### The t Distributions

```The Single-Sample
t Test
Chapter 9
The t Distributions
> Distributions of Means When the
Parameters Are Not Known
> Using t distributions
• Estimating a population standard deviation
from a sample
Sample Standard Deviation
SD 
2
(
X

M
)

N
Population Standard Deviation
s
2
(
X

M
)

( N  1)
Calculating the Estimated
Population SD
> Step 1: Calculate the sample mean
> Step 2: Use the sample mean in the
corrected standard deviation formula
s
(X  M )
( N  1)
2
Steps to calculating s:
(8  12  16  12  14)
M
 12.4
5
s
2
(
X

M
)

( N  1)
35.2

(5  1)
= 8.8 = 2.97
Calculating Standard Error for the
t Statistic
> Using the standard error
s
SM 
N
> The t statistic
(M  M )
t
SM
Steps to calculating t statistic
using standard error:
> From previous example:
s
2
(
X

M
)

( N  1)
= 2.97
SM 
s
2.97

 1.33
N
5
> Assume population mean is 11:
( M  M ) (12.4  11)
t

 1.05
SM
1.33
The t Statistic
> When sample size increases, s
approaches σ and t and z become
more equal
> The t distributions
• Distributions of differences between means
Wider and Flatter t Distributions
> When would you use a z test? Give an
example.
> When would you use a t test? Give an
example.
Hypothesis Tests: The Single
Sample t Test
> The single sample t test
• When we know the population mean, but
not the standard deviation
• Degrees of freedom
df = N - 1 where N is sample size
Stop and think. Which is more conservative: one-tailed or
two-tailed tests? Why?
> The t test
• The six steps of hypothesis testing
> 1. Identify population, distributions, assumptions
> 2. State the hypotheses
> 3. Characteristics of the comparison distribution
> 4. Identify critical values
df =N-1
> 5. Calculate
> 6. Decide
Example: Single Sample t Test
STEP 1: Identify population, distribution,
assumptions
Population 1: All clients at this counseling center who sign a
contract to attend at least 10 session
Population 2: All clients at this counseling center who do not
sign a contract to attend at least 10 sessions
• The comparison distribution will be a distribution of means
• Use a single-sample t test because there is one sample
and we know the population mean but not the population
standard deviation
• Assumptions?
Calculating the Single Sample t
Test
STEP 2: State the hypotheses
H0 : 1 = 2
H1: 1  2
STEP 3: Determine the characteristics
of the comparison distribution.
t Test Calculation Continued
STEP 4: Determine the critical values, or
cutoffs
df = N -1 = 5 -1 = 4
t Test Calculation Completed
STEP 5: Calculate the test statistic
( M  M ) (7.8  4.6)
t

 2.873
SM
1.114
STEP 6: Make a decision
Calculating Confidence Intervals
> Draw a picture of the distribution
> Indicate the bounds
> Look up the t statistic
> Convert the t value into a raw mean
Example Confidence Interval
STEP 1: Draw a picture of a t distribution that
includes the confidence interval
STEP 2: Indicate the bounds of the confidence
interval on the drawing
Confidence Interval Continued
STEP 3: Look up the t statistics that fall at
each line marking the middle 95%
Confidence Interval Example
STEP 4: Convert the t statistics back into
raw means.
Confidence Interval Completed
STEP 5: Check that the confidence interval
makes sense
The sample mean should fall exactly in the middle
of the two ends of the interval:
4.71-7.8 = -3.09 and 10.89 - 7.8 = 3.09
The confidence interval ranges from 3.09 below the
sample mean to 3.09 above the sample mean.
Interpretation of Confidence
Interval
If we were to sample five students from
the same population over and over, the
95% confidence interval would include
the population mean 95% of the time.
Calculating Effect size
(M   )
d
s
For the counseling center data:
(M  ) (7.8  4.6)
d

 1.29
s
2.490
Dot Plots
> The dot plot is a graph that displays all
the data points in a sample, with the
range of scores along the x-axis and a
dot for each data point above the
appropriate value.
> Dot plots serve a similar function to
stem-and-leaf plots.
> The three steps to creating a dot plot
STEP 1: We determine the lowest score and highest
score of the sample
STEP 2: We draw an x-axis and label it, including the
values from the lowest through highest scores
STEP 3: We place a dot above the appropriate value
for every score.
Example Dot Plot
Stop and Think
> When would you use a z test over a t
test?
> When would you use an independent
sample t test? Think of a specific study.
```