# 11.2 Goodness-of-fit test  (Page 6/20)

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## Formula review

$\sum _{k}\frac{{\left(O-E\right)}^{2}}{E}$ goodness-of-fit test statistic where:
O : observed values
E : expected values

k : number of different data cells or categories

df = k − 1 degrees of freedom

Determine the appropriate test to be used in the next three exercises.

An archeologist is calculating the distribution of the frequency of the number of artifacts she finds in a dig site. Based on previous digs, the archeologist creates an expected distribution broken down by grid sections in the dig site. Once the site has been fully excavated, she compares the actual number of artifacts found in each grid section to see if her expectation was accurate.

An economist is deriving a model to predict outcomes on the stock market. He creates a list of expected points on the stock market index for the next two weeks. At the close of each day’s trading, he records the actual points on the index. He wants to see how well his model matched what actually happened.

a goodness-of-fit test

A personal trainer is putting together a weight-lifting program for her clients. For a 90-day program, she expects each client to lift a specific maximum weight each week. As she goes along, she records the actual maximum weights her clients lifted. She wants to know how well her expectations met with what was observed.

Use the following information to answer the next five exercises: A teacher predicts that the distribution of grades on the final exam will be and they are recorded in [link] .

A 0.25
B 0.30
C 0.35
D 0.10

The actual distribution for a class of 20 is in [link] .

A 7
B 7
C 5
D 1

$df=$ ______

3

State the null and alternative hypotheses.

χ 2 test statistic = ______

2.04

p -value = ______

At the 5% significance level, what can you conclude?

We decline to reject the null hypothesis. There is not enough evidence to suggest that the observed test scores are significantly different from the expected test scores.

Use the following information to answer the next nine exercises: The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in [link] .

Ethnicity Number of Cases
White 2,229
Hispanic 1,157
Black/African-American 457
Asian, Pacific Islander 232
Total = 4,075

The percentage of each ethnic group in Santa Clara County is as in [link] .

Ethnicity Percentage of total county population Number expected (round to two decimal places)
White 42.9% 1748.18
Hispanic 26.7%
Black/African-American 2.6%
Asian, Pacific Islander 27.8%
Total = 100%

If the ethnicities of AIDS victims followed the ethnicities of the total county population, fill in the expected number of cases per ethnic group.
Perform a goodness-of-fit test to determine whether the occurrence of AIDS cases follows the ethnicities of the general population of Santa Clara County.

H 0 : _______

H 0 : the distribution of AIDS cases follows the ethnicities of the general population of Santa Clara County.

H a : _______

Is this a right-tailed, left-tailed, or two-tailed test?

right-tailed

degrees of freedom = _______

χ 2 test statistic = _______

88,621

p -value = _______

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p -value.

Let α = 0.05

Decision: ________________

Reason for the Decision: ________________

Conclusion (write out in complete sentences): ________________

Graph: Check student’s solution.

Decision: Reject the null hypothesis.

Reason for the Decision: p -value<alpha

Conclusion (write out in complete sentences): The make-up of AIDS cases does not fit the ethnicities of the general population of Santa Clara County.

Does it appear that the pattern of AIDS cases in Santa Clara County corresponds to the distribution of ethnic groups in this county? Why or why not?

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