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T Test For Two Independent Samples
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- Week 9:Independent t -test t test for Two Independent Samples 1
- Independent Samples t - test ? The reason for hypothesis testing is to gain knowledge about an unknown population. ? Independent samples t-test is applied when we have two independent samples and want to make a comparison between two groups of individuals. The parameters are unknown. ? How is this different than a Z-test and One Sample t- test? 2
- Independent t - test ? We are interested in the difference between two independent groups. As such, we are comparing two populations by evaluating the mean difference. ? In order to evaluate the mean difference between two populations, we sample from each population and compare the sample means on a given variable. ? Must have two independent groups (i.e.samples) and one dependent variable that is continuous to compare them on. 3
- Examples: ? Do males and females significantly differ on their level of math anxiety? IV: Gender (2 groups: males and females) DV: Level of math anxiety ? Do older people exercise significantly less frequently than younger people? IV: Age (2 groups: older people and younger people) DV: Frequency of getting exercise 4
- Examples: ? Do 8th graders have significantly more unexcused absences than 7th graders in Toledo junior highs? IV: Grade (2 groups: 8th grade and 7th grade) DV: Unexcused absences ? Note that Independent t-test can be applied to answer each research question when the independent variable is dichotomous with only two groups and the dependent variable is continuous. 5
- Generate examples of research questions requiring an Independent Samples t-test: ? What are some examples that you can come up with? Remember- you need two independent samples and one dependent variable that is continuous. 6
- Assumptions ? The two groups are independent of one another. ? The dependent variable is normally distributed. ? Examine skewness and kurtosis (peak) of distribution ? Leptokurtosis vs. platykurtosis vs. mesokurtosis ? The two groups have approximately equal variance on the dependent variable. (When n1 = n2 [equal sample sizes] ,the violation of this assumption has been shown to be unimportant.) 7
- Steps in Independent Samples t-test 8
- Step 1: State the hypotheses Ho: The null hypothesis states that the two samples come from the same population. In other words, There is no statistically significant difference between the two groups on the dependent variable. Symbols: Non-directional: Ho: ?1 = ?2 Directional: H 0:µ ? µ1 2 or H 0:µ ? µ 1 2 • If the null hypothesis is tenable, the two group means differ only by sampling fluctuation – how much the statistic’s value varies from sample to sample or chance. 9
- Ha: The alternative hypothesis states that the two samples come from different populations. In other words, There is a statistically significant difference between the two groups on the dependent variable. Symbols: Non-directional: H 1:µ ? µ 1 2 Directional: H 1:µ > µ 1 2 H 1:µ < µ 1 2 10
- Step 2: Set a Criterion for Rejecting Ho ? Compute degrees of freedom ? Set alpha level ? Identify critical value(s) ? Table C. 3 (page 638 of text) 11
- Computing Degrees of Freedom ?Calculate degrees of freedom (df) to determine rejection region. n n df = 1 + 2 ? 2 -2 sample size for sample1+ sample size for sample2 • df describe the number of scores in a sample that are free to vary. • We subtract 2 because in this case we have 2 samples. 12
- More on Degrees of Freedom • In an Independent samples t-test, each sample mean places a restriction on the value of one score in the sample, hence the sample lost one degree of freedom and there are n-1 degrees of freedom for the sample. 13
- Set alpha level ? Set at .001, .01 , .05, or .10, etc. 14
- Identify critical value(s) ? Directional or non-directional? ? Look at page 638 Table C.3. ? To determine your CV(s) you need to know: ? df – if df are not in the table, use the next lowest number to be conservative ? directionality of the test ? alpha level 15
- Step 3: Collect data and Calculate t statistic t= x ?x 1 2 ? ( ? 1) + ( ? 1) ?? 1 2 2 ? variance ?s n 1 1 s n ?? + 1 2 2 ? ? n +n ?2 ?? n n ? ? 1 2 ?? 1 2? Whereby: n: Sample size s2 = variance df x :Sample mean subscript1 = sample 1 or group 1 subscript2 = sample 2 or group 2 16
- Step 4: Compare test statistic to criterion df = 18 ? = .05 , two-tailed test in this example • critical values are ± 2.101 in this example 17
- Step 5: Make Decision Fail to reject the null hypothesis and conclude that there is no statistically significant difference between the two groups on the dependent variable, t = , p > ?. OR Reject the null hypothesis and conclude that there is a statistically significant difference between the two groups on the dependent variable, t = , p < ?. • If directional, indicate which group is higher or lower (greater, or less 18 than, etc.).
- Interpreting Output Table: Mean APGAR Sample size SCORE Levene’s tests the assumption of equal variances – if p < .05, then variances t-value Degrees of are not equal and use a different test freedom to modify this: Here, we have met the assumption so use first row. CI p - value Observed difference 19 Retrieved on July 12, 2007 from SPSSShortManual.html between the groups
- Interpreting APA table: 20
- Variable Math anxiety t Gender Male 3.66 Female 3.98 3.35*** Age Under 40 years 3.32 Over 41 years 3.64 2.67** Note. **p < .01. ***p < .001. 21
- Examples and Practice ? See attached document. ? Create the following index cards from this lecture: ? When to conduct a t-test (purpose, conditions, and assumptions) ? t-test statistic formula for computation ? t-test statistic formula ? df formula 22
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