Confidence Intervals and Hypothesis Testing I

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Confidence Intervals and Hypothesis Testing I

• PS585
• 5/18/99

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Get right into it

• Pick up your homework
• Extra credit projects available

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Review
How accurate are our estimates/statistics? How
good is our sample?
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Confidence intervals

• Probability estimates of the true parameter value
  (p.124)
• How good are our estimates?
• Formula

Population Standard Deviation
Sample mean
Sample size
Confidence level
Margin of error
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What if we dont know d(pop. Std. Dev.)

• Use the sample standard deviation (s)
• Very similar to using the d, but use t, not Z
• Be sure to use the two-tailed column!
• T takes sample size into account
• For this, degrees of freedom sample size - 1

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Lets try one
Mean (x)43 Standard Deviation (s)12 n36 Confide
nce Level95 (Alpha level .05)

• We are 95 sure that the mean is between 38.92
  and 47.08.

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Hypothesis Testing

• Much of this is like KKV
• Null Hypothesis there is no statistically
  significant1 difference between the means
• Research Hypothesis there is a statistically
  significant1 difference between the means
• We accept one, and reject the other

1. We use alpha levels here
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2x2 Contingency Table of Type I and II Errors
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Which Difference of Means Test Do I Use???
The choice of difference of means test depends on
a few things, such as your goal (do you want to
compare a sample to a population, or not?), the
type of information you have about the
population, and the type of sample data you have.
This flowchart is designed to help you decide
which test is the most appropriate.
Do you wish to compare a sample mean to a
population mean?
YES
NO
Do you know the mean AND the standard deviation
for the population?
Do you have data from two samples?
YES
NO
YES
NO
Use a Difference (Matched or Paired) t test
Use a Z test
Use a Dependent-Sample t test
Use an Independent-Sample t test
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Z test (Friar Jacques...)

• Is the difference between the means significant?
• 10gt2.58 so reject the null

• Ingredients Required
• population mean
• pop. std. dev.
• sample size and mean

(mu)
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Practice Exercise 1

• 6gt1.96 so?
• (Z obtainedgtZ critical)
• What error if we kept null?

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Dependent-sample t test

• Is the difference between the means significant?
• 2.5gt2.492 so reject the null

• Ingredients Required
• population mean
• sample size, s.d., and mean

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Practice Exercise 3

• Is the difference between the means significant?
• 3.75gt2.042 so reject the null

• Ingredients Required
• population mean
• sample size, s.d., and mean

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Independent-samples t test

• Is the difference between the means significant?

• Ingredients Required
• sample stuff only

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Independent-samples t test
Group 1 mean8 s1.5 n19 Group 2 mean
7 s3 n21 (alpha.05)
t obtained
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Independent-samples t test
Group 1 mean8 s1.5 n19 Group 2 mean
7 s3 n21 (alpha.05) (t critical2.042)
t obtained
Which hypothesis do we reject?
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Practical exercise 4
Group 1 mean18 s2.5 n15 Group 2 mean
14 s1.5 n12 alpha.001 t critical3.275
t obtained
4.87gt3.275 so reject null
df n1 n2 -2
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Difference (matched or paired) t test

• Uses different means from the same group
• Test -gtexperiment -gt Test
• or Testcondition1 and Testcondition2

Where...
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Difference (matched or paired) t test

• O T1 T2
• 1 34 37
• 2 31 43
• 3 18 27
• 4 36 33
• 5 31 39
• 6 30 30
• 7 38 41

• Uses different means from the same group
• Test -gtexperiment -gt Test

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Practical exercise 6
Tcritical 2.306 and 3.01gt2.306 so disprove null
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Which one do I use?
Depends on what information you have!!!
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Which Difference of Means Test Do I Use???
The choice of difference of means test depends on
a few things, such as your goal (do you want to
compare a sample to a population, or not?), the
type of information you have about the
population, and the type of sample data you have.
This flowchart is designed to help you decide
which test is the most appropriate.
Do you wish to compare a sample mean to a
population mean?
YES
NO
Do you know the mean AND the standard deviation
for the population?
Do you have data from two samples?
YES
NO
YES
NO
Use a Difference (Matched or Paired) t test
Use a Z test
Use a Dependent-Sample t test
Use an Independent-Sample t test