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Chapter 8: Problem 58

Critical values What critical value t* from Table B should be used for a confidence interval for the population mean in each of the following situations? (a) A 90% confidence interval based on n 12 observations. (b) A 95% confidence interval from an SRS of 30 observations.

Short Answer

Expert verified
(a) \( t^* \approx 1.796 \); (b) \( t^* \approx 2.045 \).

Step by step solution

01

Understanding the Critical Value

The critical value, denoted as \( t^* \), is a value that we use to calculate the margin of error for a confidence interval. It is based on the desired level of confidence and the degrees of freedom (\( df \)) in the t-distribution. The table to use is often referred to as a \( t \)-table, or Table B.
02

Determine Degrees of Freedom for Part (a)

For part (a), we have 12 observations. The degrees of freedom \( df \) is equal to the sample size minus 1. Thus, \( df = 12 - 1 = 11 \).
03

Finding the Critical Value for 90% Confidence Interval

For a 90% confidence interval and 11 degrees of freedom, we refer to Table B to find the critical \( t^* \) value, which typically matches the row with \( df = 11 \) and the column for 90% confidence. This value is usually around \( t^* = 1.796 \).
04

Determine Degrees of Freedom for Part (b)

For part (b), we have 30 observations. The degrees of freedom \( df \) is \( 30 - 1 = 29 \).
05

Finding the Critical Value for 95% Confidence Interval

For a 95% confidence interval with \( df = 29 \), we refer again to Table B. We look for the value corresponding to 95% confidence level with 29 degrees of freedom. This value is usually around \( t^* = 2.045 \).

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Key Concepts

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

Critical Value
In the world of statistics, a *critical value* is crucial for calculating the margin of error in a confidence interval. It is symbolized as \( t^* \). To determine \( t^* \), you need two pieces of information: the confidence level that you are aiming for and the degrees of freedom of your sample data. A critical value helps determine how far your sample mean could deviate from the population mean.
  • Higher confidence levels result in higher critical values, implying a wider confidence interval.
  • The critical value ensures that the confidence interval contains the true population parameter a specified percentage of the time.
Understanding the properties of the critical value is essential, as it varies depending on the confidence interval and sample size.
Confidence Interval
A *confidence interval* provides a range of values that, with a certain degree of confidence, contains the population parameter. It's a way of expressing statistical certainty. For instance, a 90% confidence interval means you can be 90% certain the interval includes the true population mean. Crafting an accurate confidence interval involves understanding several components:
  • Sample mean: The average value of the sample data.
  • Standard deviation: A measure of data variability.
  • Sample size: Number of observations in your sample.
By using the critical value, sample mean, and standard deviation, you can compute the margin of error and the confidence interval for more precise estimations.
Degrees of Freedom
*Degrees of freedom* (df) is a statistical concept that determines the number of values in a calculation that are free to vary. It's an integral part of finding the critical value in a t-distribution. For most applications involving t-distributions, degrees of freedom are computed as the sample size minus one \( (n-1) \). This calculation reflects the number of independent values that elaborate on your data set. Degrees of freedom are fundamental:
  • They adjust the critical value of \( t^* \) according to the size of the sample and level of confidence.
  • Higher degrees of freedom tend to shrink the critical value, leading to narrower confidence intervals.
  • Understanding degrees of freedom helps in selecting the right \( t^* \) from statistical tables, like Table B.
With this concept, you can more reliably infer results from sample data to the larger population.
Sample Size
When examining statistical data, the *sample size* is a central piece of information. Represented by \( n \), the sample size tells you how many data points are considered in your analysis. Larger samples provide more reliable estimates with smaller margins of error. It directly affects the degrees of freedom, leading to:
  • Increased sample size generally increases degrees of freedom, narrowing the confidence intervals.
  • It enhances the preciseness of estimates made about the population.
The question of how big a sample size should be is often tied to the desired level of confidence and the population's variability. Keeping these factors in mind ensures your confidence intervals are as informative as possible.

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Most popular questions from this chapter

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