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Introduction to the tdistribution tNormal distribution vs. tdistribution t1 Sample ttest tExample The tDistribution Introduction
When do we ever know the standard deviation of the population ()? (
Can use M when testing hypotheses about the sampling means
1Sample Z test Introduction
When we don't know , need to use the best don' estimator
Estimated Standard Error (sM) Assumption: our sample is a random sample from the population
Variance of the sample should reflect the variance of the population 1 Introduction
Potential problem
Biased estimate Introduction
We can calculate an unbiased estimate of the population variance i.e., an unbiased estimate of the standard error of the means sx =
2 sx = 2 sx (X  M )
(n  1) 2 or 2 sx = Population Distribution Sample Distribution Distribution of Means SS df Where SS = Sums of Squares & df = degrees of freedom Normal Distribution vs. tDistribution Normal distribution vs. t distribution
Shape of the tdistribution depends on degrees tof freedom (df)
Entire "family" of tdistributions (each with a family" tdifferent df) Shape of the normal distribution is unaffected by N
This is because the normal distribution is the population distribution Leptokurtic = Bigger tails 2 Family of tdistributions Normal distribution vs t distribution
With df 100, tdistribution almost normal ttdistribution differs most from normal curve when N is small The greater the value of df is for a sample
the better s2 represents 2 & the better the tscore represents the zscore tz 1 Sample ttest
When sM is used to estimate M, then the significance test uses the tdistribution instead of the normal distribution The test statistic is called a 1 Sample ttest t1. 2. Assumptions:
1 Sample t
Normality of the population Scores are independent
Independent observations 3 When to use:
1 Sample ttest tCompare a single sample mean to some hypothesized value of the population mean Know the population mean but do not know the population variance Interval/ratio data 1 sample tdistribution
Similar to zscores, need to find a tscore to zfind the likelihood (or probability) of finding that score or a score more extreme given the null hypothesis is true (i.e., the p value). t= M data & hypothesis t = obtained difference between Standard distance expected by chance sM Hypothetical Example
New attention paradigm Looking to see how your response of interest compares to the population of similar attention paradigms.
Follow the car... car... 4 Hypothetical Example
Sample 100 students startle response (e.g., change in pupil size ratio data) to the film clip. Have a population mean from similar startle paradigms, but not a population standard deviation. Need to use your sample standard deviation as an unbiased estimate of the population standard deviation...
Viola! 1Sample tTest! 1t Degrees of Freedom (df):
1 Sample t
Our df will be (n1) (nWhy? Putting it into numbers...
S# 1 2 3 4 5 6 7 8 9 X 30 29 12 18 16 24 19 20 (XM) (X10 9 8 2 4 4 1 0 8 (X(XM) 10 19 11 9 5 9 8 8 0 M = 20 n=9 df = (n1) (ndf = 8 S# 1 2 3 4 5 6 7 8 9 X 30 29 12 18 16 24 19 20 12 (XM) (X10 9 8 2 4 4 1 0 8 (X(XM) 10 19 11 9 5 9 8 8 ? 0 Cannot be anything but... 5 Example
Researchers want to investigate whether the mean score of fifth graders on a reading achievement test differs from the national mean of 76. Example
Step 1 = Hypotheses H0: M = (or M  = 0) H1: M (or M  0) Step 2 = Nature of DVs 72 78 85 88 82 69 76 97 76 91 98 78 84 79 69 87 66 86 82 74 Ratio: Scores on the test Example
Step 3: Test selection
We will use a 1Sample t test because we know and 1we know M and sx. We do not know therefore we need to use sx to figure out sM. Example
Step 5: Sample size (or figure out power) = d MEI n
= 0.50 20 = 2.24 power = 0.60 Step 4: Error rates = 0.05 =? 6 Example
Step 3: Test selection
We will use a 1Sample t test because we know and 1we know M. We do not know therefore we need to use s to figure out sM. Example
Step 6: Collect data = DONE Step 7: Conduct 1sample ttest 1t t ( n  1) = = (M  M ) sm Step 4: Error rates = 0.05 = 0.40 (80.85  76) 8.875 20 = 2.444 Example
Step 8 Observed effect size d OBS = =
Significance, p < Reject Ho (M  m ) sx dOBS > dMEI Practical Significance! (80.85  76) 8.875 = 0.55 7 Example
Step 9: Decision = 76 M = 80.85 The mean score of the class did differ from the population such that the class scored significantly higher (i.e., better) than the population, t (19) =
2.44, p < 0.05, dOBS = 0.55. Now let's try it with CIs... First do the 95% CI around the expected mean [Mean] +/ (tCRIT)(sM) +/ (tCRIT)(sM) +/ Given = 76 M = 80.85 n = 20 Sx = 8.875 tcrit = +/2.093 (tcrit )( sm )
8.875 ) 20 [71.8464, 80.1536] 76 (2.093)(
76 71.8464 80.85 80.1536 Our sample mean falls outside of the 95% confidence interval around the population mean. The mean performance score for the class did differ significantly from the general population such that the class scored higher. 8 Now for extra practice... Try doing the 95% CI around the observed mean on your own. [Mean] +/ (tCRIT)(sM) 9 ...
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 Spring '08
 Dicorcia
 Normal Distribution, Standard Deviation, 76 m, 10 19 11 9 5 9 8 8 0 M

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