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Chapter 16: Problem 28

Why are larger samples better? Statisticians prefer large samples. Describe briefly the effect of increasing the size of a sample on the margin of error of a \(95 \%\) confidence interval.

Short Answer

Expert verified
Larger samples decrease the margin of error in confidence intervals, making estimates more precise.

Step by step solution

01

Understanding the Margin of Error

The margin of error (MOE) in the context of a confidence interval represents the range of values that will likely contain the population parameter. A smaller margin of error indicates more precise estimates. For a given confidence level, the MOE is primarily influenced by the standard deviation and the sample size.
02

The Formula for Margin of Error

The margin of error for a 95% confidence interval is calculated using the formula: \( MOE = Z \times \left( \frac{\sigma}{\sqrt{n}} \right) \), where \( Z \) is the Z-score corresponding to the confidence level (approximately 1.96 for 95%), \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
03

Effect of Increasing Sample Size

When sample size \( n \) increases, the term \( \frac{\sigma}{\sqrt{n}} \) decreases, reducing the entire margin of error. This means that with larger samples, the range of values predicted to contain the true population parameter becomes narrower, leading to more precise estimates.
04

Statisticians' Preference for Larger Samples

Statisticians prefer larger samples because they yield more reliable estimates of the population parameter. A reduced margin of error means that the estimate is closer to the true population parameter, thereby increasing the validity and credibility of the results.

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Key Concepts

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

Margin of Error
The Margin of Error (MOE) is a vital concept in statistics, especially when working with confidence intervals. Imagine MOE as a safety buffer. It is a range that captures the uncertainty inherent in collecting data from just a sample of a population. A smaller MOE indicates that our estimates of the true population value are more precise.
The MOE for a 95% confidence interval is calculated with the formula: \[ MOE = Z \times \left( \frac{\sigma}{\sqrt{n}} \right) \]Here, each component plays a specific role:
  • **Z-score**: Corresponds to the chosen confidence level (1.96 for 95%).
  • **Standard Deviation (蟽)**: How spread out the values in the population are.
  • **Sample Size (n)**: The number of observations in the sample. Increasing this reduces the MOE.
When you increase the sample size, the denominator (\(\sqrt{n}\)) becomes larger, meaning the MOE shrinks. This reduction signifies that our confidence in the sample as a representation of the population increases, hence enhancing statistical precision.
Confidence Interval
A Confidence Interval (CI) provides a range of values which is likely to contain the true population parameter. It is like saying, 鈥渋f I were to take 100 different samples and compute a CI from each, I'd expect about 95 of those intervals to contain the true parameter, assuming a 95% confidence level.鈥�
The formula for a CI is:\[ \text{CI} = \bar{x} \pm \text{MOE} \]Where:
  • **\( \bar{x} \)**: Is the sample mean, representing the midpoint of the interval.
  • **MOE**: Adds and subtracts from the mean to create the interval's boundary.
The width of a CI gives us insight into the precision of our estimate. A narrower interval often suggests a more precise estimate, achieved by increasing the sample size. Larger samples tend to "trap" the parameter more reliably within that interval, due to a decreased MOE.
Statistical Precision
Statistical Precision refers to the exactness of a statistical estimate, such as the mean or proportion derived from sample data. With more precise estimates, we base conclusions about the population on smaller error margins.
In practical terms:
  • **Reduced MOE**: A smaller margin of error signals higher precision, achieved effectively by larger samples.
  • **Consistent estimates**: With high precision, repeated samples would generate similar results.
  • **Confidence in results**: More precise measurements give statisticians stronger confidence that sample findings reflect the true nature of the population.
Hence, statisticians advocate for larger samples. Larger samples diminish the MOE, thereby refining the predictive accuracy of research findings. Ultimately, statistical precision means the difference between a rough guess and a sharp, trustworthy prediction.

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Most popular questions from this chapter

Pulling wood apart. How heavy a load (in pounds) is needed to pull apart p盲eces of Douglas fir 4 inches long and \(1.5\) inches square? Here are data from students doing a laboratory exercise: wood $$ \begin{array}{lllll} 33,190 & 31,860 & 32,590 & 26,520 & 33,280 \\ 32,320 & 33,020 & 32,030 & 30,460 & 32,700 \\ 23,040 & 30,930 & 32,720 & 33,650 & 32,340 \\ 24,050 & 30,170 & 31,300 & 28,730 & 31,920 \end{array} $$ (a) We are willing to regard the wood pleces prepared for the lab session as an SRS of all similar pieces of Douglas fir. Engineers also commonly assume that characteristics of materials vary Normally. Make a graph to show the shape of the distribution for these data. Does it appear safe to assume that the Normality condition is satisfied? Suppose that the strength of pieces of wood like these follows a Normal distribution with standard deviation 3000 pounds. (b) Give a \(95 \%\) confidence interval for the mean load required to pull the wood apart.

Addicted to Coffee. A Gallup Poll in July 2015 found that \(26 \%\) of the 675 coffee drinkers in the sample said they were addicted to coffee. Gallup announced, "For results based on the sample of 675 coffee drinkers, one can say with \(95 \%\) confidence that the maximum margin of sampling error is \(\pm 5\) percentage points." (a) Confidence intervals for a percent follow the form $$ \text { estimate } \pm \text { margin of error } $$ Based on the information from Gallup, what is the \(95 \%\) confidence interval for the percent of all coffee drinkers who would say they are addicted to coffee? (b) What does it mean to have \(95 \%\) confidence in this interval?

Confidence Level and Margin of Error. Example \(16.1\) (page 377) described NHANES survey data on the body mass index (BMI) of 654 young women. The mean BMI in the sample was \(x^{-} \bar{x}=26.8\). We treated these data as an SRS from a Normally distributed population with standard deviation \(\sigma=7.5\). (a) Give three confidence intervals for the mean BMl \(\mu\) in this population, using \(90 \%, 95 \%\), and \(99 \%\) confidence. (b) What are the margins of error for \(90 \%, 95 \%\), and \(9 \% \%\) confidence? How does increasing the confidence level change the margin of error of a confidence interval when the sample size and population standard deviation remain the same?

To give a \(99.9 \%\) confidence interval for a population mean \(\mu\), you would use the critical value (a) \(z^{*}=1.960\). (b) \(z^{*}=2.576\). (c) \(x^{+}=3.291\). Use the following information for Exercises \(16.12\) through 16.14. A laboratory scale is known to have a standard deviation of \(\sigma=0.001\) gram in repeated weighings. Scale readings in repeated weighings are Normally distributed, with mean equal to the true weight of the specimen. Three weighings of a specimen on this scale give \(3.412,3.416\), and \(3.414\) grams.

I want more muscle. Many young men in North America and Europe (but not in Asia) tend to think they need more muscle to be attractive. One study presented 200 young American men with 100 images of men with various levels of muscle. \({ }^{7}\) Researchers measure level of muscle in kilograms per square meter \(\left(\mathrm{kg} / \mathrm{m}^{2}\right)\) of fat-free body mass. Typical young men have about \(20 \mathrm{~kg} / \mathrm{m}^{2}\). Each subject chose two images, one that represented his own level of body muscle and one that he thought represented "what women prefer." 'The mean gap between self- image and "what women prefer" was \(2.35 \mathrm{~kg} / \mathrm{m}^{2}\). Suppose that the "muscle gap" in the population of all young men has a Normal distribution with standard deviation \(2.5 \mathrm{~kg} / \mathrm{m}^{2}\). Give a \(90 \%\) confidence interval for the mean amount of muscle young men think they should add to be attractive to women. (They are wrong: women actually prefer a level close to that of typical men.)
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