Notes7 - 1 The t-test Introduction to Using Statistics for...

1
The t -test
Introduction to Using Statistics for
Hypothesis Testing
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Overview of the t -test
• The t -test is used to help make decisions
about population values. We’ll start with
how to do it, and come back to why later.
• There are two main forms of the t -test, one
for a single sample and one for two samples.
• The one sample t -test is used to test whether a
population has a specific mean value, e.g.,
whether the USF mean SAT-V is greater than
500.
• The two sample t -test is used to test whether
population means are equal, e.g., do training
and control groups have the same mean.
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Review of the Confidence
Interval
• 95%CI =
• The confidence interval is the mean
plus or minus a critical value of t times
the standard error of the mean.
• The standard error of the mean is
• The standard error is just the standard
deviation divided by the square root of
N. The standard deviation is:
X
S t X
05 .
±
N
SD
S
X
=
1
) (
2
−
−
=
∑
N
X X
SD
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2
One-sample t -test
We can use a confidence interval to “test” or decide
whether a population mean has a given value. For
example, suppose we want to test whether the mean
height of women at USF is less than 68 inches.
Suppose we randomly sample 50 women students
at USF. We find that their mean height is 63.05
inches. The SD of height in the sample is 5.75
inches. Then we find the standard error of the
mean by dividing SD by sqrt(N) = 5.75/sqrt(50) =
.81. The critical value of t with (50-1) df is
2.01(find this in a t -table). Our confidence interval
is, therefore, 63.05 plus/minus 1.63. See the graph.
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One-sample t -test Example
80 70 60 50
Height in Inches
= 63.05
=5.75
Pop Mean = 68
.01
Histogram of Sample Height
Take a sample, set a confidence interval around the
sample mean. Does the interval contain the
hypothesized value?
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One-sample t -test Example
70 Height in Inches
The sample mean is roughly six standard deviations (St. Errors)
from the hypothesized population mean. If the population mean
is really 68 inches, it is very, very unlikely that we would find a
sample with a mean as small as 63.05 inches.
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3
Second Example
• Mean height = 66
in. (hypoth = 68 in.)
• SD height = 4 in.
• N= 25 women at
USF
• t
.05
= 2.06
• Standard error = ?
• CI = ?
• Test value of t = ?
• Result?
8 .
25
4 = = =
N
SD
S
X
648 . 67 352 . 64
648 . 1 66 ) 8 (. 06 . 2 66
to
CI ± = ± =
5 . 2
8 .
68 66
=
−
=
−
=
X
S
X
t
µ
Critical value of t is 2.06. Sample was not drawn
from population with mean of 68 inches.
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Six Steps for Significance
Tests
• 1 Set α (alpha or p level)
• 2 State null and alternative hypotheses
• 3 Calculate test statistic
• 4 Determine critical value
• 5 State decision rule
• 6 State conclusion
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Example of Six Steps (1)
1. Set α . Alpha or p level is the probability of
a Type I error (I will explain this later). By
convention, alpha is typically set at .05 or .01.
This tells you where to look for critical values
in tables (like z , t and F ).