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Chapter 8: Problem 22

A researcher plans to use a random sample of families to estimate the mean monthly family income for a large population. The researcher is deciding between a sample of size \(n=500\) and a sample of size \(n=1000 .\) Compared to using a sample size of \(n=500,\) a \(95 \%\) confidence interval based on a sample size of \(n=1000\) will be (a) narrower and would involve a larger risk of being incorrect. (b) wider and would involve a smaller risk of being incorrect. (c) narrower and would involve a smaller risk of being incorrect. (d) wider and would involve a larger risk of being incorrect. (e) narrower and would have the same risk of being incorrect.

Short Answer

Expert verified
(c) narrower and would involve a smaller risk of being incorrect.

Step by step solution

01

Determine Effect of Sample Size on Confidence Interval Width

The width of the confidence interval is dependent on the sample size and standard deviation. Larger sample sizes tend to produce narrower confidence intervals because the standard error (SE= \(\frac{\sigma}{\sqrt{n}}\)) decreases as the sample size \(n\) increases. Thus, a confidence interval based on \(n=1000\) will be narrower than one based on \(n=500\).
02

Determine the Effect of Sample Size on Risk of Being Incorrect

The risk of being incorrect here refers to the level of confidence, which is 95% for both sample sizes. Larger samples tend to provide more precise estimates and involve a smaller risk of being incorrect about the mean estimate. Therefore, a larger sample of \(n=1000\) generally involves a smaller risk of being incorrect compared to \(n=500\).
03

Select the Correct Option

Based on the insights from Step 1 and Step 2: the correct answer choice must be one where the confidence interval is narrower for \(n=1000\) than \(n=500\) and the risk of being incorrect is smaller for the larger sample size. Thus, the correct answer is option (c) "narrower and would involve a smaller risk of being incorrect."

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Key Concepts

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

Sample Size
When conducting research, the sample size you choose plays a crucial role in the accuracy and precision of your results. The sample size, denoted as \(n\), is the number of observations or data points collected. In larger samples, more data on the population is available, leading to more reliable statistical findings.

In our exercise, we compare two different sample sizes: 500 and 1000. A larger sample size generally means that the collected data is more representative of the population, reducing biases and increasing confidence in the results.
  • A sample size of 1000 provides a more precise estimate than 500.
  • A larger sample size improves the reliability of statistical analyses.
  • In practical scenarios, the availability of resources often determines the feasible sample size.
Remember, while increasing the sample size improves results, it must be balanced with the available budget and resources.
Standard Error
Standard Error (SE) is a statistical term that provides a measure of the accuracy of a sample mean's estimate compared to the population mean. It tells us how much variability exists in the sample mean if we were to take multiple samples from the same population.

SE is calculated as the standard deviation (蟽) of the population divided by the square root of the sample size \( SE = \frac{\sigma}{\sqrt{n}} \).
  • A larger sample size \( n \) decreases the standard error, leading to more precise estimates.
  • Lower standard error implies less variability in sample means across different samples.
  • While SE provides insight into estimate precision, it doesn't directly measure the width of confidence intervals.
In our example, increasing the sample size from 500 to 1000 decreases the standard error, resulting in a more accurate mean estimate.
Statistical Risk
Statistical risk, in the context of confidence intervals, refers to the probability of making an incorrect inference about the population parameter. Often, this is framed in terms of confidence levels, such as 95%, which means there is a 5% risk of concluding the wrong population parameter from the sample data.

A larger sample generally decreases the statistical risk because the estimates become more precise:
  • Risk decreases with larger samples, as they offer better estimates of the population parameters.
  • In the exercise, a larger sample size of 1000 reduces the risk compared to 500.
  • Managing statistical risk is crucial in studies where accurate decision-making is required.
Ultimately, with a sample size of 1000, the statistical risk related to our confidence interval becomes smaller.
Mean Estimate
The mean estimate is the average value calculated from a sample data set and is used to infer about the population mean. It serves as a central value around which the confidence interval is constructed.

With different sample sizes, the mean estimate's precision can vary:
  • Larger sample sizes provide a mean estimate closer to the population mean.
  • Smaller sample sizes may yield mean estimates with greater variability and potential bias.
  • In our exercise, using a sample size of 1000 results in a mean estimate that is likely more accurate than that from a sample size of 500.
Understanding how the mean estimate works is critical for interpreting confidence intervals and their implications on population parameters.

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Most popular questions from this chapter

A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. Which of the following would result in the smallest margin of error in estimating the mean salt content \(\mu ?\) (a) \(90 \%\) confidence; \(n=25\) (b) \(90 \%\) confidence; \(n=50\) (c) \(95 \%\) confidence; \(n=25\) (d) \(95 \%\) confidence; \(n=50\) (e) \(n=100\) at any confidence level

Does living near power lines cause leukemia in children? The National Cancer Institute spent 5 years and \(\$ 5\) million gathering data on this question. The researchers compared 638 children who had leukemia with 620 who did not. They went into the homes and measured the magnetic fields in children's bedrooms, in other rooms, and at the front door. They recorded facts about power lines near the family home and also near the mother's residence when she was pregnant. Result: no connection between leukemia and exposure to magnetic fields of the kind produced by power lines was found. (a) Was this an observational study or an experiment? Justify your answer. (b) Does this study show that living near power lines doesn't cause cancer? Explain.

Tonya wants to estimate what proportion of her school's seniors plan to attend the prom. She interviews an SRS of 50 of the 750 seniors in her school and finds that 36 plan to go to the prom. (a) Identify the population and parameter of interest. (b) Check conditions for constructing a confidence interval for the parameter. (c) Construct a \(90 \%\) confidence interval for \(p\). Show your method. (d) Interpret the interval in context.

A $$ 95 \%$$ confidence interval for the mean body mass index (BMI) of young American women is \(26.8 \pm 0.6\). Discuss whether each of the following explanations is correct. (a) We are confident that \(95 \%\) of all young women have BMI between 26.2 and 27.4 . (b) We are \(95 \%\) confident that future samples of young women will have mean BMI between 26.2 and 27.4 . (c) Any value from 26.2 to 27.4 is believable as the true mean BMI of young American women. (d) If we take many samples, the population mean BMI will be between 26.2 and 27.4 in about \(95 \%\) of those samples. (e) The mean BMI of young American women cannot be 28 .

How heavy a load (pounds) is needed to pull apart pieces of Douglas fir 4 inches long and 1.5 inches square? A random sample of 20 similar pieces of Douglas fir from a large batch was selected for a science class. The Fathom boxplot below shows the class's data. Explain why it would not be wise to use a \(t\) critical value to construct a confidence interval for the population mean \(\mu\)
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