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Confidence Interval

Suppose you have a sample from a population

You know the sample mean is an unbiased estimate of population mean

Question: What is the population mean?

Answer: You will never really know

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But ... you can determine, with some degree of certainty, a range which contains the mean

• Range is called the Confidence Interval of the Mean

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Definition

A Confidence Interval is a statement concerning a range of values which is likely to include the population mean based upon a sample from the population.

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Calculation:

CI = M ± t sM

And to use the CI

CI = M - t sM < μ < M + t sM

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Some Important Notes:

• For an interval estimate, you use a range of values as your estimate of an unknown quantity.

• When an interval estimate is accompanied by a specific level of confidence (or probability), it is called a confidence interval.

The general goal of estimation is to determine how much effect a treatment has.

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The general goal of estimation is to determine how much effect a treatment has.

• Whereas, the purpose of a confidence interval is to use a sample mean or mean difference to estimate the corresponding population mean or mean difference.

• Also, for independent-measures t-statistics, the values used for estimation is the difference between two population means.

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DATA:

• 13, 10, 8, 13, 9, 14, 12, 10, 11, 10, 15, 13, 7, 6, 15, 10

• SS ?

• Var?

• SM?

• df?

• 90% CI ?

• 95% CI ?

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Between Groups ANOVA

• Next step: Comparing three or more samples

• Nothing really new, just extending what is already learned

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For t-statistic: Single alternative hypothesis (H1)

Nondirectional (two-tail)

Directional

(one-tail)

For F-statistic: Many alternative hypotheses (H1's)

Always nondirectional

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Design: Between Groups ANOVA

• Partition the total variance of a sample into two separate sources (hence name of test)

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Partition the total variance of a sample into two separate sources (hence name of test)

Total variance – Variance associated with treatments and error

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Total variance – Variance associated with treatments and error

– Variance associated with just error

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Calculations: Between Groups ANOVA

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Treatment 1 Treatment 2 Treatment 3

4 8 3

5 8 2

4 9 1

6 10 3

n=

Σx=

Σx²=

Treatment mean

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Computational Formula for SSBG

SSBG =(ΣX1)²

n1+

(ΣX2)²

n2+

(ΣX3)²

n3..+

(ΣXk)²

nk[

(ΣXT)²

nT]

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Computational Formula for SSW

SSW= ΣX² [(ΣX1)²

n1+

(ΣX2)²

n2

+

(ΣX3)²

n3..+

(ΣXK)²

nK]

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ANOVA Summary

Source SS df MS F-Ratio

Treaments

SSBG

Error

SSW

Total

SSTotal

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Evaluating F-obtained: Between Groups ANOVA

Evaluate F-obtained value using an F-table

Similar to t-table except………

• Determining F value requires two separate degrees of freedom entries – Degrees of freedom for MS Between to locate the

correct column – Degrees of freedom for MS Within to locate the

correct row

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• Body of table typically gives

• values for p < .05 and p < .01

• Reject null hypothesis if:

• Obtained value exceeds tabled value

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Formal Properties: Between Groups ANOVABetween groups F-statistic is appropriate when Independent measure is

– Between subjects • Quantitative • Qualitative

– Design includes three or more treatment groups

Dependent measure is – Quantitative

– Scale of measurement is interval or better

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Between groups F-statistic assumes

Treatment groups are – Normally distributed – Homogeneity of within group variance

Subjects are:– Randomly and Independently selected from

population

Randomly assigned to treatment groups

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Comparing Treatments: Between Groups ANOVA

• Problem with multiple t-tests to compare treatment effects

• Multiple t-tests would yield some significant decisions by chance

• Can correct by making comparisons with a statistic that accounts for, "corrects for" multiple comparisons

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Number of different tests

Fisher’s LSD Test (Least Significant Difference)

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Tukey's HSD (Honest Significant Difference)

Where: CD = Absolute critical difference q = Studentized range value obtain from table entered

with – k groups signifying appropriate column – df for within treatments MS signifying row

n = number of observations per group

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Other Post –Hocs comparisions

• Scheffe

• Newman-Keuls

• Duncan

• Bonferroni