Statistics 303

1
Statistics 303

• Chapter 9
• Two-Way Tables

2
Relationships Between 2 Variables

• Relationships between two categorical variables
• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• In this case, we look at the percentages of one
  variable for each level of the other variable.
• Examples
• Gender and Soda Preference
• Country of Origin and Marital Status
• Smoking Habits and Socioeconomic Status

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Relationships Between 2 Variables

• Relationships between two categorical variables
• A two-way table can summarize the data for
  relationships between two categorical variables.
• Example Gender and Highest Degree Obtained

4
Relationships Between 2 Variables

• Example
• Percents

5
Two-Way Tables

• Two-way tables come about when we are interested
  in the relationship between two categorical
  variables.
• One of the variables is the row variable.
• The other is the column variable.
• The combination of a row variable and a column
  variable is a cell.

6
Two-Way Tables

• Example

7
Two-Way Tables

• To test whether or not there is a relationship
  between the row variable and the column variable
  we use the chi-square statistic (X2). We call it
  Pearsons Chi-square test.
• The null hypothesis (H0) is no relationship among
  the two variables, i.e. the variables are
  independent.
• The alternative hypothesis (Ha) is that there is
  a relationship, i.e. the variables are not
  independent.
• For 2x2 Tables, we require that all four expected
  cell counts be 5 or more.
• For tables larger than 2x2, we will use this
  approximation whenever the average of the
  expected counts is 5 or more and the smallest
  expected count is 1 or more.

8
Two-Way Tables

• The degrees of freedom for the chi-square test
  are (rows 1)(columns 1).
• For the example we have (2-1)(2-1) 1.
• The P-value is determined by looking in
  Chi-square Table, or computer output.

9
Two-Way Tables

• A comparison of the proportion of successes in
  two populations leads to a 2x2 table. We can
  compare two population proportions either by the
  chi-square test or by the two-sample z test from
  section 8.2
• These tests always give exactly the same result.
• The chi-square statistic is equal to the square
  of the z statistic and X2(1) critical values are
  equal to the squares of the corresponding N(0,1)
  critical values.
• Advantage of the z test We can test either
  one-sided or two-sided alternatives
• Chi-square test always tests the two-sided
  alternative
• Advantage of chi-square We can compare more than
  two populations
• z-Test compares only two populations

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Two-Way Tables

• The P-value is thus found as follows

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Two-Way Tables

• Analysis in SPSS gives us

The P-value is 0.103. Because this is larger
than 0.05 we fail to reject H0 and conclude there
is no significant relationship between gender and
tomato enjoyment.
12
Link between Diabetes and Heart Disease?

• Background
• Contradictory opinions
•  1. A diabetics risk of dying after a first
  heart attack is the same as that of someone
  without diabetes. There is no link between
  diabetes and heart disease.
• Vs.
• 2. Diabetes takes a heavy toll on the body and
  diabetes patients often suffer heart attacks and
  strokes or die from cardiovascular complications
  at a much younger age.
• So we use hypothesis test based on the latest
  data to see whats the right conclusion.
• There are a total of 5167 managed-care patients,
  among which 1131 patients are non-diabetics and
  4036 are diabetics. Among the non-diabetic
  patients, 42 of them had their blood pressure
  properly controlled (therefore its 475 of 1131).
  While among the diabetic patients only 20 of
  them had the blood pressure controlled (therefore
  its 807 of 4036).

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Link between Diabetes and Heart
Disease?DataDiabetes 1Not have diabetes,
2Have DiabetesControl 1Controlled,
2Uncontrolled
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Link between Diabetes and Heart Disease?

• Data

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Link between Diabetes and Heart Disease?
16
Link between Diabetes and Heart Disease?

• Hypothesis test
• 1.) Ho There is no link between diabetes and
  heart disease. (There is no relationship between
  diabetes and heart disease. Diabetes and heart
  disease are independent.)
• 2.)Ha There is link between diabetes and heart
  disease. (There is a relationship between
  diabetes and heart disease. Diabetes and heart
  disease are dependent.)
• 3.) Assume a significance level of .05

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Link between Diabetes and Heart Disease?SPSS
Output
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Link between Diabetes and Heart Disease?

• 4.)The computer gives us a Chi-Square Statistic
  of 229.268
• 5.)The computer gives us a p-value of .000
• 6.) Because our p-value is less than alpha, we
  would reject the null hypothesis.
• 7.) There IS sufficient evidence that there is
  link between diabetes and heart disease.

19
Is there a relationship between exposure to
R-rated movies and adolescent smoking?

• The study attempted to examine the relationship
  between exposure to R-Rated movies and smoking
  habits among adolescents. Smoking in R-rated
  movies is higher than any other movie-rating
  category. Therefore, the objective of this study
  was to determine if an association existed
  between parental restrictions on movies and
  adolescent cigarette use.
•  

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Is there a relationship between exposure to
R-rated movies and adolescent smoking?
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Is there a relationship between exposure to
R-rated movies and adolescent smoking?

• Hypothesis Test
• 1.)Ho There is no relationship between exposure
  to R-rated movies and tobacco use among
  adolescents
• 2.)Ha There is a relationship between the
  occurrence of tobacco use and the exposure to
  R-rated movies among adolescents
• 3.)alpha.05
• 4.) The computer gives us a chi-square test
  statistic of 469.003

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Is there a relationship between exposure to
R-rated movies and adolescent smoking?

• SPSS Output

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Is there a relationship between exposure to
R-rated movies and adolescent smoking?

• 5.) The computer output gives us a p-value that
  is .000
• 6.) Decision Rule
• If p-value
• If p-value alpha, we fail to reject Ho
• Because our p-value is less than our
  significance level (alpha), we would reject the
  null hypothesis
• 7.) Because we rejected Ho, we can conclude that
  there IS significant evidence that a relationship
  between exposure to r-rated movies and adolescent
  tobacco use exists.