Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

DEMONSTRATION 17.1 595
FOCUS ON PROBLEM SOLVING
DEMONSTRATION 17.1
1. The expected frequencies that you calculate must satisfy the constraints of the sample. For
the goodness-of-fit test, Σf e
= Σf o
= n. For the test of independence, the row totals and
column totals for the expected frequencies should be identical to the corresponding totals
for the observed frequencies.
2. It is entirely possible to have fractional (decimal) values for expected frequencies.
Observed frequencies, however, are always whole numbers.
3. Whenever df = 1, the difference between observed and expected frequencies ( f o
= f e
) will
be identical (the same value) for all cells. This makes the calculation of chi-square easier.
4. Although you are advised to compute expected frequencies for all categories (or cells),
you should realize that it is not essential to calculate all f e
values separately. Remember
that df for chi-square identifies the number of f e
values that are free to vary. Once you
have calculated that number of f e
values, the remaining f e
values are determined. You can
get these remaining values by subtracting the calculated f e
values from their corresponding
row or column totals.
5. Remember that, unlike previous statistical tests, the degrees of freedom (df) for a
chi-square test are not determined by the sample size (n). Be careful!
TEST FOR INDEPENDENCE
A manufacturer of watches would like to examine preferences for digital versus analog
watches. A sample of n = 200 people is selected, and these individuals are classified by age
and preference. The manufacturer would like to know whether there is a relationship between
age and watch preference. The observed frequencies ( f o
) are as follows:
Digital Analog Undecided Totals
Younger than 30 90 40 10 140
30 or Older 10 40 10 60
Column totals 100 80 20 n = 200
STEP 1
STEP 2
State the hypotheses, and select an alpha level The null hypothesis states that there is no
relationship between the two variables.
H 0
: Preference is independent of age. That is, the frequency distribution of preference
has the same form for people younger than 30 as for people 30 or older.
The alternative hypothesis states that there is a relationship between the two variables.
H 1
: Preference is related to age. That is, the type of watch preferred depends on a
person’s age.
We will set alpha to α = .05.
Locate the critical region Degrees of freedom for the chi-square test for independence are
determined by
df = (C – 1)(R – 1)
For these data,
df = (3 – 1)(2 – 1) = 2(1) = 2