The Manga Guide to Statistics

188 Chapter 7

Exercise and Answer

Exercise

The table below is the same as the cross tabulation found on page 138.

Using a chi-square test of independence, estimate if the Cramer’s coefficient for type of food often

ordered and preference for coffee or tea in the population “people of age 20 or older residing in Japan” is

greater than 0. This is the same as estimating whether there is a correlation between type of food often

ordered and preference for coffee or tea. Use 0.01 as the significance level.

Answer

Step 1

Define the population. The population in this case is “people of age 20 or older

residing in Japan.”

Step 2

Set up a null hypothesis and an

alternative hypothesis.

The null hypothesis is “type of food often ordered and

preference for coffee or tea are not correlated.” The

alternative hypothesis is “type of food often ordered and

preference for coffee or tea are correlated.”

Step 3

Choose which hypothesis test

to do.

A chi-square test of independence will be applied.

Step 4

Determine the significance level. The significance level is 0.01.

Step 5

Calculate the test statistic from the

sample data.

A chi-square test of independence is being used in this

exercise. Therefore, the test statistic is Pearson’s chi-

square test statistic (χ

0

2

). The value of χ

0

2

in this exercise

has already been calculated on page 141. χ

0

2

= 3.3483

Step 6

Determine whether the test

statistic obtained in step 5 is in the

critical region.

The test statistic χ

0

2

is 3.3483. Because the significance

level (a) is 0.01, the critical region is 9.2104 or above,

according to the table of chi-square distribution on

page 103. The test statistic is not within the critical

region.

Step 7

If the test statistic is in the criti-

cal region in step 6, reject the null

hypothesis. If not, fail to reject the

null hypothesis.

The test statistic was not within the critical region. Thus,

the null hypothesis “type of food often ordered and pref-

erence for coffee or tea are not correlated” cannot be

rejected.

Type of food

often ordered

Sum

Japanese

European

Chinese

Preference for coffee or tea

Coffee

43

51

29

123

Tea

33

53

41

127

Sum

76

104

70

250

Let’s Explore the Hypothesis Tests 189

Summary

A hypothesis test is an analysis technique used to estimate whether the analyst’s

hypothesis about the population is correct using the sample data.

The formal name for a hypothesis test is statistical hypothesis testing.

Test statistics are obtained from a function that calculates a single value from the

sample data.

In general, 0.05 or 0.01 is used as the significance level.

The critical region is an area that corresponds to the significance level (also called the

alpha value and expressed by the symbol a).

A chi-square test of independence is an analysis technique used to estimate whether

the Cramer’s coefficient for a population is 0. It can also be said that it is an analysis

technique used to estimate whether the two variables in a cross tabulation are cor-

related.

If the Cramer’s coefficient for a population is 0, Pearson’s chi-square test statistic fol-

lows a chi-square distribution.

The P-value in a test of independence is a probability that gives a Pearson’s chi-square

test statistic equal to or greater than the value earned in the case when the null

hypothesis is true.

When making a conclusion in a hypothesis test, there are two bases of judgment:

1. Whether the test statistic is in the critical region

2. Whether the P-value is smaller than the significance level

The process of analysis in any hypothesis test is the same as the process for the test of

independence or any other kind of test. The actual procedure is:

Step 1

Define the population.

Step 2

Set up a null hypothesis and an alternative hypothesis.

Step 3

Choose which hypothesis test to do.

Step 4

Determine the significance level.

Step 5

Calculate the value of the test statistic from the sample data.

Step 6

Determine whether the test statistic obtained in step 5 is in the critical

region.

Step 7

If the test statistic is in the critical region in step 6, reject the null

hypothesis. If not, fail to reject the null hypothesis.

Step 6p

Determine whether the P-value corresponding to the test statistic

obtained in step 5 is smaller than the significance level.

Step 7p

If the P-value is smaller than the significance level in step 6p, reject the

null hypothesis. If not, fail to reject the null hypothesis.

•

•

•

•

•

•

•

•

•

•