1. Exam 5 Review
GOVT 201

2. Exam 5: Review Review Topics: Chi-Square Tests
a. Calculating Observed and Expected Frequencies
b. Calculate Chi-Square Formula (with and without Yates Correction).
c. Calculate Degrees of Freedom
2. Median Tests
3. Regression
a. Regression Model
1. Sum of X and Y
2. Calculate X and Y means
3. Regression Coefficient (Calculate Sum of Square and Product)
4. Y-Intercept

3. Nonparametric Tests: Two Nonparametric Tests:
The Chi-Square Test: concerned with the distinction between expected frequencies and observed frequencies.
The Median Test: A chi-square based test that also evaluates whether the scores fall above or below the median.

4. Nonparametric Tests:
Some things to know about chi square: 1) It compares the distribution of one variable (DV) across the category of another variable (IV) 2) It makes comparisons across frequencies rather than mean scores. 3) It is a comparison of what we expect to what we observe. Null versus Research Hypotheses: The null hypotheses states that the populations do not differ with respect to the frequency of occurrence of a given characteristic, whereas a research hypothesis asserts that sample difference reflects population difference in terms of the relative frequency of a given characteristic.

5. Nonparametric Tests:
Chi Square: Example: Political Orientation and Child Rearing Expected and Observed Frequencies: The chi-square test of significance is defined by Expected and Observed Frequencies. Expected Frequencies (fe) is the frequency we would expect to get if the hull hypothesis is true, that is there is no difference between the populations. Observed Frequencies (fo) refers to results we actually obtain when conducting a study (may or may not vary between groups). Only if the difference between expected and observed frequencies is large enough do we reject the null hypothesis and decide that a population difference does exist.

6. Political Orientation Child-Rearing Methods
Nonparametric Tests:
Chi Square: Political Orientation and Child Rearing: Observed Frequencies
Row Marginal
Political Orientation
Child-Rearing Methods
Liberals
Conservatives
Total 13 7
Permissive 20
Not Permissive 20 20 20
N = 40
Total
Col. Marginal

7. Political Orientation Child-Rearing Methods
Chi Square: Example: Political Orientation and Child Rearing Since the marginals are all equal, it is easy to calculate the expected frequencies: 10 in each cell.
Political Orientation
Child-Rearing Methods
Liberals
Conservatives
Total 10
Permissive 20
Not Permissive 20 20 20
N = 40
Total
It is unusual for a study to produce row and column marginals that are evenly split.

8. Political Orientation Child-Rearing Methods
Chi Square: Example: Political Orientation and Child Rearing Calculating expected frequencies when the marginals are not even:
Row Marginal
Political Orientation
Child-Rearing Methods
Liberals
Conservatives
Total 15 10 5
Permissive 25
Not Permissive 15 20 20
N = 40
Total
To determine if these frequencies depart from what is expected (null) by chance alone, we have to calculate the expected frequencies.

9. Calculating Expected Frequencies
fe = (column marginal)(row marginal) N
Example:
fe = (20)(25) 40
= 500
= 12.5

10. Political Orientation Child-Rearing Methods
Example: fe = (25)(20) 40
= 500
= 12.5
Political Orientation
Child-Rearing Methods
Liberals
Conservatives
Total
15 (12.5)
10 (12.5)
5 (7.5)
10 (7.5)
Permissive
25 (62.5%)
Not Permissive 15 20 20
N = 40
Total
The answer is 12.5 (62.5% of 20 or .625 x 20). We then know that the expected frequency for non permissive is 7.5 (20 – 12.5).

11. Political Orientation Child-Rearing Methods
If 25 of 40 respondents are permissive, than 62.5 % of them are permissive. To then determine the expected frequency, which asserts that Libs and Cons are the same (null) we have to calculate what would be 62.5% of 20 Libs and 20 Cons (the number of each that are in the study.
Political Orientation
Child-Rearing Methods
Liberals
Conservatives
Total
15 (12.5)
10 (12.5)
5 (7.5)
10 (7.5)
Permissive
25 (62.5%)
Not Permissive 15 20 20
N = 40
Total
The answer is 12.5 (62.5% of 20 or .625 x 20). We then know that the expected frequency for non permissive is 7.5 (20 – 12.5).

12. The Chi-Square Test Formula
Once we have the observed and expected frequencies we can use the following formula to calculate Chi-square.
Where:
fo = observed frequency in any cell
fe = expected frequency in any cell

13. Nonparametric Tests: Chi-Square Tests
Observed
Expected
Subtract
Square
Divide by fe
Sum
After obtaining fo and fe, we subtract fe from fo, square the difference, divide by the fe and then add them up.

14. Nonparametric Tests: Chi-Square Tests
Formula for Finding the Degrees of Freedom
df = (r-1)(c-1)
Where
r = the number of rows of observed frequencies
c = the number of columns of observed frequencies

15. Formula for Finding the Degrees of Freedom
Formula for Finding the Degrees of Freedom Since there are two rows and two columns of observed frequencies in our 2 x 2 table df = (r-1)(c-1) df = (2-1)(2-1) = (1)(1) = 1 Next Step, Table E, where we will find a list of chi-square scores that are significant at .05 and .01 levels. Table E (.05, df = 1): 3.84 Obtained X = 2.66 Retain null 2

16. Step by Step Chi-Square Test:
Step by Step Chi-Square Test: 1) Subtract each expected frequency from its corresponding observed frequency 2) Square the difference 3) Divide by the expected frequency, and then 4) Add up these quotients for all the cells to obtain the chi-square value

17. Correcting for Small Frequencies
Generally, chi square should be used with great care whenever some of the frequencies are below Five (5).
Though, this is not a hard and fast rule.

18. Yate’s Correction
HOWEVER, when working with a 2x2 table where any expected frequency is less than 10 but greater than 5, use Yate’s correction which reduces the difference between the expected and observed frequencies.
The vertical indicate that we must reduce the absolute value (ignoring minus signs) of each fo – fe by .5

19. Yate’s Correction Smoking Status Nationality American Canadian
Nonsmokers
Smokers
15 (11.67)
6 (9.33)
5 (8.33)
10 (6.67) 20 16
N = 36
Total 21 15

20. Requirements for the use of Chi-Square
A comparison between two or more samples.
Nominal data must be used.
Samples should have been randomly selected.
The expected cell frequencies should not be too small.

21. Regression Analysis Regression Analysis:
Regression analysis makes the importance of the variance more clear.
Goal of Research: Explain Variation
Example: Judge A versus Judge B
Why do some defendants get longer sentences than others?

22. Regression Analysis Judge Example: Sentences
What if a specific judge handed down the following sentence (in months) :
12, 13, 15, 19, 26, 27, 29, 31, 40, 48
How do we explain the variation? What factors contributed to some defendants getting 48 months while others got only 12 months?
Mean = 26 Months

23. Regression Analysis Judge Example: Sentences
The mean sentence tells us something about the judge’s sentencing pattern, but it does not help explain the wide variation in sentences.
Variance: S2
Is measured by calculating “the mean of the squared deviation.”
S2= Σ(X - χ )2 N
S2 = 125 months

24. Regression Analysis Judge Example: Sentences: Prior Convictions?
How much of this variance is the consequence of a defendant’s prior convictions?
Regression:
It enables it us to quantify the relative importance of any proposed factor or variable, in this case prior convictions.
Cause (IV): Prior Convictions
Effect (DV): Sentence Length.
Prior Convictions (IV: Cause) Sentence Length (DV: effect).

25. Regression Analysis Regression Model: Y = a + bX + e
Y = DV: Sentence Length (response variable).
X = IV: Prior Convictions (predictor variable).
a = Y-intercept: base-line: No Priors (What Y (DV: sentence length) is when X (IV: priors)= zero).
b = Slope (regression coefficient) for X. (Amount that Y (DV: sentence length) changes for each change in one unit of X (IV: priors)).
e = error term (what is unpredictable).

26. Y-Intercept (baseline) (Regression coefficient)
Regression Analysis
How much is Sentence (DV) effected by the number of a defendants prior convictions (IV: Cause)?
Regression Model:
DV: Effect
Y-Intercept (baseline)
Slope
(Regression coefficient) IV
(Cause)
Error
Term
Y = a + bX + e
Sentence Length.
No Priors. (Y when X=0)
Amount Y changes for change in X
Number of Priors
Unpredictable

27. Regression Analysis
Regression Model: Y = a + bX + e: Calculating each variable:
Priors (IV)
(X)
Sentence Length (DV)
(y) 3 1 6 5 4 10 8
N = 10, X = 40, Mean X =4 12 13 15 19 26 27 29 31 40 48
y = ?, Mean Y =26 (months)

28. Research Questions: Regression Model: Y = a + bX + e
Plug into regression formula: (adding mean for X and Mean for Y)
= 4 (mean of priors)
= 26 (mean of sentences (mean of Y))
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] =
a [y-intercept] =

29. Research Questions: Regression Model: Y = a + bX + e
Plug into regression formula: (adding mean for X and Mean for Y)
= 4 (mean of priors)
= 26 (mean of sentences (mean of Y))
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] =
a [y-intercept] =
Regression Formula

30. Research Questions: Regression Model: Y = a + bX + e
Plug into regression formula: (adding mean for X and Mean for Y)
= 4 (mean of priors)
= 26 (mean of sentences (mean of Y))
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] =
a [y-intercept] =
Adding mean for X (priors) and Y (sentence)

31. Research Questions: Regression Model: Y = a + bX + e
Calculate b [regression coefficient]:
= 4 (mean of priors)
= 26 (mean of sentences)
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] = Σ (X – χ)(Y – y) = 300 = 3
Σ(X – χ)
a [y-intercept] =

32. Regression Analysis
Calculating: b [regression coefficient]: Y = a + bX + e
= 4 (mean of priors) = 26 (mean of sentences (mean of Y))
300 = 3
100

33. Regression Model Regression Model: Y = a + bX + e
Calculating a [y-intercept]
= 4 (mean of priors)
= 26 (mean of sentences)
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] = Σ (X – χ)(Y – y) = 300 = 3
Σ(X – χ)
a [y-intercept] = – (3)(4) = 14
Y = a + bX + e
Y = X (Y = DV: Sentence, X = Prior Convictions)

34. Regression Analysis: Alternative Method
Regression Model: Y = a + bX + e
Alternative way to Calculate b [regression coefficient]
= 4 (mean of priors)
= 26 (mean of sentences)
Y = DV: Sentence Length.
X = IV: Prior Convictions.
b [regression coefficient] =
a [y-intercept] = SP
SSx
or:
b =

35. Regression Analysis 300 = 3 (this approach is implied in long-hand
Calculating: b [regression coefficient] Y = a + bX + e
= 4 (mean of priors) = 26 (mean of sentences (mean of Y))
300 = 3 (this approach is implied in long-hand
calculations.)

36. Regression Analysis
Calculating Regression Coefficient: (Sum of Products and Sum of Squares for x) Hh

37. Regression Analysis Calculating: a [Y-intercept] with SP and SSx data:4
Y = a + bX

38. Regression Analysis Regression Line:
It is a line that “falls closest to all the points in a scatter plot.” It crosses the Y axis at the Y-intercept and traces the slope (b) for the independent variable (X: Priors).
Y = a + bX + e
Y = X …

39. Regression Analysis Predicted and Actual Values (262)
A regression line represents a “predicted rather than an actual value.”

40. Regression Analysis Defining Error: Residual
X (IV) values will give you a predicted value for Y (DV)
which may in fact be different from the actual value of Y.
Ý = Predicted Y
Y = a + bX
Y = Observed Y
Residual is the Difference Between Ý and Y.
e = Y – Ý

41. Regression Analysis Plotting a Regression Line:
To plot a regression line you need to locate and then connect at least two points.
Easiest Line: Y-intercept and χ and y Mean
The easiest way to do this is to draw a line from the y-intercept (a) (X = 0, Y = a) and then through the χ (IV: priors) and y mean (average sentence (DV)).
a = (Y-intercept: base-line: No Priors) = 14
χ = (mean of IV: prior convictions) = 4
y = (mean of DV: sentences (in months)) = 26

42. Regression Analysis Plotting a Regression Line:
To plot a regression line you need to locate and then connect at least two points.
Easiest Line: Y-intercept and χ and y Mean
The easiest way to do this is to draw a line from the y-intercept (a) (X = 0, Y = a) and then through the χ (IV: priors) and y mean (average sentence (DV)).
a = (Y-intercept: base-line: No Priors) = 14
χ = (mean of IV: prior convictions) = 4
y = (mean of DV: sentences (in months)) = 26
Means for X and Y

43. Regression Analysis

44. Regression Analysis Mean for Y (Sentence) = 26 a [y intercept]= 14
Mean for X (Priors) = 4

45. Regression Analysis Plotting the Regression Line: Figure 11.2
If the Y-intercept and X and Y means are two close to plot a line, you can insert a larger value for X and then plug it into the equation.
Example: 10 Priors
Y (Ý) = DV: Sentence Length: ?
X = IV: Prior Convictions: 10
a = Y-intercept: base-line: No Priors: 14
b = Slope (regression coefficient) for X = 3
Y = a + bX
Ý = X
Ý = (10) = 44

46. Regression Analysis Plotting the Regression Line: Figure 11.2
If the Y-intercept and X and Y means are two close to plot a line, you can insert a larger value for X and then plug it into the equation.
Example: 13 Priors
Y (Ý) = DV: Sentence Length: ?
X = IV: Prior Convictions: 13
a = Y-intercept: base-line: No Priors: 14
b = Slope (regression coefficient) for X = 3
Y = a + bX
Ý = X
Ý = (13) = 53

47. Regression Analysis
The chart itself can predict how changes in X (priors) will effect Y (sentence):
13 Priors = 53 months.

48. Regression Analysis Requirements of Regression:
It is assumed that both variables are measured at the interval level.
Regression assumes a straight-line relationship. Extremely deviant cases in scatter plot are removed from the analysis.
Sample members must be chosen randomly in order to employ tests of significance.
To test the significance of the regression line, one must also assume normality for both variables or else have a large sample.

49. Regression Analysis: Review
Interpreting the Regression Line:
Regression analysis allows make predictions about one variable (IV: cause (X)) will effect another (DV: effect (Y)).
Example: Priors Convictions and Sentence Length
The Y-intercept tells us what Y (DV) is when the X (IV) is zero. If you have no priors (X (IV)), than the average sentence is 14 months.
The regression coefficient b tells us how much Y (DV) (sentence length) will increase or decrease of unit change in X (IV) (prior).
As such, we can also predict what the sentence length of will be for a defendant based on their number of prior convictions.

50. Regression Analysis Interpreting the Regression Line:
Regression analysis allows make predictions about one variable (IV: cause (X)) will effect another (DV: effect (Y)).
Example: 5 Priors
Y = a + bX
Ý = X
Ý = (5) = 29

51. Extra: Regression Analysis
Example of perfect Linear Relationship
Take the examples of Teachers’ salaries and seniority where seniority determines salaries.
Y = a + bX + e
Y = DV: Salary
X = IV: Seniority
a = Y-intercept: base-line: starting salary (What Y is when X = zero).
b = Slope (regression coefficient) for X. (Amount that Y
changes for each change in one unit of X).

52. Extra: Regression Analysis
Example of perfect Linear Relationship
Using this formula we can determine what an individual teacher’s salary (DV: Effect) will be starting with a baseline (a) of $12,000 and an extra $2000 for each year on the job (X: IV: Cause).
Y = a + bX
becomes:
Y = 12, X
(because this is a deterministic argument -seniority determines salary- there is no error).

53. Extra: Regression Analysis
Seniority and Salary: Figure Y = 12, (7) Y = $26,000