Difference Between Means Test (“t” statistic)

Analysis of Variance (“F” statistic)

Difference Between Means (“t”) Test • So far we’ve examined several statistics that can be used to test hypotheses:

– Chi-Square (X 2), which requires all variables be categorical– Regression (R2), which requires all variables be continuous– Logistic regression (b and Exp b), which requires a nominal dependent variable

• The difference between the means test (t) is used to test hypotheses with categorical independent and continuous dependent variables– Gender height– Gender cynicism (1-5 scale)

• We compare the means of two randomly drawn samples– The null hypothesis can be rejected if the difference, expressed through the t

statistic, exceeds sampling error by an mount so large that there are less thanfive chances in 100 (p> .05) that the relationship is due to chance

– This sampling error is the “standard error of the difference between means” – the difference between all possible pairs of means, due to chance alone

• When using the t table we must know whether the hypothesis is 1-tailed (direction of effect predicted) or 2-tailed (direction not predicted)

• Major advantage: Remember that weak real-life effects can produce significant results? – When comparing means, we know their actual values. This lets us recognize

situations where differences are real but, in the real world, trivial.

Calculating t1. Obtain the “pooled sample variance” Sp

2

(Simplified method – midpoint between the two sample variances)

2.Compute the S.E. of the Diff. Between Means

3.Compute the t statistic Actual (“obtained”) difference between means

Predicted difference due to sampling error

• The result is a ratio: the smaller the predicted error, and the greater the obtained difference between means, the larger the t coefficient

• The larger the t, the more likely we are to reject the null hypothesis, that any difference between means is due to chance alone

• We use a table to determine whether the t is large enough to reject the null hypothesis (see next slide). We can reject the null if the probability that the difference between means is due to chance is less than five in one-hundred (p< .05).

• If the probability that the difference between the means is due to chance is greater than five in one-hundred (p> .05), the null hypothesis is true.

Classroom exercise - Jay’s police departmentH1: Male officers more cynical than females (1 - tailed)

H2: Officer gender determines cynicism (2 - tailed)

• Draw one sample of male officers, one of females• Compute each sample’s variance, then obtain the pooled sample variance

• Compute the S.E. of the Difference Between Means

• Calculate the t coefficient

• Check the t table (next slide) for significance. Confirm the hypothesis if there are less than 5 chances in 100 that it is correct. Be sure to use the correct significance row (1-tailed or 2-tailed).

Note: One-tailed hypotheses (direction of the effect on the dependent variable is predicted) require a smaller t to reach statistical significance. Why? Because they only using one side of the t statistic’s probability distribution. t-table

1. Is hypothesis one-tailed (direction of change in the DV predicted) or two-tailed (direction not predicted)?

H1: Males more cynical than females.This is one-tailed, so use the top row.

H2: Males and females differ in cynicism.This is two-tailed, so use the second row.

2. df, “Degrees of Freedom” represents sample size – add the numbers of cases in both samples, then subtract two: df = (n1 + n2) – 2

3. To call a t “significant” (thus reject the null hypothesis) the coefficient must be as large or larger than what is required at the .05 level; that is, we cannot take more than 5 chances in 100 that the difference between means is due to chance.

•For a one-tailed test, use the top row, then slide over to the .05 column. For a two-tailed test, use the second row, then slide to .05 column. If the t is smaller than the number at the intersection of the .05 column and the appropriate df row, it is non-significant.

•If the t is that size or larger, it is significant. Slide to the right to see if it is large enough to be significant at a more stringent level.

Parking lot exercise

Higher income More expensive car

1. Transfer your panel’s data from

the other coding sheet

2. Compute each sample’s variance, then calculate the pooled sample variance

3. Compute the S.E. of the Difference Between Means

4. Calculate the t coefficient

5. Check the t table for significance. Is the working hypothesis confirmed?

More complex mean comparisons:

Analysis of Variance

Dependent variable: continuousIndependent variable(s): categorical

Example: does officer professionalism vary between cities? (scale 1-10)

Calculate the “F” statistic, look up the table. An “F” statistic that is sufficiently large can overcome the null hypothesis that the differences between the means are due to chance.

When there are more than two groups:Analysis of Variance

City L.A. S.F. S.D.

Mean 8 5 3

• Stratified independent variable(s)

• F statistic is a ratio of “between-group” to “within” group differences. To overcome the null hypothesis, the differences in scores between groups (between cities and, overall, between genders) should be much greater than the differences in scores within cities Between group variance (error + systematic effects of ind. variable)

Within group variance (how scores disperse within each city)

“Two-way” Analysis of Variance

City L.A. S.F. S.D.

Mean – M 10 7 5

Mean - F 6 3 2

Between

Homework

Two random samples of 10 patrol officers from the XYZ Police Department, each officer tested for cynicism (continuous variable, scale 1-5)

Sample 1 scores: 3 3 3 3 3 3 3 1 2 5 -- Variance = .99

Sample 2 scores: 2 1 1 2 3 3 3 3 4 2 -- Variance = .93

Homework assignment

Pooled sample variance Sp2

Simplified method: midpoint between the two sample variances

Sp2 =

Standard error of the difference between means x1 -x2

x1 -x2 = Sp2 ( )

T-Test for significance of the difference between means

x1 -x2

t = --------------

x1 -x2

s21 + s2

2

2

1 1

n1 n2

+

CALCULATIONS

Pooled sample variance: .96

Standard error of the difference between means: .44

t statistic: 1.14

df – degrees of freedom: (n1 + n2) – 2 = 18

Would you use a ONE-tailed t-test OR a TWO-tailed t-test?

Depends on the hypothesis

Two-tailed (does not predict direction of the change): Gender cynicismOne-tailed (predicts direction of the change): Males more cynical than females

Can you reject the NULL hypothesis? (probability that the t coefficient could have been produced by chance must be less than five in a hundred)

NO – For a ONE-tailed test need a t of 1.734 or higher

NO – For a TWO-tailed test need a t of 2.101 or higher

Final exam practice

• You will be given scores and variances for two samples and asked to decide whether their means are significantly different.

• You will be asked to state the null hypothesis. You will then compute the t statistic. You be given formulas, but should know the methods by heart. Please refer to week 15 slide show.

• To compute the t you will compute the pooled sample variance and the standard error of the difference between means.

• You will then compute the degrees of freedom (adjusted sample size) and use the t table to determine whether the coefficient is sufficiently large to reject the null hypothesis.

– Print and bring to class: http://www.sagepub.com/fitzgerald/study/materials/appendices/app_f.pdf

– Use the one-tailed test if the direction of the effect is specified, or two-tailed if not

• You will be asked to express using words what the t-table conveys about the significance (or non-significance) of the t coefficient

• Sample question: Are male CJ majors significantly more cynical than female CJ majors? We randomly sampled five males and five females. Males: 4, 5, 5, 3, 4 Females: 4, 3, 4, 4, 5

– Null hypothesis: No significant difference between cynicism of males and females

– Variance for males (provided): 0.7 Variance for females (provided): 0.5

– Pooled sample variance = .6 SE of the difference between means = .49 t = .41 df = 8

– Check the “t” table. Can you reject the null hypothesis? NO

– Describe conclusion using words: The t must be at least 1.86 (one-tailed test) to reject the null hypothesis of no significant difference in cynicism, with only five chances in 100 that it is true.