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Chapter 16: Problem 20

Hiring In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a. How many randomly selected employers must we contact in order to create an estimate in which we are \(98 \%\) confident with a margin of error of \(5 \% ?\) b. Suppose we want to reduce the margin of error to \(3 \%\). What sample size will suffice? C. Why might it not be worth the effort to try to get an interval with a margin of error of only \(1 \% ?\)

Short Answer

Expert verified
a. The sample size needed for a 5% margin of error at 98% confidence level is approximately 843. b. For a 3% margin of error at the same confidence level, the needed sample size is about 2346. c. It might not be worth to aim for a 1% margin of error because the increased precision may not justify the significantly larger sample size needed, which can result in higher costs, more time, and more resources.

Step by step solution

01

Calculate the sample size for a 5% margin of error

The common formula for sample size when population standard deviation is unknown is \(n = (z^2 * p * (1-p)) / E^2\). Since we don't have an estimate for \(p\), we should use the conservative estimate of \(p = 0.5\), which maximizes the sample size. Here \(z\) is Z-value, which is 2.33 for 98% confidence, \(E\) is the Margin of Error (0.05 for 5%), and \(p\) is the estimated proportion of the attribute present in the population. Substituting the given values into the formula, we get \((2.33^2 * 0.5 * (1 鈥� 0.5)) / 0.05^2\).
02

Calculate the sample size for a 3% margin of error

For the margin of error of \(3 \%\) we go through the same steps but change the margin of error, \(E\), in our formula to \(0.03\). So now the formula becomes \((2.33^2 * 0.5 * (1-0.5)) / 0.03^2\).
03

Discuss the trade-offs in reducing the margin of error to 1%

Reducing the margin of error to \(1 \%\) requires a larger sample size, which might exponentially increase the resources needed for the same. Now, would that be worth it? It might not be if the cost or feasibility of collecting a larger sample size outweighs the benefits. It's a tedious task and takes a lot of time, money, resources, and effort. Additionally, reducing the margin of error from 3% to 1% doesn't vastly improve the precision of estimates, since a 3% margin is already quite accurate in most practical cases.

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Key Concepts

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

Sample Size Calculation
When it comes to estimating how many businesses plan to hire in the upcoming months, calculating the right sample size is crucial. Sample size determines how many businesses you need to survey to achieve accurate results. To calculate sample size, use the formula:
  • \[ n = \left( \frac{z^2 \times p \times (1-p)}{E^2} \right) \]
In this formula:
  • \( n \) is the sample size.
  • \( z \) is the Z-value corresponding to your desired confidence level. A 98% confidence level has a \( z \)-value of 2.33.
  • \( p \) is the estimated population proportion, often taken as 0.5 if unknown, to maximize sample size.
  • \( E \) is the margin of error. For a 5% margin, \( E = 0.05 \).
By plugging in these values, you can compute how many businesses need to be contacted to get a reliable estimate of hiring plans.
Confidence Interval
A confidence interval provides a range of values which is likely to contain the population parameter. It gives a better idea on where the true proportion of businesses planning to hire might fall. For example, a 98% confidence interval means that if you repeated the sampling numerous times, 98% of those intervals would contain the true proportion.
The width of a confidence interval depends on:
  • Your desired confidence level (98% in this estimation).
  • The sample size. Larger samples lead to narrower intervals.
  • The margin of error around the estimate. Smaller margins of error produce tighter intervals but require larger sample sizes.
When aiming for a tighter confidence interval by reducing the margin of error, remember that an increased sample size is necessary, which could involve significant resources.
Population Proportion Estimation
Estimating the population proportion of businesses planning to hire is at the heart of this problem. The population proportion \( p \) represents the part of the total population with a specific attribute, which in this case, is the intention to hire employees.
Without prior knowledge of what \( p \) might be, it's standard to use 0.5 in calculations to ensure the largest possible sample size. This conservative approach helps in obtaining a reliable estimator for \( p \) when data is scarce.
Once the sample proportion is calculated from the businesses surveyed, it acts as an approximation of the actual proportion for the whole population. This is why the accuracy of your sample size calculation directly impacts the reliability of your population proportion estimation for decision-making in economic reports or policy formulation.

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Most popular questions from this chapter

Death penalty, again In the survey on the death penalty you read about in the Step-by-Step Example, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, "Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?" The other 510 were asked, "Generally speaking, do you believe the death penalty is applied unfairly or fairly in this country today?" Seems like the same question, but sometimes the order of the choices matters. Suppose that for the second way of phrasing it, \(64 \%\) said they thought the death penalty was fairly applied. (Recall that \(53 \%\) of the original 510 thought the same thing.) a. What kind of bias may be present here? b. If we combine them, considering the overall group to be one larger random sample of 1020 respondents, what is a \(95 \%\) confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied? c. How does the margin of error based on this pooled sample compare with the margins of error from the separate groups? Why?

Confidence intervals Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? a. For a given sample size, higher confidence means a smaller margin of error. b. For a specified confidence level, larger samples provide smaller margins of error. c. For a fixed margin of error, larger samples provide greater confidence. d. For a given confidence level, halving the margin of error requires a sample twice as large.

Margin of error A TV newscaster reports the results of a poll of voters, and then says, "The margin of error is plus or minus \(4 \%\)." Explain carefully what that means.

Junk mail Direct mail advertisers send solicitations (a.k.a. "junk mail") to thousands of potential customers in the hope that some will buy the company's product. The acceptance rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people randomly selected from their mailing list of over 200,000 people. They get orders from 123 of the recipients. a. Create a \(90 \%\) confidence interval for the percentage of people the company contacts who may buy something. b. Explain what this interval means. c. Explain what "90\% confidence" means. d. The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least a \(5 \%\) return. What does your confidence interval suggest? Explain.

34\. Still living online The Pew Research poll described in Exercise 5 ? found that \(56 \%\) of a sample of 1060 teens go online several times a day. (Treat this as a Simple Random Sample.) a. Find the margin of error for this poll if we want \(95 \%\) confidence in our estimate of the percent of American mteens who go online several times a day. b. Explain what that margin of error means. c. If we only need to be \(90 \%\) confident, will the margin of error be larger or smaller? Explain. d. Find that margin of error. e. In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?
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