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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
A chi-square distribution is a probability distribution used primarily in hypothesis testing. Its main properties are that it is asymmetric and its skewness decreases as the degrees of freedom increase. Its only parameter, the degrees of freedom, is typically equal to n-1 (where n represents the number of independent, standard normal variables), and influences the shape of the distribution.

Step by step solution

01

Definition

The chi-square distribution is a theoretical probability distribution. It is primarily used in hypothesis testing, particularly with chi-square tests - which can be used to test the goodness of fit and the independence or homogeneity of two criteria. It is also used in the estimation of variance in a normal distribution.
02

Properties

The chi-square distribution has only one parameter: the degree of freedom (df). It is a family of curves, each characterized by a positive integer that indicates the degrees of freedom. The distribution is asymmetrical, skewed to the right, and the skewness decreases as the degrees of freedom increase. The mean of the distribution is equal to the degrees of freedom, and the variance is twice the degrees of freedom.
03

Parameter of Chi-square Distribution

The single parameter of a chi-square distribution is the degrees of freedom. The degrees of freedom are typically equal to n-1, where n represents the number of independent, standard normal variables. They also play a role in shaping the distribution: the larger the degrees of freedom, the closer the distribution approximates a normal distribution.

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Most popular questions from this chapter

Consider the following contingency table, which is based on a sample survey. \begin{tabular}{lrcr} \hline & Column 1 & Column 2 & Column 3 \\ \hline Row 1 & 137 & 64 & 105 \\ Row 2 & 98 & 71 & 65 \\ Row 3 & 115 & 81 & 115 \\ \hline \end{tabular} A. Write the null and alternative hypotheses for a test of independence for this table. b. Calculate the expected frequencies for all cells, assuming that the null hypothesis is true. c. For \(\alpha=.01\), find the critical value of \(\chi^{2}\). Show the rejection and nonrejection regions on the chi-square distribution curve. d. Find the value of the test statistic \(\chi^{2}\). e. Using \(\alpha=.01\), would you reject the null hypothesis?

A sample of 25 observations selected from a normally distributed population produced a sample variance of 35 . Construct a confidence interval for \(\sigma^{2}\) for each of the following confidence levels and comment on what happens to the confidence interval of \(\sigma^{2}\) when the confidence level decreases. a. \(1-\alpha=.99\) b. \(1-\alpha=.95\) c. \(1-\alpha=.90\)

A sample of seven passengers boarding a domestic flight produced the following data on weights (in pounds) of their carry-on bags. \(\begin{array}{lllllll}46.3 & 41.5 & 39.7 & 31.0 & 40.6 & 35.8 & 43.2\end{array}\) a. Using the formula from Chapter 3 , find the sample variance, \(s^{2}\), for these data. b. Make the \(98 \%\) confidence intervals for the population variance and standard deviation. Assume that the population from which this sample is selected is normally distributed. c. Test at the \(5 \%\) significance level whether the population variance is larger than 20 square pounds.

Determine the value of \(\chi^{2}\) for 14 degrees of freedom and an area of \(.10\) in the left tail of the chisquare distribution curve.

A drug company is interested in investigating whether the color of their packaging has any impact on sales. To test this, they used five different colors (blue, green, orange, red, and yellow) for the boxes of an over-the- counter pain reliever, instead of their traditional white box. The following table shows the number of boxes of each color sold during the first month.
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