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Chapter 11: Problem 1

Describe the chi-square distribution. What is the parameter (parameters) of such a distribution?

Short Answer

Expert verified
The chi-square distribution is a type of skewed probability distribution mainly used in hypothesis testing. The distribution is defined by one parameter, the degrees of freedom, denoted as 'k'.

Step by step solution

01

Definition of chi-square distribution

The chi-square distribution is a type of probability distribution that's widely used in inferential statistics and hypothesis testing. It's used extensively in the Pearson's chi-square test and the chi-square goodness-of-fit test.
02

Characteristics of the chi-square distribution

The chi-square distribution is an asymmetrical distribution that has a minimum value of 0. The shape of the chi-square distribution changes significantly based on the degrees of freedom.
03

Parameters of the chi-square distribution

The chi-square distribution has one parameter, labeled as k, which represents the degree of freedom. The degrees of freedom refers to the number of free choices left after a statistical constraint is imposed.

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Most popular questions from this chapter

Determine the value of \(\chi^{2}\) for 23 degrees of freedom and an area of \(.990\) in the left tail of the chisquare distribution curve.

Of all students enrolled at a large undergraduate university, \(19 \%\) are seniors, \(23 \%\) are juniors, \(27 \%\) are sophomores, and \(31 \%\) are freshmen. A sample of 200 students taken from this university by the student senate to conduct a survey includes 50 seniors, 46 juniors, 55 sophomores, and 49 freshmen. Using a \(2.5 \%\) significance level, test the null hypothesis that this sample is a random sample. (Hint: This sample will be a random sample if it includes approximately \(19 \%\) seniors, \(23 \%\) juniors, \(27 \%\) sophomores, and \(31 \%\) freshmen.)

A random sample of 25 students taken from a university gave the variance of their GPAs equal to \(.19 .\) a. Construct the \(99 \%\) confidence intervals for the population variance and standard deviation. Assume that the GPAs of all students at this university are (approximately) normally distributed. b. The variance of GPAs of all students at this university was \(.13\) two years ago. Test at a \(1 \%\) significance level whether the variance of current GPAs at this university is different from \(13 .\)

To make a goodness-of-fit test, what should be the minimum expected frequency for each category? What are the alternatives if this condition is not satisfied?

Determine the value of \(\chi^{2}\) for 13 degrees of freedom and a. 025 area in the left tail of the chi-square distribution curve b. \(.995\) area in the right tail of the chi-square distribution curve
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