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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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