T-Tests
Lecture: Nov. 6, 2002
Review
•
So far we have learned how to test
hypothesis for two types of data:
1. Binomial data using the binomial
distribution (test statistic: p)
2. Comparison of a sample mean to a known
population mean and variance.
(test statistic: z)
T-Tests
• Z-tests require that we know the population
standard deviation, which is often NOT
known.
• In these cases, we have to use the sample
variance and standard deviation, as
estimates of the population variance and
standard deviation.
T-Tests
•
•
•
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Population standard error (sX) = s2 / n
Use sample variance (df = N – 1)
Sample standard error (sX) =
s2 / n
This creates a new statistic, the t statistic.
t = X – m / sX
T-Tests
• The only difference between a t-test and a
z-test is the use of sample variance, not
population variance in the formula.
• The degrees of freedom for a t-test is N-1.
• The greater the degrees of freedom (i.e.,
the greater the N), the more accurate the tstatistic represents the z-score.
T-tests and Distributions
• Each statistic has an underlying
distribution.
• Binomial p = binomial distribution which
changes depending on p, N, and r.
• z-scores = normal distribution
• t-statistics = t-distribution
– Changes depending on the degrees of freedom.
– Larger the degrees of freedom, the more it
approximates the normal distribution.
T-statistics and Distributions
• The t distribution is bell-shaped, but more
spread out.
• Greater the N, and, thus, the degrees of
freedom, the less spread out the t
distribution.
Determining Probabilities of t
• Just as you look up the probability of
obtaining a z score using the unit normal
table, we simply look up given t-statistic
using a table for the t distribution (page
693 in textbook).
• However, we must also know the degrees
of freedom.
Determining Probabilities of t
Proportion in One Tail
.05
df
.025
.01
.005
Proportion in Two Tails
.10
.05
.02
.01
1
6.314
12.706 31.821 63.657
2
3
4
2.920
4.303
6.965
9.925
2.353
3.182
4.541
5.841
2.132
2.776
3.747
4.604
Hypothesis Testing with
the t-test
Example: A researcher knows that in 1990
the average age at which 25-year-olds
report having their first drink was 16. The
researcher predicts that that ten years later,
25-year-olds would report a significantly
younger first age of drinking. She recruits
20 25-year-olds for her study and asks
when each participant had his/her first
drink.
Hypothesis Testing with
the t-test
1. First she constructs her hypotheses.
H0: mage of first drink = 16
Ha: mage of first drink = 16
Hypothesis Testing with
the t-test
2. Create a decision rule.
•
•
Set the alpha level; a = .05.
Find the “critical t” in on our t-table.
•
•
•
•
Because non-directional hypothesis, we must
find t-value that for which that t-value or higher
is < .05 using two-tails.
Must look up based on 19 degrees of freedom
(N – 1).
This t-statistic is 2.093.
So, if we get a t-statistic > 2.086, we can reject
the null hypothesis.
Hypothesis Testing with
the t-test
3. Collect data and calculate a t-statistic.
•
•
Assume researcher found that in her sample
of 20 adults, the average age of first drink
was 14, and the variance of 4.
The standard error of the t-statistic.
s2/n
=
4/20 = .45
The t-statistic is:
X – m / sX = 14-16/.45 = -2/.45 = - 4.44
Hypothesis Testing with
the t-test
4. Apply the decision rule.
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–
Is our obtained t-statistic equal to or larger than our
“critical t”?
Obtained t: -4.44
Critical t: 2.093
The absolute value of our t-statistic is more extreme.
In other words, there is less than a 5% probability
that if the population mean is 16 that we would
obtain a sample mean of 14 with a variance of 4.
Reject the null hypothesis; accept the alternative
hypothesis.
Assumptions of the t-test
1. All observations must be independent.
2. The population distribution of scores
must be normally distributed.
Reporting a t-test
• When reporting a t-statistic you should
provide the sample mean and sample
standard deviation somewhere in the paper.
• You should also report the t-statistic, the
degrees of freedom, whether it fell within
the region of rejection, and whether it was
a one- or two-tailed test.
Reporting a t-test
Example:
The 20 participants reported the age at which they
had their first drink (M = 14, SD = 2). This age
was significantly different from the average age
of 16 reported a decade earlier, t(19) = -4.44, p <
.05, two-tailed.