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Hypothesis Testing Review. χ 2 & GAMMA. Example: Men and Women and support for Gun Control. H 0 : No difference between men and women in support for gun control. H 1 : Men are not equal to women , men could be more or less supportive, at one tail of the distribution or the other.

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Example: Men and Women and support for Gun Control

H0 : No difference between men and women in support for gun control.

H1 : Men are not equal to women , men could be more or less supportive, at one tail of the distribution or the other.

H2: Men > Women; men more favorable than women

H3: Men < Women; men less favorable.


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Compute the Critical Ratio Test

Note the similarities between this t score and the t score we used to compute confidence intervals for means


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Make a Decision

  • Determine which is greater– the t score from the table or the t score from your sample computations

  • If the t from your sample is greater than the t* from the table, you are significant at the level of α (risk) chosen

    State Results

  • State results in terms of the Null Hypothesis

  • If t* < t, we can reject the null hypothesis

  • If t* > t, we fail to reject the null


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Example: Are Conservative parents more strict with their children than liberal parents? A study surveyed 25 Liberal parents and 25 Conservative parents. Parents were rated on a scale from 1 (strict) to 100 (permissive) . Liberal parents had a mean permissiveness score of 60 with a std. dev. of 12. Conservative Parents have a mean score of 49 with a std. dev. of 14.

Liberal Conservative

X1 = 60 X2 = 49

s1 = 12 s2 = 14

N1 = 25 N2 = 35


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Decision making steps children than liberal parents? A study surveyed 25 Liberal parents and 25 Conservative

  • Step 1: State Hypothesis.

  • What is the null?

    • The null hypothesis is Liberal and Conservative parents are not different.

  • What is the Alternative?

    • The null hypothesis is Liberal Parents are more permissive (t* < t)

      Step 2: Determine Distribution. When comparing Sample Means, use t distribution (normal distribution for difference of means).


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Step 3: Determine Critical Value For the Distribution. Set critical value t* at 95%, alpha = 5%.

Step 4: Compute the Critical Ratio Test. To test the difference between the sample mean and the null hypothesis we first compute the deviation by the standard error of the sample mean.


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Step 4 Continued: (Compute) critical value t* at 95%, alpha = 5%.


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Step 5: Make Decision- critical value t* at 95%, alpha = 5%.

Recall Ha was t* < t (one-tailed) with df = n1 + n2 - 2 = 60 – 2  58 at .05 level

Computed t was 3.12

t* from t-table at .05 is 1.671

Step 6: State Conclusion

1.671 < 3.12

Therefore, we can reject the null hypothesis that liberals and conservatives are equally permissive, since t-value of 3.12 is greater than the critical value of t of 1.68


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CHI SQUARE ( critical value t* at 95%, alpha = 5%.χ2)


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Why Chi ? ( critical value t* at 95%, alpha = 5%.χ2)

  • We want to compare two variables, but…

  • Not all variables are interval-level, so we cannot use regression.

  • Hypothesis Tests for Difference of Means and Difference of Proportions only allow us to compare two groups with one value.

  • We need something else. . .


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Imagine a a bag that contained 90 white marbles and 10 black marbles.

If you drew 10 marbles, how many would you expect to come up white, and how many black?

We expect 9 white marbles and 1 black. But there is some probability that we will get 8/2 and some probability we will get 7/3 …


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What do we do? marbles.

  • We can compare what we would expect by chance to what we actually observed.

  • We can make a probabilistic statement about the chances of observing what we did based on our expectations.

  • Finally, we test the hypothesis that there is no real difference between what we observed and what we expected (using the 6 steps of hypothesis testing.


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Basic Assumption of the Null Hypothesis marbles.

  • There is no difference in the population, the difference you observe is just the chance variation of your sample.

  • We are comparing observed values (“frequency actually observed in our sample, written “fo”) to some set of expected by chance frequencies (written “fe”). The expected values being chance – with the null hypothesis holding that the difference between fo minus fe is really zero and the observed vs. expected difference is due to sampling error.


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Chi Square ( marbles.χ2)

  • The test statistic for testing hypothesis comparing 2 or more nominal categories

  • The Chi Square Statistic compares nominal values in a cross-tabulation table, making what are called row by column comparisons or “r x c” tables.

  • A Nominal variable …

  • … is a categorical variable with mutually exclusive categories. For example gender where male = 1 and female = 2.



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The formula for marbles.c2 is:

OR, sometimes written:

Where fo is the observed frequency of each category in each cell of a table.


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O or f marbles.o is what we observe from our sample, the observed frequency. NOTE that c2 works with frequencies in each cell.

E or fe is the expected frequency, the number of people who would show up in each cell IF the null hypothesis were true, if there was no racial difference in approval, if the frequencies were due solely to chance.


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For each cell in the table we compare what we observe to what we expect by chance:

  • Subtract the value of the hypothetical expectancy (fe) from the observed frequency (fo) for each cell.

  • Square each of these deviations.

  • Divide each of the squared differences by the expected value of each cell.

  • Finally, take the sum of the squared fo- f e differences to get χ2 .


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The Chi Square statistic tests : what we expect by chance:

  • Whether the difference between what you observe and what chance would predict is due to sampling error.

  • The greater the deviation of what we observe to what we would expect by chance, the greater the probability that the difference is NOT due to chance.



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Hypothesis Testing what we expect by chance:

  • Step 1: State Hypothesis

    • What is the Null?

    • What is the Alternative (Research) Hypothesis?



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FIRST STEP: COMPUTE PERCENTAGE TABLE what we expect by chance:

  • Row margins:

    • 335/908 = 36.9% Disapprove

    • 573/908 = 63.1% Approve

  • NOTE: Most Citizens Approve of Clinton

  • BUT: We Are Testing for a Gender Effect:Are Women more Supportive Than Men?


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PERCENTAGIZE TABLE what we expect by chance:


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INTERPRETATION what we expect by chance:

  • Row Marginal: Most Citizens (63%) Approved of Clinton.

  • Column Marginal: There Are More Men in the Sample (54%).

  • Based on the percent table looks like women are more supportive of Clinton than are men – 69% vs. 58%.

    Step #1: Hypotheses:

  • Null Hypothesis:H0: Women – Men = 0 + Chance

  • Cell Percents Show Women More Supportive Null Hypothesis Challenged.


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STEP # 2: WHAT IS THE DISTRIBUTION? what we expect by chance:

Categorizing same individuals in two ways:Approval and Gender

Looking at the Effect of an Independent Variable (Gender) on Dependent Variable (Approval).

This is a classic application for the c2 test.

· 1. We have nominal data in both variables - men vs women, approve vs disapprove.

·  2. The data are in the form of frequencies and

·  3. We are looking to see if there is an relationship between the two variables.


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Step 3: DETERMINE LEVEL OF SIGNIFICANCE what we expect by chance:

We set as our standard 95% confidence that the difference we observe in our study is not due to chance This is equivalent of setting alpha at risk level of 5% (=.05) for 95% confidence.

STEP 4: DETERMINE CRITICAL VALUE OF 2*

  • Degrees of Freedom:

    • (# rows – 1) * (# Columns – 1)

    • Here: (2 – 1) * (2 – 1) = 1 * 1 = 1

  • Look up Critical Value of 2* at 5% risk with 1 df and find: 2* = 3.84


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CHI-SQUARE DISTRIBUTION what we expect by chance:


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STEP 5: MAKE DECISION what we expect by chance:

  • Question: Is the proportion of men and women approving Clinton different from what you would expect by sampling error in more than 5% of all samples?

    Assuming the null hypothesis is true what would be the expected values?

  • What we need now are the expected values against which to compare our observed values. What would you expect by assuming the null is true?


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CALCULATING EXPECTED VALUES what we expect by chance:

  • Look at the Marginal Values:Note: 63% of all respondents approve

  • Therefore, assuming the null is true — that there is no gender difference — what would you expect the cell percentages to look like?

  • 37% of women should disapprove as well as 37% of men, + sampling error

  • 63% of women as well as 63% of men should approve, + sampling error

  • The proportions of men and women approving and disapproving should be the same + sampling error


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CONSTRUCT A “NULL” TABLE of f what we expect by chance:e Values


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TWO METHODS FOR COMPUTING EXPECTED VALUES what we expect by chance:

  • Method (Easiest):

    Row Margin * Column Margin

    Total N

    Cell a: 335 * 418 / 908 = 140,030 / 908 = 154

    Cell b: 335 * 490 / 908 = 164,150 / 908 = 181

    Cell c: 573 * 418 / 908 = 239,514 / 908 = 264

    Cell d: 573 * 490 / 908 = 280,770 / 908 = 309


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2. what we expect by chance:Null Percentage Method:If null is true, the Percentage of Men and Women should be the same. Then compute the frequency based on that percentage. Purely for Mathematical Completeness

  • Cell a: .369 * 418 = 154

  • Cell b: .369 * 490 = 181

  • Cell c: .631 * 418 = 264

  • Cell d: .631 * 490 = 309



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KEY QUESTIONS what we expect by chance:

  • How closely do fo values match fe values?

  • Do the squared fo – fe differences fit the null hypothesis?

  • Or, are the differences between observations and chance expectations so deviant as to justify rejection of the null?


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CALCULATE CHI-SQUARE what we expect by chance:


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Compare Chi-Square Values: what we expect by chance:

  • Look in table for alpha at .05 with 1 df

  • Critical value: 3.84

  • Chi-square computed from data: 10.07

STEP 6: STATE CONCLUSION

  • Computed value of chi-square greater than critical value, therefore, reject the null hypothesis.

  • Substantive interpretation?: The difference between groups on the IV is statistically different from the null hypothesis.


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Summing up the properties of the what we expect by chance:c2 Distribution:

  • c2 distribution ranges from zero to some positive value, i.e., ‘no difference’ to some ‘big difference’.

  • c2 distribution is not symmetrical, but skewed to the right, from zero to a large positive c2. Chi square looks at differences from zero. Itsvalue depends on the number of comparisons made, that is, the number of df. Note that the critical value of chi square gets bigger as the df get bigger, just because the more comparisons made the more likely you are to find differences, so df corrects for this.

  • There are many differentc2 distributions. Like the t distribution, c2 varies with degrees of freedom.


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