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Transcript
Chapter 7
Introduction to the t Test
Part 1: One-sample t test
Oct. 1, 2013
t Test for a Single Sample
• Z test requires that you know  from pop
• Use a t-test when you don’t know the
population standard deviation.
• One sample t-test:
– Compare a sample mean to a population with a
known mean but an unknown variance
Hypothesis testing procedure for t-test
• Same general procedure:
– Assume null hypothesis is true, relative to an alternate
(research hypothesis)
– Compute observed t statistic from sample data based
on sampling distribution of the mean
– Determine cutoff point (now a critical t) in comparison
distribution based on 
– Reject the null hypothesis if your observed t value falls
in critical region
• |t observed| > |t critical|
One–sample t Test
• Must estimate the population variance from the
sample scores
– Unbiased estimate of the population variance (S2)
S
2
(X  M )


Use N-1 to make
S2 bigger than sample
N 1
2
SS

N 1
Sum of squared
deviations
t Test for a Single Sample
• Degrees of freedom
df  N  1
– Number of scores that are
“free to vary”
– Formula for S2 using degrees of freedom
S 
2
2
(
X

M
)

df
SS

df
Note: S2 indicates we estimated this from a sample
( always indicates population info)
t Test for a Single Sample
• Also need to find standard deviation of
distribution of means (SM)
• The variance of the distribution of means
S
2
SM 
N
2
• The standard deviation of the distribution of
means
2
SM  SM
t Test for a Single Sample
• For t-tests, relevant comparison distribution is t
distribution (not the normal curve used before in z test)
• The t distribution
One-sample t Test
• State null & research hypotheses
• Assuming null is true, compute observed t
statistic for your sample mean
M 
t
SM
• Find correct critical t value based on your df
• When do we reject the null?
Note on t-table
• When > 30 df, critical values only given for 35,
40, 45 df, etc.
• If your df are in between these groups, be more
conservative and use the lower df
• Do example in class…
One Sample t-test in SPSS
• Use menus for:
Analyze  Compare Means  One sample t
Gives pop-up menu…need 2 things:
– select variable to be tested/compared to population
mean (use “self-esteem”)
– Notice “test value” window at bottom. Enter the
population/comparison mean here (use x.xx from past
semester)
– Hit OK, get output and find sample mean, observed t,
df, significance value (if < .05, reject null)