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Chapter 7: Problem 7

Consider a \(90 \%\) confidence interval for \(\mu\). Assume \(\sigma\) is not known. For which sample size, \(n=10\) or \(n=20\), is the critical value \(t_{c}\) larger?

Short Answer

Expert verified
The critical value \(t_{c}\) is larger for \(n=10\).

Step by step solution

01

Understanding the Confidence Interval

A confidence interval is a range of values that is likely to contain the population parameter with a certain level of confidence. In this exercise, we are looking at a 90% confidence interval for the mean \(\mu\). Since the population standard deviation \(\sigma\) is not known, we will use the t-distribution to find the critical value \(t_{c}\).
02

Examining the t-distribution

The t-distribution is used when the population standard deviation is unknown and the sample size is small. The shape of the t-distribution depends on the degrees of freedom, which is \(n-1\), where \(n\) is the sample size. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
03

Calculating Degrees of Freedom for Both Sample Sizes

For a sample size \(n=10\), the degrees of freedom \(df = n - 1 = 9\). For a sample size \(n=20\), the degrees of freedom \(df = n - 1 = 19\).
04

Comparing Critical Values

The critical value \(t_{c}\) is determined by the t-distribution and the degrees of freedom. Generally, smaller degrees of freedom result in a larger critical value because fewer observations lead to greater variability. Therefore, \(t_{c}\) for \(n=10\) (\(df=9\)) will be larger than \(t_{c}\) for \(n=20\) (\(df=19\)).
05

Conclusion

The critical value \(t_{c}\) is larger for a sample size of \(n=10\) compared to \(n=20\), due to the smaller degrees of freedom resulting in greater variability in the t-distribution.

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Key Concepts

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

t-distribution
When dealing with statistical data, especially when estimating population parameters, the t-distribution becomes crucial.
Unlike the normal distribution, which requires knowledge of the population standard deviation, the t-distribution is used when the standard deviation is unknown and for small sample sizes.
  • **Origin:** The t-distribution was introduced by William Sealy Gosset under the pseudonym "Student," so it is sometimes called the Student's t-distribution.
  • **Shape:** This distribution is similar to the normal distribution but has thicker tails, which allows for the variability found within smaller samples.
  • **Usage:** It's particularly useful for creating confidence intervals and conducting hypothesis tests when the sample size is less than 30.
As more sample data is collected, the t-distribution begins to resemble a normal distribution, primarily because of increasing degrees of freedom.
degrees of freedom
In statistics, degrees of freedom (df) are vital as they define the shape of several probability distributions, including the t-distribution.
Essentially, degrees of freedom refer to the number of independent values or quantities which can vary in an analysis without violating any constraints.
  • **Explanation:** For example, if a sample size is 10, and we've calculated its mean, only 9 of the values remain independent (as they are free to vary).
  • **Context in t-distribution:** In a t-distribution, degrees of freedom are calculated as (n-1), where n is the sample size. Hence, more degrees of freedom equal more data and less uncertainty.
The degrees of freedom are crucial since they affect the critical value, and ultimately, the width of the confidence interval.
critical value
The critical value plays a pivotal role in statistical analysis, primarily when constructing confidence intervals.
It essentially marks the boundary or cutoff point in a distribution which determines the likelihood that a given parameter lies outside this range.
  • **Definition:** In the context of the t-distribution, the critical value is the point on the t-distribution that we reference to get our confidence level (e.g., 90%, 95%).
  • **Relationship with degrees of freedom:** With fewer degrees of freedom (like in smaller samples), the t-distribution curve is wider, resulting in a larger critical value.
Thus, the critical value inversely correlates to the degrees of freedom; as the degrees of freedom increase, the critical value decreases, leading to narrower confidence intervals.

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Most popular questions from this chapter

Archaeology: Pottery Santa Fe black-on-white is a type of pottery commonly found at archaeological excavations in Bandelier National Monument. At one excavation site a sample of 592 potsherds was found, of which 360 were identified as Santa Fe black-on-white (Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo and Casa del Rito, edited by Kohler and Root, Washington State University). (a) Let \(p\) represent the population proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate for \(p\). (b) Find a \(95 \%\) confidence interval for \(p .\) Give a brief statement of the meaning of the confidence interval. (c) Check Requirements Do you think the conditions \(n p>5\) and \(n q>5\) are satisfied in this problem? Why would this be important?

Confidence Intervals: Sample Size A random sample is drawn from a population with \(\sigma=12\). The sample mean is 30 . (a) Compute a \(95 \%\) confidence interval for \(\mu\) based on a sample of size 49 . What is the value of the margin of error? (b) Compute a \(95 \%\) confidence interval for \(\mu\) based on a sample of size 100 . What is the value of the margin of error? (c) Compute a \(95 \%\) confidence interval for \(\mu\) based on a sample of size 225 . What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the sample size increases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the sample size increases, does the length of a \(90 \%\) confidence interval decrease?

Confidence Intetvals: Values of \(\sigma\) A random sample of size 36 is drawn from an \(x\) distribution. The sample mean is \(100 .\) (a) Suppose the \(x\) distribution has \(\sigma=30\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (b) Suppose the \(x\) distribution has \(\sigma=20\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (c) Suppose the \(x\) distribution has \(\sigma=10\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the standard deviation decreases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a \(90 \%\) confidence interval decrease?

Pro Football and Basketball: Heights of Players Independent random samples of professional football and basketball players gave the following information (References: Sports Encyclopedia of Pro Football and Official NBA Basketball Encyclopedia). Note: These data are also available for download at the Online Study Center. $$ \begin{aligned} &\text { Heights (in } \mathrm{ft} \text { ) of pro football players: } x_{1} ; n_{1}=45\\\ &\begin{array}{llllllllll} 6.33 & 6.50 & 6.50 & 6.25 & 6.50 & 6.33 & 6.25 & 6.17 & 6.42 & 6.33 \\ 6.42 & 6.58 & 6.08 & 6.58 & 6.50 & 6.42 & 6.25 & 6.67 & 5.91 & 6.00 \\ 5.83 & 6.00 & 5.83 & 5.08 & 6.75 & 5.83 & 6.17 & 5.75 & 6.00 & 5.75 \\ 6.50 & 5.83 & 5.91 & 5.67 & 6.00 & 6.08 & 6.17 & 6.58 & 6.50 & 6.25 \\ 6.33 & 5.25 & 6.67 & 6.50 & 5.83 & & & & & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Heights (in } \mathrm{ft} \text { ) of pro basketball players: } x_{2} ; n_{2}=40\\\ &\begin{array}{llllllllll} 6.08 & 6.58 & 6.25 & 6.58 & 6.25 & 5.92 & 7.00 & 6.41 & 6.75 & 6.25 \\ 6.00 & 6.92 & 6.83 & 6.58 & 6.41 & 6.67 & 6.67 & 5.75 & 6.25 & 6.25 \\ 6.50 & 6.00 & 6.92 & 6.25 & 6.42 & 6.58 & 6.58 & 6.08 & 6.75 & 6.50 \\ 6.83 & 6.08 & 6.92 & 6.00 & 6.33 & 6.50 & 6.58 & 6.83 & 6.50 & 6.58 \end{array} \end{aligned} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 6.179, s_{1} \approx 0.366, \bar{x}_{2} \approx 6.453\), and \(s_{2} \approx 0.314 .\) (b) Let \(\mu_{1}\) be the population mean for \(x_{1}\) and let \(\mu_{2}\) be the population mean for \(x_{2}\). Find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the \(90 \%\) level of confidence, do professional football players tend to have a higher population mean height than professional basketball players? (d) Check Requirements Which distribution (standard normal or Student's \(t)\) did you use? Why? Do you need information about the height distributions? Explain.

Basic Computation: Confidence Interval for \(p\) Consider \(n=100\) binomial trials with \(r=30\) successes. (a) Check Requirements Is it appropriate to use a normal distribution to approximate the \(\hat{p}\) distribution? (b) Find a \(90 \%\) confidence interval for the population proportion of successes \(p .\) (c) Interpretation Explain the meaning of the confidence interval you computed.
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