The two sample t
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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

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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


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)?


Critical Values/Region for the z test( = .05)


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:

      ALPHATESTNt(Critical)

      .052-tailed57

      .011-tailed25

      .102-tailed32

      .051-tailed15


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


Answer #1 Conceptually

  • 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


    Answers

    • 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?


    Answers

    • 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


    Answers

    • 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?


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