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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χ2 & GAMMA
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.
Note the similarities between this t score and the t score we used to compute confidence intervals for means
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.
X1 = 60 X2 = 49
s1 = 12 s2 = 14
N1 = 25 N2 = 35
Decision making steps children than liberal parents? A study surveyed 25 Liberal parents and 25 Conservative
Step 2: Determine Distribution. When comparing Sample Means, use t distribution (normal distribution for difference of means).
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.
Step 4 Continued: (Compute) critical value t* at 95%, alpha = 5%.
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
CHI SQUARE ( critical value t* at 95%, alpha = 5%.χ2)
Why Chi ? ( critical value t* at 95%, alpha = 5%.χ2)
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 …
What do we do? marbles.
Chi Square ( marbles.χ2)
The formula for marbles.c2 is:
OR, sometimes written:
Where fo is the observed frequency of each category in each cell of a table.
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.
For each cell in the table we compare what we observe to what we expect by chance:
The Chi Square statistic tests : what we expect by chance:
Step #1: Hypotheses:
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.
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*
Assuming the null hypothesis is true what would be the expected values?
Row Margin * Column Margin
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
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
STEP 6: STATE CONCLUSION
Summing up the properties of the what we expect by chance:c2 Distribution: