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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
Observed Frequencies are the actual occurrences recorded in the data. Expected Frequencies are what we'd anticipate seeing if a particular assumption is correct. In a Goodness-of-fit test, these expected and observed frequencies are compared to verify the hypothesis.

Step by step solution

01

Defining Observed Frequency

The observed frequency is the number of times an event or characteristic occurs in a sample. For example, in a sample of 100 students, if 30 are left-handed, the observed frequency of left-handed students is 30.
02

Defining Expected Frequency

The expected frequency is the number of times we expect an event or characteristic to occur if a certain hypothesis is true. It is usually calculated by multiplying the sample size by the probability of the event. For instance, if we know that approximately 10% of the population is left-handed, in a sample of 100, we expect to find 10 left-handed students. The expected frequency would hence be 10.
03

Connecting with Goodness-of-Fit Test

A goodness-of-fit test compares the observed frequencies to the expected frequencies to see if they agree. If they are notably different, we reject the hypothesis that specifies the expected ratio. For example, if we observe 30 left-handed students where we expected 10, we might conclude that our hypothesis was incorrect and that the rate of left-handedness is higher.

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Most popular questions from this chapter

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

A chemical manufacturing company wants to locate a hazardous waste disposal site near a city of 50,000 residents and has offered substantial financial inducements to the city. Two hundred adults (110 women and 90 men) who are residents of this city are chosen at random. Sixty percent of these adults oppose the site, \(32 \%\) are in favor, and \(8 \%\) are undecided. Of those who oppose the site, \(65 \%\) are women; of those in favor, \(62.5 \%\) are men. Using the \(5 \%\) level of significance, can you conclude that opinions on the disposal site are dependent on gender?

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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. 8\. \(n=12\) b. \(n=16\) c. \(n=25\)

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