t tests for Single Samples

1
t - tests for Single Samples

• Unit 7
• Ch 9 1-5, 7, 9, 11, 13, 15, 17 (pp. 253-256)

2
Evaluating Hypotheses About Means

• Evaluating hypothesis about population
• taking a single sample
• Does it likely come from population?
• Test statistics
• z test if s known
• t test if s unknown

3
s and Hypothesis Tests

• s is known
• use z test

• s is NOT known
• Cannot calculate standard error
• Use estimated standard error of mean

4
Sample Standard Deviation (s)

• Square root of variance
• typical distance of scores from mean

5
t statistic

• Cannot use z statistic

• How good an estimate is it?
• depends on sample size
• tcv is adjusted for sample size
• Table B.2, p A-27
• df n - 1

6
t statistic
7
Steps in Hypothesis Evaluation

• 1. State null alternative hypotheses
• 2. Set criterion for rejecting H0
• 3. collect sample compute sample statistic
  test statistic
• 4. Interpret results

8
1. State null alternative hypotheses

• Survey college students study 21 hr/wk
• Do Coe students study 21 hrs/week?
• Select sample (n 16)
• s unknown
• Nondirectional hypothesis
• H0 m 21 H1 m ¹ 21
• reject H0 if increase or decrease

9
2. Set Criterion for Rejecting H0

• Determine critical value of test statistic
• Directionality
• a .05
• df
• tCV
• Defines critical region

10
Critical regions
f
11
3. Collect sample compute statistics

• Compute sample statistics
• assume

12
3. Collect sample compute statistics

• Use sample statistics to compute test statistic

13
Critical regions
f
14
4. Interpret Results

• tobs in rejection region?
• Yes. 2.38 gt 2.131
• Reject H0
• Coe students study gt 21 hrs/wk
• There is a significant difference
• Is there practical significance?
• Reporting in literature
• t(15) 2.38, plt.05, two-tailed

15
Changing Level of Significance

• If a .01

• Decision?
• Fail to reject H0 (accept H0)

16
A Directional Hypothesis

• s unknown same question
• evidence from prior surveys that Coe students
  study less than 21 hrs per week
• H1 experimental hypothesis
• can use directional hypotheses

17
A Directional Hypothesis

• 1. State H0 H1
• H0 m gt 21
• Coe students study more than or equal to 21 hrs
  per week
• H1 m lt 21
• Coe students study less than 21 hrs per week

18
A Directional Hypothesis

• 2. Set criterion for rejecting H0
• a .05, level of significance
• directional (one-tailed) test
• df 15

19
Critical Region
1-tailed
a .05
tCV
-1.753
f
20
3. Collect sample compute statistics

• Compute sample statistics
• assume

21
3. Collect sample compute statistics

• Use sample statistics to compute test statistic

22
Critical Region
1-tailed
a .05
f
23
4. Interpret Results

• Is tobs is beyond tCV?
• Is it in rejection region?
• NO. 2.38 gt -1.753
• Fail to reject H0
• Coe students study do not study less than 21
  hrs/wk