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Chapter 11: Problem 12

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?

Short Answer

Expert verified
In a goodness-of-fit test, each category should ideally have a minimum expected frequency of at least 5. If this condition is not satisfied, the test might yield unreliable results. Alternatives include combining categories to lift their expected frequencies, or pursuing other statistical approaches like exact methods or non-parametric statistics.

Step by step solution

01

Minimum Frequency Criteria

During a goodness-of-fit test, the conditions specify that the expected frequency for every possible outcome or category should meet a minimum. Generally, it is recommended that this minimum expected frequency is at least 5 for each category to ensure a reliable result. This is typically referred to as the 'Rule of Five'.
02

Consequences If Not Satisfied

If this condition isn't met, that means some expected frequencies are too low. This might result in a less reliable test, as the goodness-of-fit model assumes that the data follows certain statistical distributions. When the Rule of Five cannot be met, it indicates that the data may not follow the assumed distribution adequately.
03

Alternative Methods

If the conditions for a goodness-of-fit test can鈥檛 be met due to insufficient expected frequencies, alternative methods can be used. These may involve combining categories using logical rules until the Rule of Five is met or alternatively using exact statistical methods or non-parametric statistics that do not rely on expected frequency assumptions.

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Most popular questions from this chapter

Home Mail Corporation sells products by mail. The company's management wants to find out if the number of orders received at the company's office on each of the 5 days of the week is the same. The company took a sample of 400 orders received during a 4 -week period. The following table lists the frequency distribution for these orders by the day of the week. $$ \begin{array}{l|ccccc} \hline \text { Day of the week } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thu } & \text { Fri } \\ \hline \text { Number of orders received } & 92 & 71 & 65 & 83 & 89 \\ \hline \end{array} $$ Test at a \(5 \%\) significance level whether the null hypothesis that the orders are evenly distributed over all days of the week is true.

The makers of Flippin' Out Pancake Mix claim that one cup of their mix contains 11 grams of sugar. However, the mix is not uniform, so the amount of sugar varies from cup to cup. One cup of mix was taken from each of 24 randomly selected boxes. The sample variance of the sugar measurements from these 24 cups was \(1.47\) grams. Assume that the distribution of sugar content is approximately normal. a. Construct the \(98 \%\) confidence intervals for the population variance and standard deviation. b. Test at a \(1 \%\) significance level whether the variance of the sugar content per cup is greater than 1.0 gram.

Describe in your own words a test of independence and a test of homogeneity. Give one example of each.

Construct the \(98 \%\) confidence intervals for the population variance and standard deviation for the following data, assuming that the respective populations are (approximately) normally distributed. $$ \text { a. } n=21, s^{2}=9.2 \quad \text { b. } n=17, s^{2}=1.7 $$

The manufacturer of a certain brand of lightbulbs claims that the variance of the lives of these bulbs is 4200 square hours. A consumer agency took a random sample of 25 such bulbs and tested them. The variance of the lives of these bulbs was found to be 5200 square hours. Assume that the lives of all such bulbs are (approximately) normally distributed. a. Make the \(99 \%\) confidence intervals for the variance and standard deviation of the lives of all such bulbs. b. Test at a \(5 \%\) significance level whether the variance of such bulbs is different from 4200 square hours.
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