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The Two Sample t

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The Two Sample t

Review significance testing

Review t distribution

Introduce 2 Sample t test / SPSS

- State a Null Hypothesis
- Calculate the odds of obtaining your sample finding if the null hypothesis is correct
- Compare this to the odds that you set ahead of time (e.g., alpha)
- If odds are less than alpha, reject the null in favor of the research hypothesis
- The sample finding would be so rare if the null is true that it makes more sense to reject the null hypothesis

- Find the “critical value” of the test statistic for your sample outcome
- Z tests always have the same critical values for given alpha values (e.g., .05 alpha +/- 1.96)
- Use if N >100

- t values change with sample size
- Use if N < 100
- As N reaches 100, t and z values become almost identical

- Z tests always have the same critical values for given alpha values (e.g., .05 alpha +/- 1.96)
- Compare the critical value with the obtained value Are the odds of this sample outcome less than 5% (or 1% if alpha = .01)?

- Research hypothesis must be directional
- Predict how the IV will relate to the DV
- Males are more likely than females to…
- Southern states should have lower scores…

- Predict how the IV will relate to the DV

- Nondirectional
- Ho: there is no effect:
(X = µ)

- H1: there IS an effect:
(X ≠ µ)

- APPLY 2-TAILED TEST
- 2.5% chance of error in each tail

- Ho: there is no effect:
- Directional
- H1: sample mean is larger than population mean
(X > µ)

- Ho x ≤ µ
- APPLY 1-TAILED TEST
- 5% chance of error in one tail

- H1: sample mean is larger than population mean

-1.96 1.96

1.65

- Find the t (critical) values in App. B of Healey
- “degrees of freedom”
- # of values in a distribution that are free to vary
- Here, df = N-1
Practice:

ALPHATESTNt(Critical)

.052-tailed57

.011-tailed25

.102-tailed32

.051-tailed15

- A random sample of 16 UMD students completed an IQ test. They scored an average of 104, with a standard deviation of 9. The IQ test has a national average of 100. IS the UMD students average different form the national average?
- Professor Fred hypothesizes that Duluthians are more polite than the average American. A random sample of 50 Duluth residents yields an average score on the Fred politeness scale (FPS) of 31 (s = 9). The national average is known to be 29. Write the research an null hypotheses. What can you conclude?
- USE ALPHA = .05 for both

- Under the null hypothesis (no difference between means), there is more than a 5% chance of obtaining a mean difference this large.

Sampling distribution for one sample t-test

(a hypothetical plot of an infinite number of mean differences, assuming null was correct)

t (obtained) = 1.72

Critical Region

Critical Region

-2.131

(t-crit, df=15)

2.131(t-crit, df=15)

- Apply when…
- You have a hypothesis that the means (or proportions) of a variable differ between 2 populations

- Components
- 2 representative samples – Don’t get confused here (usually both come from same “sample”)
- One interval/ratio dependent variable
- Examples
- Do male and female differ in their aggression (# aggressive acts in past week)?
- Is there a difference between MN & WI in the proportion who eat cheese every day?

- The 2 pops. are not different in terms of the dependent variable

- Assumptions:
- Random (probability) sampling
- Groups are independent
- Homogeneity of variance
- the amount of variability in the D.V. is about equal in each of the 2 groups

- The sampling distribution of the difference between means is normal in shape

- We rarely know population S.D.s
- Therefore, for 2-sample t-testing, we must use 2 sample S.D.s, corrected for bias:
- “Pooled Estimate”

- Therefore, for 2-sample t-testing, we must use 2 sample S.D.s, corrected for bias:
- Focus on the t statistic:
t (obtained) = (X – X)

σx-x

- we’re finding the
difference between the two means…

…and standardizing this difference with the pooled estimate

- 2-Sample Sampling Distribution
- – difference between sample means (closer sample means will have differences closer to 0)

- t-test for the difference between 2 sample means:
- Does our observed difference between the sample means reflects a real difference in the population means or is due to sampling error?

- t critical 0 t critical

ASSUMING THE NULL!

- Example:
- Research Hypothesis (H1):
- Soc. majors at UMD drink more beers per month than non-soc. majors
- Random sample of 205 students:
- Soc majors: N = 100, mean=16, s=2.0
- Non soc. majors: N = 105, mean=15, s=2.5
- Alpha = .01
- FORMULA:
t(obtained) = X1 – X2

pooled estimate

- Research Hypothesis (H1):

- Null Hypothesis (Ho): Soc. majors at UMD DO NOT drink more beers per month than non-soc. Majors
- Directional 1 tailed test
- tcritical for 1 tailed test with 203 df and = .05 (use infinity) = 1.645
- tobtained = 3.13

- 3.13 standard errors separates soc and non-soc majors with respect to their average beer consumption

- Dr. Phil believes that inmates with tattoos will get in more fights than inmates without tattoos.
- Tattooed inmates N = 25, s = 1.06, mean = 1.00
- Non-Tattooed inmates N = 37, s =.5599, mean = 0.5278

- Null hypothesis?
- Directional or non?
- tcritical?
- Difference between means?
- Significant at the .01 level?

- Null hypothesis:
- Inmates with tattoos will NOT get in more fights than inmates without tattoos.

- Use a 1 or 2-tailed test?
- One-tailed test because the theory predicts that inmates with tattoos will get into MORE fights.

- tcritical (60 df) = 2.390
- tobtained = 2.23

- Reject the null?
- No, because the t(obtained) (2.23) is less than the t(critical, one-tail, df=398) (1.658)
- This t value indicates there are 2.23 standard error units that separate the two mean values
- Greater than a 1% chance of getting this finding under null

- This t value indicates there are 2.23 standard error units that separate the two mean values

- No, because the t(obtained) (2.23) is less than the t(critical, one-tail, df=398) (1.658)
- SPSS DEMO

- Independent Samples t Test Output:
- Testing the Ho that there is no difference in number of adult arrests between a sample of individuals who were abused/neglected as children and a matched control group.

- Difference in mean # of adult arrests between those who were abused as children & control group

- t statistic, with degrees of freedom

- “Sig. (2 tailed)”
- gives the actual probability of obtaining this finding if the null is correct
- a.k.a. the “p value” – p = probability
- The odds are NOT ZERO (if you get .ooo, interpret as <.001)

- gives the actual probability of obtaining this finding if the null is correct

- Number under “Sig.” column is the exact probability of obtaining that t-value (finding that mean difference) if the null is true
- When probability > alpha, we do NOT reject H0
- When probability < alpha, we DO reject H0

- As the test statistics (here, “t”) increase, they indicate larger differences between our obtained finding and what is expected under null
- Therefore, as the test statistic increases, the probability associated with it decreases

H0: There is no relationship between whether an individual has a

college degree and his or her age when their first child is born.

- What is the mean difference in age?
- What is the probability that this t statistic is due to sampling error?
- Do we reject H0 at the alpha = .05 level?
- Do we reject H0 at the alpha = .01 level?