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Chapter 9: Problem 48

You wish to estimate the mean number of travel days per year for salespeople. The mean of a small pilot study was 150 days, with a standard deviation of 14 days. If you want to estimate the population mean within 2 days, how many salespeople should you sample? Use the \(90 \%\) confidence level.

Short Answer

Expert verified
You should sample 133 salespeople.

Step by step solution

01

Determine Z-Score for Confidence Level

For a 90% confidence level, the Z-score can be found using statistical tables. The Z-score that corresponds to a 90% confidence level is approximately 1.645.
02

Identify Required Values

From the problem, we have the standard deviation (\(\sigma = 14\)) days, the margin of error (\(E = 2\)) days, and the Z-score (\(Z = 1.645\)).
03

Use the Sample Size Formula

The formula to calculate the sample size needed is given by:\[ n = \left(\frac{Z \times \sigma}{E}\right)^2 \]
04

Substitute Values into Formula

Substitute the values for Z, \(\sigma\), and \(E\) into the sample size formula:\[ n = \left(\frac{1.645 \times 14}{2}\right)^2 \]
05

Calculate Sample Size

First, calculate the numerator: \(1.645 \times 14 = 23.03\).Then divide this by the margin of error: \(\frac{23.03}{2} = 11.515\).Finally, square the result to find the sample size:\(11.515^2 = 132.497\).
06

Round Up to Nearest Whole Number

Since the sample size must be a whole number, round 132.497 up to 133. Therefore, you need to sample 133 salespeople.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
A confidence interval is a range of values used to estimate the true value of a population parameter, in this case, the mean number of travel days per year for salespeople. It provides an interval within which we expect the true mean to lie, with a certain level of confidence. In the exercise, we want to be 90% confident that the true mean lies within this interval.
  • The confidence level: In this example, it's set at 90%, meaning there's a 90% chance that the true population mean is within our calculated interval.
  • Determining the confidence interval involves using statistical tables to find a Z-score associated with the desired confidence level.
  • This concept is essential in research and statistics because it tells us how much we can trust our sample data to reflect what is true in the actual population.
Margin of Error
The margin of error defines how accurate we believe our estimate to be, in this case, the estimate of the mean number of travel days. It indicates the maximum amount we expect our sample mean will differ from the true population mean. In the given problem, we are specifying a margin of error of 2 days.
  • The margin of error, together with the confidence level, controls how precise our estimate is expected to be. A smaller margin of error means a more precise estimate.
  • In our solution, it was crucial to determine the sample size needed to achieve this margin of error, ensuring our estimate stays within 2 days of the true mean.
  • It's important to understand that reducing the margin of error typically requires increasing the sample size, which could involve more resources or time.
Z-score
A Z-score, in the context of confidence intervals, is a statistical measure that describes a value's relation to the mean of a group of values. It's particularly used to determine how many standard deviations an element is from the mean.
  • In our problem, the Z-score is calculated for a 90% confidence level, which yields a value of approximately 1.645. This value comes from statistical tables that summarize the area under the normal curve.
  • The Z-score is crucial because it allows us to calculate the necessary sample size to achieve our desired confidence interval and margin of error.
  • This concept helps convert the confidence level into a usable form for practical calculations, like determining sample sizes.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the members of a data set differ from the average value of that set. In our exercise, the standard deviation is given as 14 days.
  • A larger standard deviation indicates more variation or spread in the data, while a smaller one suggests data points are closer to the mean.
  • Knowing the standard deviation is essential when calculating the sample size needed to estimate the population mean within a specified margin of error.
  • It affects the width of the confidence interval. More variability in data (i.e., a higher standard deviation) increases the size of the confidence interval.
Understanding these key concepts ensures that we make accurate estimates and decisions based on sample data, providing a clearer picture of the population we are studying.

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Most popular questions from this chapter

The Buffalo. New York, Chamber of Commerce wants to estimate the mean time workers who are employed in the downtown area spend getting to work. A sample of 15 workers reveals the following number of minutes spent traveling. $$\begin{array}{llllllll}14 & 24 & 24 & 19 & 24 & 7 & 31 & 20 \\ 26 & 23 & 23 & 28 & 16 & 15 & 21 &\end{array}$$ Develop a \(98 \%\) confidence interval for the population mean. Interpret the result.

Past surveys reveal that 30% of tourists going to Las Vegas to gamble spend more than \(\$ 1,000 .\) The Visitor's Bureau of Las Vegas wants to update this percentage. a. How many tourists should be randomly selected to estimate the population proportion with a 90% confidence level and a 1% margin of error? b. The Bureau feels the sample size determined above is too large. What can be done to reduce the sample? Based on your suggestion, recalculate the sample size.

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the \(95 \%\) confidence level and a margin of error of \(2 \% .\) A pilot survey reveals that 5 of the 50 sampled hold two or more jobs. How many in the workforce should be interviewed to meet your requirements?

Pharmaceutical companies promote their prescription drugs using television advertising. In a survey of 80 randomly sampled television viewers, 10 indicated that they asked their physician about using a prescription drug they saw advertised on TV. Develop a \(95 \%\) confidence interval for the proportion of viewers who discussed a drug seen on TV with their physician. Is it reasonable to conclude that \(25 \%\) of the viewers discuss an advertised drug with their physician?

An important factor in selling a residential property is the number of times real estate agents show a home. A sample of 15 homes recently sold in the Buffalo, New York, area revealed the mean number of times a home was shown was 24 and the standard deviation of the sample was 5 people. Develop a \(98 \%\) confidence interval for the population mean.
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