Chi-squared statistic
(Chapter 11) F statistic
(Chapter 12)
Definition:
What is t, Chi-squared, and F? Example of a research situation to employ this test statistic?
Level of Measurement:
Identify level of measurement for each test?
Five Steps in Hypothesis Testing:
Review the five steps in hypothesis testing for each test statistic (t, Chi-squared, and F)
Step 1:
What assumptions are made for the t/Chi-squared/F statistic?
Step 2:
What are the null and research hypothesis for the t/Chi-squared/F test?
Step 3:
What test statistic is used? Is there any assumption for sampling distribution or other assumptions?
Step 4:
How to compute t/Chi-squared/F statistic? List all the steps to obtain the test statistics. Don’t forget to include formulas.
Step 5:
What can be concluded by comparing the test statistic obtained and test statistic critical? How to read the t/Chi-squared/F distribution in the Appendices?
Chapter 13
Answer the following questions:
1. In what research situation do we need to employ correlation and regression? Example?
2. How to construct a scatter diagram or scatterplot? Example (practice drawing)?
3. Why do we need the best-fitting regression line?
4. How to obtain the regression equation? List all statistics and formulas needed.
5. What is the difference between r-squared and r?
CHAPTER 11:
THE CHI-SQUARE TEST (PG. 347-363)
HA TRINH
SOC3112
REVIEW: FIVE STEPS HYPOTHESIS TESTING
1.Making assumptions
2.Stating the research and null hypotheses and selecting alpha
3.Selecting the sampling distribution and specifying the test
statistic
4.Computing the test statistic
5.Making a decision and interpreting results
CHAPTER OBJECTIVES
Understanding hypothesis testing with chi-square
Defining statistical independence
Recognizing the limitations of Chi-square test: sample
size and statistical significance
CHI‐SQUARE AS A STATISTICAL TEST
Chi-square test: an inferential statistics technique designed to test for
significant relationships between two variables organized in a
bivariate table
Requires no assumptions about the shape of the population
distribution from which a sample is drawn
Can be applied to nominal or ordinal variables
STATISTICAL INDEPENDENCE
Independence (statistical): the absence of association
between two crosstabulatedvariables. The percentage
distributions of the dependent variable within each category of
the independent variable are identical
HYPOTHESIS TESTING WITH CHI‐SQUARE
1.Making assumptions (random sampling)
2.Stating the research and null hypotheses and selecting alpha
3.Selecting the sampling distribution and specifying the test statistic
4.Computing the test statistic
5.Making a decision and interpreting the results
HYPOTHESIS TESTING:
STEP 1: MAKING ASSUMPTIONS
The chi-square test requires no assumptions about the shape of the
population distribution from which the sample was drawn.
Like all inferential techniques it assumes random sampling
It can be applied to variables measured at a nominal and/or an ordinal
level of measurement
HYPOTHESIS TESTING:
STEP 2: STATING RESEARCH AND NULL HYPOTHESIS
The research hypothesis (H
1
) proposes that the two
variables are related in the population
The null hypothesis (H
0
) states that no association exists
between the two crosstabulatedvariables in the population, and
therefore the variables are statistically independent
HYPOTHESIS TESTING:
STEP 2: SELECTING ALPHA
Alpha is set in advance: 0.05; 0.01; 0.001
HYPOTHESIS TESTING:
STEP 3: SELECTING SAMPLING DISTRIBUTION AND SPECIFYING
THE TEST STATISTIC
Both the sampling distribution and the test statistic are chi-square
HYPOTHESIS TESTING:
STEP 4: DETERMINING DEGREE OF FREEDOM
HYPOTHESIS TESTING:
STEP 4:THE CONCEPT OF EXPECTED FREQUENCIES
Expected frequencies f
e
: the cell frequencies that would be
expected in a bivariate table if the two variables were
statistically independent
Observed frequencies f
o
: the cell frequencies actually
observed in a bivariate table
HYPOTHESIS TESTING:
STEP 4:THE CONCEPT OF EXPECTED FREQUENCIES
To obtain the expected frequencies for any cell in any cross-tabulation in
which the two variables are assumed independent: multiply the row and
column totals for that cell and divide the product by the total number
of cases in the table
HYPOTHESIS TESTING:
STEP 4:THE CHI-SQUARE (OBTAINED)
The test statistic that summarizes the differences between the observed (fo) and
the expected (fe) frequencies in a bivariate table
HYPOTHESIS TESTING:
STEP 4:THE CHI-SQUARE SAMPLING DISTRIBUTION
The sampling distribution of chi-square tells the probability of getting values of chi-square,
assuming no relationship exists in the population
The chi-square sampling distributions depend on the degrees of freedom
The χ2 sampling distribution is not one distribution, but is a family of distributions
The distributions are positively skewed. The research hypothesis for the chi-square is
always a one-tailed test
Chi-square values are always positive. The minimum possible value is zero, with no upper
limit to its maximum value
As the number of degrees of freedom increases. the χ2 distribution becomes more
symmetrical
HYPOTHESIS TESTING:
STEP 4:THE CHI-SQUARE SAMPLING DISTRIBUTION
HYPOTHESIS TESTING:
STEP 4:THE CHI-SQUARE TABLE (PG. 486)
HYPOTHESIS TESTING:
STEP 5: MAKING DECISION & INTERPRET RESULTS
Chi-square score and the corresponding p-value
Comparing p-value with alpha:
Larger than alpha: unable to reject null hypothesis of no relationship between two
variables
Lesser than alpha: reject null hypothesis of no relationship between two variables
LIMITATIONS OF THE CHI‐SQUARE TEST
The chi-square test does not give us much information about the strength of the
relationship or its substantive significance in the population
The chi-square test is sensitive to sample size. The size of the calculated chi-square
is directly proportional to the size of the sample, independent of the strength of
the relationship between the variables
The chi-square test is also sensitive to small expected frequencies in one or
more of the cells in the table
EXAMPLE(CONT. FROM CHAPTER 10)
PEOPLE HAVING TATTOOSIN SALT LAKE CITY AND PROVO
Having Tattoos? Salt Lake City Provo
Yes 120 50
No 480 450
Having Tattoos? Salt Lake City Provo Total
Yes 120 50 170
No 480 450 930
Total 600 500 1100
EXAMPLE:
STEP 1: MAKING ASSUMPTIONS
Random Sampling
Variables “Having Tattoos” and “City of Residence”
are nominal level of measurement
EXAMPLE:
STEP 2: RESEARCH-. NULL-HYPOTHESIS. AND ALPHA
H
1
: There is relationship between city of residence and
having tattoos
H
0
: There is no relationship between city of residence
and having tattoos
Alpha = 0.05
EXAMPLE:
STEP 3: SELECTING SAMPLING DISTRIBUTION AND SPECIFYING THE
TEST STATISTIC
Both the sampling distribution and the test statistic are chi-square
EXAMPLE:
STEP 4: CALCULATING THE CHI-SQUARE STATISTIC
1. Determining Degree of Freedom (df)
df = (r-1)*(c-1) = (2-1)*(2-1) = 1
2. Calculating Expected Frequencies (f
e
)
Having
Tattoos?
Salt Lake City
(f
0
)
f
e
Provo f
e
Total
Yes 120 170*600/1100
=92.73
50 170*500/1100
=77.28
170
No 480 930*600/1100
=507.28
450 930*500/1100
=422.73
930
Total 600 500 1100
EXAMPLE:
STEP 4: CALCULATING THE CHI-SQUARE STATISTIC
Having Tattoos? fo fe fo-fe (fo-fe)² (fo-fe)²/fe
Salt Lake City: Yes 120 92.73 27.27 743.65 8.01
Salt Lake City: No 480 507.28 -27.28 744.19 1.46
Provo: Yes 50 77.28 -27.28 744.19 9.62
Provo: No 450 422.73 27.27 743.65 1.75
Chi-square 20.87
EXAMPLE:
STEP 5: MAKING DECISION & INTERPRET RESULTS
df= 1
Chi-square = 20.87
Appendix D: p-value < 0.001, alpha = 0.05
Comparing: p-value < alpha
Conclusion: reject the null hypothesis of having no
relationship between having tattoos and city of residence
ScholarMatic: Explanation & Answer
Your ready answer from a verified tutor is just a click away for as little as $14.99
Click Order Now to get 100% Original Answer Customized to your instructions!
