Transcript Chapter 8 - Bakersfield College

Chapter 8
HYPOTHESIS TESTING USING
THE ONE-SAMPLE t-TEST
Going Forward
Your goals in this chapter are to learn:
• The difference between the z-test and the ttest
• How the t-distribution and degrees of
freedom are used
• When and how to perform the t-test
• What is meant by the confidence interval for
m, and how it is computed
Using the t-Test
• Use the z-test when  X is known
• Use the t-test when  X is not known
and must be estimated by calculating
sX
Understanding the
One-Sample t-Test
Setting Up the Statistical Test
1. Set up the statistical hypotheses (H0 and Ha) in
precisely the same fashion as in the z-test
2. Select alpha
3. Check the assumptions for a t-test
Assumptions for a t-Test
• You have a one-sample experiment using
interval or ratio scores
• The raw score population forms a normal
distribution
• The variability of the raw score population is
estimated from the sample
Performing the One-Sample t-Test
Performing the One-Sample t-Test
1. Compute the estimated population variance
2
(ss X ) using the formula
2
(

X
)
X 2 
N
s X2 
N 1
Performing the One-Sample t-Test
2. Compute the estimated standard error of
the mean ( S X ) using the formula
sX
sX 
N
Performing the One-Sample t-Test
3. Calculate the tobt statistic using the formula
tobt
X m

sX
The t-Distribution
The t-distribution is the distribution of all
possible values of t computed for random
sample means selected from the raw score
population described by H0
Comparison of Two t-Distributions
Based on Different Sample Ns
Degrees of Freedom
• The quantity N – 1 is called the degrees of
freedom (df )
• This is the number of scores in a sample that
reflect the variability in the population
Using the t-Table
Obtain the appropriate value of tcrit from the ttables using
• The correct table depending on whether you
are conducting a one-tailed or a two-tailed
test,
• The appropriate column for the chosen a, and
• The row associated with your degrees of
freedom (df)
Interpreting the t-Test
A Two-Tailed t-Distribution for df = 8
When H0 is True and m = 10
Reaching a Decision
If tobt is beyond tcrit in the tail of the distribution:
1. Reject H0 ; accept Ha
2. Conclude there is a relationship between
your independent variable and dependent
variable
3. Describe the relationship
Reaching a Decision
If tobt is not beyond tcrit :
1. Fail to reject H0
2. Consider if your power level was sufficient
3. Conclude you have no evidence of a
relationship between your independent
variable and dependent variable
One-Tailed Tests
If you believe your sample represents a
population where the mean is greater than
some value (e.g., 25):
H0: m ≤ 25
Ha: m > 25
One-Tailed Tests
If you believe your sample represents a
population where the mean is less than
some value (e.g., 25):
H0: m ≥ 25
Ha: m < 25
Summary of the
One-Sample t-Test
1. Create the two-tailed or the one-tailed H0
and Ha
2. Compute tobt
1. Compute X and s X2
2. Compute s X
3. Compute tobt
3. Create the sampling t-distribution and use df
= N – 1 to find tcrit in the t-tables
4. Compare tobt to tcrit
Estimating m by Computing a Confidence
Interval
Estimating m
There are two ways to estimate the population
mean (m)
 Point estimation in which we describe a point
on the dependent variable at which the
population mean (m) is expected to fall
 Interval estimation in which we specify a
range of values within which we expect the
population mean (m) to fall
Confidence Intervals
• We perform interval estimation by creating a
confidence interval
• The confidence interval for a single m
describes an interval containing values of m
( s X )( tcrit )  X  m  ( s X )( tcrit )  X
Example
Use the following data set and conduct a twotailed t-test to determine if m = 12
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
Example
• H 0 : m  12; H a : m  12
• Choose a = 0.05
• Reject H0 if tobt > +2.110 or if tobt < -2.110
tobt
X  m 13.67  12 1.67



 4.40
sX
1.61
0.380
N
18
• Since 4.4 > 2.110, reject H0 and conclude m
does not equal 12