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Magazine Chapter 8: Problem 8
Consider a \(90 \%\) confidence interval for \(\mu\). Assume \(\sigma\) is not known. For which sample size, \(n=10\) or \(n=20\), is the confidence interval longer?
Short Answer
Expert verified
The confidence interval is longer for \(n=10\).
Step by step solution
01
Understand the Problem
We need to find which sample size, either \(n = 10\) or \(n = 20\), results in a longer 90% confidence interval for the population mean \(\mu\), given that the population standard deviation \(\sigma\) is unknown.
02
Formula for Confidence Interval
When \(\sigma\) is unknown, we use the t-distribution to construct the confidence interval. The 90% confidence interval for \(\mu\) is given by \( \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \frac{s}{\sqrt{n}} \), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, and \(t_{\frac{\alpha}{2}, n-1}\) is the t-critical value for \(\frac{\alpha}{2}\) and \(n-1\) degrees of freedom.
03
Impact of Sample Size on Interval Length
The width of the confidence interval is based on \(2 \times t_{\frac{\alpha}{2}, n-1} \frac{s}{\sqrt{n}}\). As \(n\) increases, \(\frac{s}{\sqrt{n}}\) decreases, narrowing the interval. Additionally, increasing \(n\) typically decreases the t-critical value for a fixed confidence level.
04
Calculating Interval Lengths for Each Sample Size
For \(n=10\), the degrees of freedom are 9, and for \(n=20\), the degrees of freedom are 19. The difference in degrees of freedom affects the t-critical value: \(t_{0.05, 9}\) > \(t_{0.05, 19}\). Smaller \(n\) results in a larger t-critical value and larger \(\frac{s}{\sqrt{n}}\), indicating a longer interval.
05
Conclusion Based on Calculations
The confidence interval is longer for \(n=10\) because both the t-critical value is larger and the \(\frac{s}{\sqrt{n}}\) term is larger compared to \(n=20\). Therefore, the confidence interval shortens as the sample size increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
t-distribution
When constructing confidence intervals for the population mean \(\mu\), particularly when the population standard deviation \(\sigma\) is unknown, the t-distribution becomes a key tool.
Unlike the normal distribution, the t-distribution is more spread out with heavier tails. This accounts for the additional uncertainty introduced by estimating \(\sigma\) with the sample standard deviation \(s\).
The shape of the t-distribution depends on the degrees of freedom, a concept we will discuss more shortly. **Why Use the t-distribution?*** The t-distribution compensates for smaller sample sizes by providing wider confidence intervals.
* These intervals ensure a higher chance of capturing the true population mean than would be possible with the normal distribution.
* As the sample size increases, the t-distribution approaches the normal distribution.
Unlike the normal distribution, the t-distribution is more spread out with heavier tails. This accounts for the additional uncertainty introduced by estimating \(\sigma\) with the sample standard deviation \(s\).
The shape of the t-distribution depends on the degrees of freedom, a concept we will discuss more shortly. **Why Use the t-distribution?*** The t-distribution compensates for smaller sample sizes by providing wider confidence intervals.
* These intervals ensure a higher chance of capturing the true population mean than would be possible with the normal distribution.
* As the sample size increases, the t-distribution approaches the normal distribution.
sample size impact
Sample size is a crucial factor in the determination of a confidence interval's width and precision.
When the sample size increases, several changes occur that impact the confidence interval. **What Happens with Larger Sample Sizes?*** The term \(\frac{s}{\sqrt{n}}\) decreases as \(n\) grows, because dividing by a larger number results in a smaller quotient.
* This reduction leads to a narrower confidence interval, indicating more precise estimates of \(\mu\).
* In addition, for a given confidence level, the t-critical value \(t_{\frac{\alpha}{2}, n-1}\) also decreases with larger samples, as the distribution resembles the normal distribution more closely. Ultimately, larger sample sizes tend to produce more reliable statistics, creating confidence intervals that are shorter and therefore more informative about the true population parameter.
When the sample size increases, several changes occur that impact the confidence interval. **What Happens with Larger Sample Sizes?*** The term \(\frac{s}{\sqrt{n}}\) decreases as \(n\) grows, because dividing by a larger number results in a smaller quotient.
* This reduction leads to a narrower confidence interval, indicating more precise estimates of \(\mu\).
* In addition, for a given confidence level, the t-critical value \(t_{\frac{\alpha}{2}, n-1}\) also decreases with larger samples, as the distribution resembles the normal distribution more closely. Ultimately, larger sample sizes tend to produce more reliable statistics, creating confidence intervals that are shorter and therefore more informative about the true population parameter.
degrees of freedom
Degrees of freedom (df) in statistics represent the number of independent values that can vary in a data set when calculating a statistical measure.
For a t-distribution used in confidence intervals, the degrees of freedom are calculated as \(n-1\), where \(n\) is the sample size. **Influence of Degrees of Freedom on Intervals*** Higher degrees of freedom result in a t-distribution that is less spread out, or more like the normal distribution.
* This means narrower confidence intervals at the same confidence level, since a higher df causes a lower t-critical value.
* Therefore, when calculating a confidence interval, sample sizes with larger degrees of freedom create shorter intervals.Understanding degrees of freedom helps to appreciate why smaller sample sizes lead to wider intervals: fewer independent pieces of information lead to increased variability, which the t-distribution accounts for.
For a t-distribution used in confidence intervals, the degrees of freedom are calculated as \(n-1\), where \(n\) is the sample size. **Influence of Degrees of Freedom on Intervals*** Higher degrees of freedom result in a t-distribution that is less spread out, or more like the normal distribution.
* This means narrower confidence intervals at the same confidence level, since a higher df causes a lower t-critical value.
* Therefore, when calculating a confidence interval, sample sizes with larger degrees of freedom create shorter intervals.Understanding degrees of freedom helps to appreciate why smaller sample sizes lead to wider intervals: fewer independent pieces of information lead to increased variability, which the t-distribution accounts for.
