Textbook Question
The procedure for constructing a confidence interval about a mean is ________, which means minor departures from normality do not affect the accuracy of the interval.
40
views
7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
3:52 minutesProblem 9.4.17
Textbook Question
Critical Values Sir R. A. Fisher, a famous statistician, showed that the critical values of a chi-square distribution can be approximated by the standard normal distribution
χ²_k = [(z_k + √(2v – 1)) / 2]²
where v is the degrees of freedom and z_k is the z-score such that the area under the standard normal curve to the right of z_k is k. Use Fisher’s approximation to find χ²_0.975 and χ²_0.025 with 100 degrees of freedom. Compare the results with those found in Table VIII.
Verified step by step guidance1
Identify the given parameters: the degrees of freedom \(v = 100\), and the significance levels \(k = 0.975\) and \(k = 0.025\). These correspond to the right-tail probabilities for which we want to find the chi-square critical values.
Find the corresponding \(z_k\) values for each \(k\). Recall that \(z_k\) is the z-score such that the area to the right under the standard normal curve is \(k\). For example, \(z_{0.975}\) is the z-score with 2.5% in the right tail, and \(z_{0.025}\) is the z-score with 97.5% in the right tail. Use a standard normal table or calculator to find these \(z_k\) values.
Apply Fisher's approximation formula for the chi-square critical values:
\[\chi^2_k = \left( \frac{z_k + \sqrt{2v - 1}}{2} \right)^2\]
Substitute the values of \(v\) and each \(z_k\) into this formula to calculate the approximate chi-square critical values.
Calculate the square root term \(\sqrt{2v - 1}\) first, then add \(z_k\) to it, divide by 2, and finally square the result to get \(\chi^2_k\) for each \(k\).
Compare the approximated \(\chi^2_{0.975}\) and \(\chi^2_{0.025}\) values obtained from Fisher's formula with the exact critical values listed in Table VIII of the chi-square distribution. Note any differences and consider the accuracy of the approximation.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Distribution
The chi-square distribution is a probability distribution used primarily in hypothesis testing and confidence interval estimation for variance. It depends on degrees of freedom (v), which typically relate to sample size. The distribution is skewed right, but becomes more symmetric as degrees of freedom increase.
Recommended video:
Guided course
07:01
Intro to Least Squares Regression
Standard Normal Distribution and Z-Scores
The standard normal distribution is a symmetric, bell-shaped distribution with mean 0 and standard deviation 1. Z-scores represent the number of standard deviations a value is from the mean. Critical z-scores correspond to specific tail probabilities, used to find cutoff points in hypothesis testing.
Recommended video:
Guided course
03:17Finding Z-Scores for Non-Standard Normal Variables
Fisher’s Approximation for Chi-Square Critical Values
Fisher’s approximation relates chi-square critical values to the standard normal distribution, simplifying calculations for large degrees of freedom. It uses the formula χ²_k = [(z_k + √(2v – 1)) / 2]², where z_k is the z-score for tail probability k and v is degrees of freedom. This approximation helps estimate chi-square critical values without consulting tables.
Recommended video:
05:50Critical Values: t-Distribution
Related Videos
Related Practice