, \(60\%\) of the employees using a FSA). Sample proportion, on the other hand, refers to the fraction or percentage of our sample that possesses that characteristic. When we select a sample randomly, we expect its proportion to be close to the population proportion, but not precisely the same. This is due to the variability inherent in random sampling.
02

Calculate the Standard Error

Next, calculate the standard error, which quantifies the expected variability in sample proportions. The formula for standard error for proportions is \(\sqrt{(p*(1-p)/n)}\), where \(p\) is the population proportion and \(n\) is the sample size. For our problem, \(p = 0.60\) and \(n = 200\). Thus, the standard error \(= \sqrt{(0.60*(1-0.60)/200)}\).
03

Improve the Precision of Sample Proportion

The precision of a sample proportion can be improved by increasing the sample size. Larger samples tend to produce statistics that are closer to the value of the population parameter, because they are less influenced by variability due to random chance. Sampling methodology can also be improved by ensuring that samples are selected randomly and are representative of the overall population.

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Most popular questions from this chapter

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Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$\begin{array}{|l|l|l|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\\\\hline & 936 & 607 \\ \hline\end{array}$$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$\begin{array}{|l|c|c|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\\\\hline & 94 & 61 \\ \hline\end{array}$$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.
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