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Chapter 9: Problem 56

56\. Population A has standard deviation \(\sigma_{\mathrm{A}}=5,\) and population \(\mathrm{B}\) has standard deviation \(\sigma_{\mathrm{B}}=10 .\) How many times larger than Population A's sample size does Population B's need to be to estimate \(\mu\) with the same margin of error? [Hint: Compute \(\left.n_{\mathrm{B}} / n_{\mathrm{A}}\right]\).

Short Answer

Expert verified
Population B's sample size needs to be 4 times larger than Population A's.

Step by step solution

01

- Understand the margin of error relationship

The margin of error (MOE) for a population is proportional to the standard deviation (蟽) divided by the square root of the sample size (n). Therefore, the formula for the margin of error can be written as: \[ MOE \text{ is proportional to } \frac{蟽}{\text{鈭殅n} \]
02

- Set up the ratio for the sample sizes

In order to compare the sample sizes of Population A and Population B for the same margin of error, set up the ratio of their respective margin of errors: \[ \frac{蟽_A}{\text{鈭殅n_A} = \frac{蟽_B}{\text{鈭殅n_B} \]
03

- Substitute the standard deviations

Substitute the given standard deviations for Population A and Population B into the equation: \[ \frac{5}{\text{鈭殅n_A} = \frac{10}{\text{鈭殅n_B} \]
04

- Isolate the sample size ratio

Solve for the ratio of the sample sizes by isolating \( \frac{n_B}{n_A} \). Start by cross-multiplying to get: \[ 5 \text{鈭殅n_B = 10 \text{鈭殅n_A \] Then divide both sides by 5: \[ \text{鈭殅n_B = 2 \text{鈭殅n_A \]
05

- Square both sides to solve for the ratio

Square both sides of the equation to remove the square root: \[ n_B = 4 n_A \]

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Key Concepts

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

standard deviation
Understanding standard deviation is essential in statistics. It measures the amount of variation or dispersion of a set of values. A low standard deviation means that values tend to be close to the mean, whereas a high standard deviation means values are spread out over a wider range. In our example, Population A has a standard deviation (\( \sigma_A \)) of 5, and Population B has a standard deviation (\( \sigma_B \)) of 10. This means that the data from Population B is more spread out around the mean than the data from Population A.
margin of error
The margin of error (MOE) illustrates the range of values, above and below the actual result, within which the true population parameter lies with a certain level of confidence. It is directly tied to the standard deviation and the sample size. The MOE can be expressed mathematically as being proportional to \( \frac{\sigma}{\sqrt{n}} \). This implies:
  • As the standard deviation gets larger, the margin of error increases if the sample size remains constant.
  • As the sample size increases, the margin of error decreases if the standard deviation remains constant.
In our exercise, for the same MOE, Population B would require a larger sample size because its standard deviation is larger.
population comparison
When comparing populations, especially in this context, we look at how their sample sizes should differ to achieve the same margin of error. Our exercise assumes two populations, A and B, requiring us to determine how much larger Population B's sample size should be to achieve the same MOE as Population A. Given the standard deviations, the samples would need to follow the equation: \( \frac{\sigma_A}{\sqrt{n_A}} = \frac{\sigma_B}{\sqrt{n_B}} \). After substituting the values from each population, we can derive the relation: \( n_B = 4 n_A \), highlighting that Population B needs a sample size four times larger than Population A to maintain identical margins of error.
sample size calculation
Calculating sample size is a key task in statistics, ensuring accurate representation and robust data. The sample size required \( n \) is determined by the standard deviation and the desired margin of error. This calculation is crucial in surveys, experiments, and any form of quantitative research. Following our problem:
  • We notice \( \frac{5}{\sqrt{n_A}} = \frac{10}{\sqrt{n_B}} \) from the margin of error relation.
  • By cross-multiplying and simplifying, we find \( n_B = 4n_A \).
Hence, precise calculation ensures that the data remains within the desired margin of error, making the findings reliable and statistically significant.

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Most popular questions from this chapter

A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates? (c) Why are the results from parts (a) and (b) so close?

True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.

A survey of 2306 adult Americans aged 18 and older conducted by Harris Interactive found that 417 have donated blood in the past two years. (a) Obtain a point estimate for the population proportion of adult Americans aged 18 and older who have donated blood in the past two years. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct a \(90 \%\) confidence interval for the population proportion of adult Americans who have donated blood in the past two years. (d) Interpret the interval.

A recent Gallup poll asked Americans to disclose the number of books they read during the previous year. Initial survey results indicate that \(s=16.6\) books. (a) How many subjects are needed to estimate the number of books Americans read the previous year within four books with \(95 \%\) confidence? (b) How many subjects are needed to estimate the number of books Americans read the previous year within two books with \(95 \%\) confidence? (c) What effect does doubling the required accuracy have on the sample size? (d) How many subjects are needed to estimate the number of books Americans read the previous year within four books with \(99 \%\) confidence? Compare this result to part (a). How does increasing the level of confidence in the estimate affect sample size? Why is this reasonable?

Jane wants to estimate the proportion of students on her campus who eat cauliflower. After surveying 20 students, she finds 2 who eat cauliflower. Obtain and interpret a \(95 \%\) confidence interval for the proportion of students who eat cauliflower on Jane's campus using Agresti and Coull's method.
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