The Two Sample t

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# The Two Sample t - PowerPoint PPT Presentation

The Two Sample t. Review significance testing Review t distribution Introduce 2 Sample t test / SPSS . Significance Testing . State a Null Hypothesis Calculate the odds of obtaining your sample finding if the null hypothesis is correct

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### The Two Sample t

Review significance testing

Review t distribution

Introduce 2 Sample t test / SPSS

Significance Testing
• 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
Significance the old fashioned way
• 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
• Compare the critical value with the obtained value  Are the odds of this sample outcome less than 5% (or 1% if alpha = .01)?
Directionality
• 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…
Non-Directional & Directional Hypotheses
• 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
• 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

-1.96 1.96

1.65

STUDENT’S t DISTRIBUTION
• 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:

ALPHA TEST N t(Critical)

.05 2-tailed 57

.01 1-tailed 25

.10 2-tailed 32

.05 1-tailed 15

Example: Single sample means, smaller N
• 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)

“2-Sample” t test
• 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?
• Null Hypothesis (Ho)
• The 2 pops. are not different in terms of the dependent variable
2-SAMPLE HYPOTHESIS TESTING
• 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
2-SAMPLE HYPOTHESIS TESTING
• 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”
• 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 HYPOTHESIS TESTING
• 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!

Applying the 2-Sample t Formula
• 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

• 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
• Reject null hypothesis: there is less than a 1% chance of obtaining a mean difference of 1 beer if the null hypothesis is true.
• 3.13 standard errors separates soc and non-soc majors with respect to their average beer consumption
Example 2
• 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
• SPSS DEMO
2-Sample Hypothesis Testing in SPSS
• 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.
Interpreting SPSS Output
• Difference in mean # of adult arrests between those who were abused as children & control group
Interpreting SPSS Output
• t statistic, with degrees of freedom
Interpreting SPSS Output
• “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)
“Sig.” & Probability
• 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
Example 2: Education & Ageat which First Child is Born

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

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

Education & Age at which 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?