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Major Points. Formal Tests of Mean Differences Review of Concepts: Means, Standard Deviations, Standard Errors, Type I errors New Concepts: One and Two Tailed Tests Significance of Differences . Important Concepts. Concepts critical to hypothesis testing Decision Type I error

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Major Points

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Major points l.jpg

Major Points

  • Formal Tests of Mean Differences

  • Review of Concepts: Means, Standard Deviations, Standard Errors, Type I errors

  • New Concepts: One and Two Tailed Tests

  • Significance of Differences


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Important Concepts

  • Concepts critical to hypothesis testing

    • Decision

    • Type I error

    • Type II error

    • Critical values

    • One- and two-tailed tests


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Decisions

  • When we test a hypothesis we draw a conclusion; either correct or incorrect.

    • Type I error

      • Reject the null hypothesis when it is actually correct.

    • Type II error

      • Retain the null hypothesis when it is actually false.


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Possible Scenarios


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Type I Errors

  • Assume there are no differences (null hypothesis is true)

  • Assume our results show that they are not same (we reject null hypothesis)

  • This is a Type I error

    • Probability set at alpha ()

      •  usually at .05

    • Therefore, probability of Type I error = .05


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Type II Errors

  • Assume there are differences (alternative hypothesis is true)

  • Assume that we conclude they are the same (we accept null hypothesis)

  • This is also an error

    • Probability denoted beta ()

      • We can’t set beta easily.

      • We’ll talk about this issue later.

  • Power = (1 - ) = probability of correctly rejecting false null hypothesis.


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Critical Values

  • These represent the point at which we decide to reject null hypothesis.

  • e.g. We might decide to reject null when (p|null) < .05.

    • Our test statistic has some value with p = .05

    • We reject when we exceed that value.

    • That value is the critical value.


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One- and Two-Tailed Tests

  • Two-tailed test rejects null when obtained value too extreme in either direction

    • Decide on this before collecting data.

  • One-tailed test rejects null if obtained value is too low (or too high)

    • We only set aside one direction for rejection.


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One- & Two-Tailed Example

  • One-tailed test

    • Reject null if number of red in Halloween candies is higher

  • Two-tailed test

    • Reject null if number of red in Halloween candies is different (whether higher or lower)


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Within subjects t tests

  • Related samples

  • Difference scores

  • t tests on difference scores

  • Advantages and disadvantages


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Related Samples

  • The same participant / thing give us data on two measures

    • e. g. Before and After treatment

    • Usability problems before training on PP and after training

    • Darts and Pros during same time period

  • With related samples, someone high on one measure probably high on other(individual variability).

Cont.


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Related Samples--cont.

  • Correlation between before and after scores

    • Causes a change in the statistic we can use

  • Sometimes called matched samples or repeated measures


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Difference Scores

  • Calculate difference between first and second score

    • e. g. Difference = Before - After

  • Base subsequent analysis on difference scores

    • Ignoring Before and After data


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Difference between Darts and Pros


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Results

  • Pros got more gains than darts

  • Was this enough of a change to be significant?

  • If no difference, mean of computed differences should be zero

    • So, test the obtained mean of difference scores against m = 0.

    • Use same test as in one sample test


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t test

D and sD = mean and standard deviation of differences.

df = 100 - 1 = 9 - 1 = 8

Cont.


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t test--cont.

  • With 99 df, t.01 = +2.62 (Table E.6)

  • We calculated t = 2.64

  • Since 6.64 > 2.62, reject H0

  • Conclude that the Pros did get significantly more than Darts


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Advantages of Related Samples

  • Eliminate subject-to-subject variability

  • Control for extraneous variables

  • Need fewer subjects


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Disadvantages of Related Samples

  • Order effects

  • Carry-over effects

  • Subjects no longer naïve

  • Change may just be a function of time

  • Sometimes not logically possible


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Between subjects t test

  • Distribution of differences between means

  • Heterogeneity of Variance

  • Nonnormality


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Pros during ups and downs in DOW

  • Effect of fluctuations in DOW: did it effect Pros

    • Different question than previously

  • Now we have two independent groups of data

    • Pros during positive DOW, and Pros during negative DOW

    • We want to compare means of two groups


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Effect of changes in DOW


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Differences from within subjects test

Cannot compute pairwise differences, since we cannot compare two random data points

We want to test differences between the two sample means (not between a sample and population)


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Analysis

  • How are sample means distributed if H0 is true?

  • Need sampling distribution of differences between means

    • Same idea as before, except statistic is (X1 - X2) (mean 1 – mean2)


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Sampling Distribution of Mean Differences

  • Mean of sampling distribution = m1 - m2

  • Standard deviation of sampling distribution (standard error of mean differences) =

Cont.


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Sampling Distribution--cont.

  • Distribution approaches normal as n increases.

  • Later we will modify this to “pool” variances.


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Analysis--cont.

  • Same basic formula as before, but with accommodation to 2 groups.

  • Note parallels with earlier t


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Degrees of Freedom

  • Each group has 5 data points.

  • Each group has n - 1 = 50 - 1 = 8 df

  • Total df = n1- 1 + n2 - 1 = n1 + n2 - 2 50 + 50 - 2 = 98 df

  • t.01(98) = +2.62 (approx.)


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Assumptions

  • Two major assumptions

    • Both groups are sampled from populations with the same variance

      • “homogeneity of variance”

    • Both groups are sampled from normal populations

      • Assumption of normality

        • Frequently violated with little harm.


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Heterogeneous Variances

  • Refers to case of unequal population variances.

  • We don’t pool the sample variances.

  • We adjust df and look t up in tables for adjusted df.

  • Minimum df = smaller n - 1.

    • Most software calculates optimal df.


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