statistics 11 confidence interval suppose you have a sample from a population you know the sample...
TRANSCRIPT
![Page 1: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/1.jpg)
Statistics 1 1
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
![Page 2: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/2.jpg)
Statistics 1 2
But ... you can determine, with some degree of certainty, a range which contains the mean
• Range is called the Confidence Interval of the Mean
![Page 3: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/3.jpg)
Statistics 1 3
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.
![Page 4: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/4.jpg)
Statistics 1 4
Calculation:
CI = M ± t sM
And to use the CI
CI = M - t sM < μ < M + t sM
![Page 5: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/5.jpg)
Statistics 1 5
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.
![Page 6: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/6.jpg)
Statistics 1 6
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.
![Page 7: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/7.jpg)
Statistics 1 7
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 ?
![Page 8: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/8.jpg)
Statistics 1 8
Between Groups ANOVA
• Next step: Comparing three or more samples
• Nothing really new, just extending what is already learned
![Page 10: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/10.jpg)
Statistics 1 10
For t-statistic: Single alternative hypothesis (H1)
Nondirectional (two-tail)
Directional
(one-tail)
For F-statistic: Many alternative hypotheses (H1's)
Always nondirectional
![Page 11: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/11.jpg)
Statistics 1 11
Design: Between Groups ANOVA
• Partition the total variance of a sample into two separate sources (hence name of test)
![Page 13: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/13.jpg)
Statistics 1 13
Partition the total variance of a sample into two separate sources (hence name of test)
Total variance – Variance associated with treatments and error
![Page 14: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/14.jpg)
Statistics 1 14
Total variance – Variance associated with treatments and error
– Variance associated with just error
![Page 16: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/16.jpg)
Statistics 1 16
Treatment 1 Treatment 2 Treatment 3
4 8 3
5 8 2
4 9 1
6 10 3
n=
Σx=
Σx²=
Treatment mean
![Page 17: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/17.jpg)
Statistics 1 17
Computational Formula for SSBG
SSBG =(ΣX1)²
n1+
(ΣX2)²
n2+
(ΣX3)²
n3..+
(ΣXk)²
nk[
(ΣXT)²
nT]
![Page 18: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/18.jpg)
Statistics 1 18
Computational Formula for SSW
SSW= ΣX² [(ΣX1)²
n1+
(ΣX2)²
n2
+
(ΣX3)²
n3..+
(ΣXK)²
nK]
![Page 19: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/19.jpg)
Statistics 1 19
ANOVA Summary
Source SS df MS F-Ratio
Treaments
SSBG
Error
SSW
Total
SSTotal
![Page 20: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/20.jpg)
Statistics 1 20
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
![Page 21: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/21.jpg)
Statistics 1 21
• Body of table typically gives
• values for p < .05 and p < .01
• Reject null hypothesis if:
• Obtained value exceeds tabled value
![Page 22: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/22.jpg)
Statistics 1 22
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
![Page 23: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/23.jpg)
Statistics 1 23
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
![Page 24: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/24.jpg)
Statistics 1 24
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
![Page 25: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/25.jpg)
Statistics 1 25
Number of different tests
Fisher’s LSD Test (Least Significant Difference)
![Page 26: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/26.jpg)
Statistics 1 26
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
![Page 27: Statistics 11 Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question:](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ebc5503460f94bc4bf4/html5/thumbnails/27.jpg)
Statistics 1 27
Other Post –Hocs comparisions
• Scheffe
• Newman-Keuls
• Duncan
• Bonferroni