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One-Sample t-test
What do these problems we have been working on have in
common?
In a population of American graduate students, individuals
earn on average $7.25/hour on alcohol, with a standard
deviation of $5. I jokingly ask whether or not graduate
students from Brooklyn earn a different wage than grad
students in general. I gather together 9 graduate students
from this program and calculate the average amount they
earn an hour: $11.00. Use an alpha level of .01.
How do I know I need to be using a z-test?
1. We are comparing a sample mean to a KNOWN population mean.
2. We KNOW the population s.
What to do if you do not know the Population
Standard Deviation (s)?
Use the best estimate of sx
You must correct for the
uncertainty of this estimate.
T-score for a single sample mean
Where sx_ =
s
What to do if you do not know the Population
Standard Deviation (s)?
_
x
_
x
Probability
Hypothesis testing with the t-statistic
Retain H0
Outcome “t”
Reject H0
t-crit
One-tailed test
One-Sample t-test
z-test is used when we know both m and s
t-test is when we know m but not s
The Sampling Distribution of the t-test
This table lists the critical value of the t-statistic for the
Degrees of freedom and a level.
Given the level and the df you can find the critical value
of the t-statistic that divides the outcomes into reject or retain the null.
The average IQ score of Americans is 100. I believe that my Statistics
class (you guys!) have different IQs than the general population. I force
all 15 of you to take an IQ test, and I calculate a mean of 110 (s= 20). Use
an alpha level of .05 to determine if this class has a different IQ than the
population.
How do I know I need to be using a one-sample t-test?
Step 1: State the null and alternative hypotheses:
H0: My stats class does not have a different IQ than the average American
H1: My stats class has a different IQ than the average American
Step 2: Find the critical value.
Go to t-table! Must figure out of this is one- or two-tailed, and df.
The Sampling Distribution of the t-test
This table lists the critical value of the t-statistic for the
Degrees of freedom and a level.
Given the level and the df you can find the critical value
of the t-statistic that divides the outcomes into reject or retain the null.
The average IQ score of Americans is 100. I believe that my Statistics
class (you guys!) have different IQs than the general population. I force
all 15 of you to take an IQ test, and I calculate a mean of 110 (s= 20). Use
an alpha level of .05 to determine if this class has a different IQ than the
population.
How do I know I need to be using a one-sample t-test?
Step 1: State the null and alternative hypotheses:
H0: My stats class does not have a different IQ than the average American
H1: My stats class has a different IQ than the average American
Step 2: Find the critical value.
Go to t-table! Must figure out of this is one- or two-tailed, and df.
+/-2.145
The average IQ score of Americans is 100. I believe that my Statistics
class (you guys!) have different IQs than the general population. I force
all 15 of you to take an IQ test, and I calculate a mean of 110 (s= 20). Use
an alpha level of .05 to determine if this class has a different IQ than the
population.
Step 3: Calculate the obtained statistic:
tobt 
x  mx
sx
=
- 100
110
________
= 10/5.17 = 1.93
20/sqrt(15)
Step 4: Make a decision.
I
I
-2.15
2.15
Step 4: Retain the null hypothesis.
The Sampling Distribution of the t-test
This table lists the critical value of the t-statistic for the
Degrees of freedom and a level.
Given the level and the df you can find the critical value
of the t-statistic that divides the outcomes into reject or retain the null.