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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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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