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Chapter 11: Problem 11

How is the expected frequency of a category calculated for a goodness-of-fit test? What are the degrees of freedom for such a test?

Short Answer

Expert verified
The expected frequency of a category in a goodness-of-fit test is calculated as the product of the total sample size and the probability of the category under the null hypothesis. The degrees of freedom for such a test is normally the number of categories minus one.

Step by step solution

01

Understanding Expected Frequency

The first part of the question requires understanding the concept of expected frequency. In the context of a goodness-of-fit test, the expected frequency of each category should be calculated given the null hypothesis. Assuming that the null hypothesis is true, each category in a probability distribution has a corresponding probability. If you multiply this probability by the total number of events or sample size, you get the expected frequency.
02

Calculation of Expected Frequency

In a goodness-of-fit test, the expected frequency of a particular category can be calculated using the formula: \(E = np\), where \(E\) is the expected frequency, \(n\) is the total number of observations or sample size, and \(p\) is the probability of the category under the null hypothesis.
03

Understanding Degrees of Freedom

The concept of degrees of freedom refers to the number of values in a study that have the freedom to vary. In the context of a goodness-of-fit test, the degrees of freedom is typically calculated as the number of categories minus one.
04

Calculation of Degrees of Freedom

Calculating degrees of freedom in a goodness-of-fit test is straightforward. If \(k\) is the number of categories, then the degrees of freedom is calculated using the formula: \(df = k - 1\)

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Most popular questions from this chapter

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. $$ \begin{array}{l|ccccc} \hline \text { Box color } & \text { Blue } & \text { Green } & \text { Orange } & \text { Red } & \text { Yellow } \\ \hline \text { Number of boxes sold } & 310 & 292 & 280 & 216 & 296 \\ \hline \end{array} $$ Using a \(1 \%\) significance level, test the null hypothesis that the number of boxes sold of each of these five colors. is the same.

Explain the difference between the observed and expected frequencies for a goodness-of-fit test.

An auto manufacturing company wants to estimate the variance of miles per gallon for its auto model AST727. A random sample of 22 cars of this model showed that the variance of miles per gallon for these cars is .62. Assume that the miles per gallon for all such cars are (approximately) normally distributed. a. Construct the \(95 \%\) confidence intervals for the population variance and standard deviation. b. Test at a \(1 \%\) significance level whether the sample result indicates that the population variance is different from \(.30\).

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. \(n=25\)
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