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Inferences About
Means of
Single Samples
Chapter 10
Homework: 1-6
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 ~

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
 If reject H0
evaluate

practical significance
Steps 1 & 2 before collecting data ~
1. State null & alternative hypotheses

s unknown
calculate X, s, & s X from sample
 use t test
 Survey: college students study 21 hr/wk
 Do Coe students study 21 hrs/week?
 Select sample (n = 16)
 Nondirectional hypothesis:
 H0 : m = 21;
H1 : m  21
 reject H0 if increase or decrease ~

What does distribution of sample
statistic look like if Ho true?
If Ho is
false?
f
-2
-1
0
1
2
2. Set Criterion for Rejecting H0
Determine critical value of test statistic
 Directionality:
 df =
 a =
 tCV =
 Defines rejection region ~

2. Set Criterion for Rejecting H0

Rejection region
 area of distribution beyond critical value
 for test statistic
Also

sample statistic
Reject H0 if tobs falls in rejection region ~
Rejection regions
f
-2
-2.131
-1
0
1
2
+2.131
3. Collect sample & compute statistics

Compute sample statistics
Mean
Standard
deviation
Standard error of mean
Observed value of test statistic
 General form

test statistic = sample statistic - population parameter
standard error of sample statistic
3. Collect sample & compute statistics
Use sample statistics to compute
test statistic
 X = 24.63; s = 7.78, s X = 1.94
 Test statistic

tobs
X m

sX
4. Interpret Results
Is tobs is beyond tCV?
 Is it in rejection region?
 NO.
1.87 < 2.131
 then “accept” H0
 Coe students study  21 hrs/wk
 No significant difference
 does not mean they are equal
 not sufficient data to reject
 Practical significance not an issue ~

A Directional Hypothesis

s unknown: same question



evidence from prior surveys that
Coe students study more than 21
hrs per week
H1 = experimental hypothesis
can use directional hypotheses ~
A Directional Hypothesis
1. State H0 & H1
 H0: m < 21
Coe
students study less than or equal
to 21 hrs per week

H1: m > 21
Coe
students study more than 21 hrs
per week ~
A Directional Hypothesis
2. Set criterion for rejecting H0
 a = .05, level of significance
 directional (one-tailed) test
 df = 15
 tCV = 1.753 ~
A Directional Hypothesis
3. Collect sample & compute statistics
 X = 24.63; s = 7.78, s X = 1.94
 test statistic = tobs
tobs
X m

sX
A Directional Hypothesis
3. Interpret results
 Is tobs in rejection region?
 tobs > tCV
1.87 > 1.753
 Reject H0
 accept H1
 Coe students study more than 21
hours per week ~
Practical Significance
Statistical significance?
 YES
 Practical significance?
 MAYBE ~

Practical Significance: Effect size
Magnitude of the result (difference)
 Raw effect size
 measured on scale of original data
 Xobs - m = 24.63 - 21 = 3.63
 Coe students study 3.63 hours per
week longer than the national
average ~

Practical Significance: Effect size

Effect size index
 compare effect size for variables using
different scales (e.g. GRE, ACT)
 divide difference by s
nondirectional
d
X obs  m
s
directional
X obs  m
d
s
Effect size index
d
= .47 standard deviations
above the mean
f
-2
-1
0
1
standard deviations
2
Practical Significance: Effect size
Is effect size practically significant?
 .5 considered moderate effect size
 e.g., Is it worth using a new statistics
textbook that test scores d = .5?
 Ultimately we must make decision ~

When s Is Known

Usually not the situation
 calculate X from sample
 use z test
 degrees of freedom not relevant
 find zCV in z table
zobs 
X m
sX
Practical Significance: Effect size

Effect size index: s is known
nondirectional
d
X obs  m
s
directional
d
X obs  m
s