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Chapter 11: Problem 10

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

Short Answer

Expert verified
In a goodness-of-fit test, observed frequencies are the actual data counts obtained from a sample, while expected frequencies are the theoretically calculated counts we would expect if the null hypothesis is true. The difference between them is used to determine whether the observed data fits the theoretical expectation.

Step by step solution

01

Definition of Observed Frequencies

Observed frequencies refer to the actual count in each category obtained from a sample data. This is actual data collected in the form of responses, behaviors, observations, etc. or the numbers we've counted in a study or experiment.
02

Definition of Expected Frequencies

Expected frequencies are the theoretical frequencies or counts that we would expect to obtain if the null hypothesis of a goodness-of-fit test is true. They are usually calculated or derived based on some theoretical expectation, assumption or model, rather than direct measurement or observation.
03

Difference between Observed and Expected Frequencies

The main difference between observed and expected frequencies lies in the source of its generation. Observed frequencies are derived directly from the data, they are what we see or measure. Expected frequencies, on the other hand, are not directly observable but calculated based on a theoretical assumption or null hypothesis. The difference between these two frequencies is used in a goodness-of-fit test to determine whether our observed data fits a particular theoretical expectation or not.

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Most popular questions from this chapter

Sandpaper is rated by the coarseness of the grit on the paper. Sandpaper that is more coarse will remove material faster. Jobs such as the final sanding of bare wood prior to painting or sanding in between coats of paint require sandpaper that is much finer. A manufacturer of sandpaper rated 220, which is used for the final preparation of bare wood, wants to make sure that the variance of the diameter of the particles in their 220 sandpaper does not exceed \(2.0\) micrometers. Fifty-one randomly selected particles are measured. The variance of the particle diameters is \(2.13\) micrometers. Assume that the distribution of particle diameter is approximately normal. a. Construct the \(95 \%\) confidence intervals for the population variance and standard deviation. b. Test at a \(2.5 \%\) significance level whether the variance of the particle diameters of all particles in 220-rated sandpaper is greater than \(2.0\) micrometers.

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 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 \(\mathrm{com}-\) ment on what happens to the confidence interval of \(\sigma^{2}\) when the confidence level decreases. a. \(1-\alpha=.99\) b. \(1-\alpha=.95\) c. \(1-\alpha=.90\)
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