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Chapter 8: Problem 24

A department store manager wants to estimate the mean amount spent by all customers at this store at a \(98 \%\) confidence level. The manager knows that the standard deviation of amounts spent by all customers at this store is \(\$ 31\). What minimum sample size should he choose so that the estimate is within \(\$ 3\) of the population mean?

Short Answer

Expert verified
The minimum sample size that the manager should choose to estimate the mean amount spent by all customers at this store at a 98% confidence level, with a standard deviation of $31 and a margin of error of $3, is 584.

Step by step solution

01

Identify and note down the known values

From the problem we know that: The standard deviation 蟽 is $31, the margin of error E is $3, and for a 98% confidence level, the Z-score is typically 2.33.
02

Substitute the known values into the sample size formula

Substitute the known values into the sample size formula: \(n = \left(\frac{Z*蟽}{E}\right)^2\) which becomes \(n = \left(\frac{2.33*31}{3}\right)^2\).
03

Calculate the sample size

Doing the calculations, we get \(n = \left(24.15\right)^2 = 583.6\).
04

Round up to the nearest whole number

Because we can't have a fraction of a sample, the sample size needs to be rounded up to the nearest whole number. So, the sample size that the manager should choose is 584.

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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 provides a range of values that is likely to contain the true population parameter. In this context, it estimates the mean amount spent by all customers at a store. The confidence level, in this case 98%, represents how confident we are that the interval contains the true mean. A higher confidence level means that the range of the interval will be broader, as we want to be more certain that the interval captures the true value.
  • To calculate a confidence interval, we need the sample mean, the Z-score for the desired confidence level, the standard deviation, and the sample size.
  • Consider the relationship between the confidence interval and sample size: As sample size increases, the estimate of the population mean becomes more precise, and the confidence interval becomes narrower.
Understanding these relationships helps in determining the required sample size to achieve a specific margin of error with a given confidence level.
Standard Deviation
Standard deviation is a measure of how spread out numbers in a data set are. In the scenario given, it represents how the amounts spent by customers vary from the average amount. A high standard deviation indicates that the values are spread out over a wider range, while a low standard deviation suggests that they are closer to the average.
  • If the standard deviation is too high, it implies considerable variability in the spending patterns, which requires a larger sample size for accurate estimation.
  • It is a key element in calculating the sample size needed to achieve a desired confidence interval width or margin of error.
Therefore, understanding standard deviation helps in assessing the variability within data, crucial for sample size calculations.
Margin of Error
The margin of error indicates the range in which the true population parameter is expected to lie, with a certain level of confidence. In the manager's case, he wants the estimate of the mean spending to be within $3 of the actual mean. The margin of error is directly correlated to the sample size. Smaller margins of error require larger sample sizes to maintain the same confidence level.
  • The formula used to calculate the sample size involves standard deviation, Z-score, and margin of error.
  • As the margin of error decreases, the necessary sample size increases, making the estimation more precise.
Understanding and working with margin of error helps ensure that the estimates made from the sample data are as close as possible to the actual population values.
Z-score
The Z-score is a statistical measure that indicates how many standard deviations a data point is from the mean. For confidence interval calculations, it represents the number of standard deviations needed to achieve a certain level of confidence. In the manager's problem, a Z-score of 2.33 is used, corresponding to a 98% confidence level.
  • The Z-score is determined by the desired confidence level鈥攈igher confidence levels have higher Z-scores.
  • It affects the margin of error and hence the sample size. A higher Z-score increases the margin of error, requiring a larger sample size.
Z-scores are crucial for determining the width of confidence intervals, affecting how confident one can be in the results of the analysis.

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Most popular questions from this chapter

Almost all employees working for financial companies in New York City receive large bonuses at the end of the year. A random sample of 65 employees selected from financial companies in New York City showed that they received an average bonus of \(\$ 55,000\) last year with a standard deviation of \(\$ 18,000 .\) Construct a \(95 \%\) confidence interval for the average bonus that all employees working for financial companies in New York City received last year.

Briefly explain the meaning of the degrees of freedom for a \(f\) distribution. Give one example.

A city planner wants to estimate the average monthly residential water usage in the city at a \(97 \%\) confidence level. Based on earlier data, the population standard deviation of the monthly residential water usage in this city is \(389.60\) gallons. How large a sample should be selected so that the estimate for the average monthly residential water usage in this city is within 100 gallons of the population mean?

At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the amounts of milk in all such cartons is always equal to \(.15\) ounce. The quality control department takes a random sample of 25 such cartons every week, calculates the mean net weight of these cartons, and makes a \(99 \%\) confidence interval for the population mean. If either the upper limit of this confidence interval is greater than \(32.15\) ounces or the lower limit of this confidence interval is less than \(31.85\) ounces, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.94\) ounces. Based on this sample, will you conclude that the machine needs an adjustment? Assume that the amounts of milk put in all such cartons have an approximate normal distribution.

A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. Assume that a preliminary study has shown that \(76 \%\) of drivers wear seat belts while driving. How large should the sample size be so that the \(99 \%\) confidence interval for the population proportion has a margin of error of \(.03\) ?
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