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Chapter 11: Problem 7

What is a goodness-of-fit test and when is it applied? Explain.

Short Answer

Expert verified
A goodness-of-fit test is a statistical method used to compare an observed distribution of data with a theoretical one. It is applied when one needs to test the hypothesis that the population is distributed in a particular way.

Step by step solution

01

Explaining Goodness-of-fit Test

A goodness-of-fit test in statistics is a way to compare an observed distribution of data with a theoretical one, which you expect your data to follow. You express the idea mathematically by setting an null hypothesis \(H_0\): the observed data follows the expected distribution, and the alternative hypothesis \(H_1\): the observed data does not follow the expected distribution.
02

Applications of Goodness-of-fit Test

The goodness-of-fit test is predominantly used when you want to check the hypothesis that the population is distributed in a particular way. Example scenarios include testing whether a dice is fair(i.e., all outcomes are equally likely), or whether a sample comes from a normal distribution. It can be computed using various methods, such as Chi-square goodness-of-fit test and Kolmogorov-Smirnov test, depending on the type of data and distribution.

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Most popular questions from this chapter

A sample of 21 units selected from a normally distributed population produced a variance of \(9.2 .\) Test at a \(5 \%\) significance level if the population variance is different from \(6.5\).

The following table lists the frequency distribution for 60 rolls of a die. $$ \begin{array}{l|cccccc} \hline \text { Outcome } & 1 \text { -spot } & 2 \text { -spot } & \text { 3-spot } & 4 \text { -spot } & \text { 5-spot } & \text { 6-spot } \\ \hline \text { Frequency } & 7 & 12 & 8 & 15 & 11 & 7 \\ \hline \end{array} $$ Test at a \(5 \%\) significance level whether the null hypothesis that the given die is fair is true.

A sample of certain observations selected from a normally distributed population produced a sample variance of \(46 .\) Construct a \(95 \%\) confidence interval for \(\sigma^{2}\) for each of the following cases and comment on what happens to the confidence interval of \(\sigma^{2}\) when the sample size increases. a. \(n=12\) b. \(n=16\) c. \(n=25\)

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

To make a test of independence or homogeneity, what should be the minimum expected frequency for each cell? What are the alternatives if this condition is not satisfied?
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