Estimation: Making Educated Guesses
• Point Estimation
• Interval Estimation
•
Hypothesis Testing
Case Ia
• Does a particular sample of observations in
this study come from a specified population
or does it represent a different population?
– “Known” population mean
– “Known” population standard deviation
The 4th Grade Case
• Suppose you are the superintendent of schools and you discover that the
average reading achievement of your 4th graders has fallen far below that
of previous years. One explanation posed by the teachers is that the
district is faced with an unusually dull group of 4th graders this year. The
teachers suggest that the average verbal IQ of this year’s 4th graders is far
different from the national average and that is why reading achievement is
so low.
• You know that IQ-test scores don’t change much from year to year unless
a school system is affected by changes in its attendance (e.g. a large
migration of new families). Your school system has remained quite stable,
but you decide to check out the teacher’s claim
• You have a limited budget and while you have extensive achievement
data on the 4th graders, you have limited IQ data available. So you
decide to test a sample of 400 4th graders rather than all 5000 of them.
The Logic of Hypothesis Testing
• Null hypotheses (H0)
• Alternative hypotheses (H1)
• Is this backwards and convoluted or what?
Hypothesis Testing: General Model
• Identify the population and population parameter of
interest
• Define the null hypothesis and alternative hypothesis
• Collect data on a random sample selected from population
of interest
• Compute a sample statistic that is an estimate of the
parameter of interest
• Decide on a criteria for evaluating the sample evidence
• Make decision to retain the null hypothesis or discard the
null hypothesis in favor of the alternative hypothesis
Error and Risk
The True Stat e of Reali ty
De cis ion
The Null
Hypoth e iss
is Tru e
The Alternative
Hypoth e iss
is Tru e
The Null Hypothesis is
True
The Alternative
Hypoth e iss is True
Correct D e cision
Proba bility = 1-
Type II Error
(Risk b)


Type I Error
(Risk a)
Correct D e cision
Proba bility = 1-b
L
J
Type I Error and Level of
Significance
• Type I error: the mistake of rejecting the
null hypothesis (H0) when in fact it is true.
• Level of Significance:
– Alpha () = .05
– Significant at the .05 level
– p < .05
Type II Error
• Type II Error: If the alternative hypothesis
(HA) is true and the decision maker decides
to stick with the null hypothesis (H0)
• Risk 
Hypothesis Testing: General Model
• Identify the population and population parameter of
interest
• Define the null hypothesis and alternative hypothesis
• Collect data on a random sample selected from population
of interest
• Compute a sample statistic that is an estimate of the
parameter of interest
• Decide on a criteria for evaluating the sample evidence
• Make decision to retain the null hypothesis or discard the
null hypothesis in favor of the alternative hypothesis
Decision Rules
• Decision Rule: the values of sample statistic
that keep you believing H0 and the values
that lead you to reject H0
Hypothetical Frequency Distribution of
1000 Samples
.3413
.3413
.1359
68%
.0214
.0214
95%
.0013
97.75
.1359
.0013
99%
98.5
99.25
100 =
population mean
100.75
101.5
102.25
“How Likely?”
• How likely is this sample mean to arise by sampling error?
• The “Sampling Distribution of Means” provides a model of what
to expect if the null hypothesis is true
Likely
IQ = 100
Population of
Scores
Unlikely
IQ = 100
Sampling Distribution
of Means
• By convention, an unlikely sample mean under the null
hypothesis occurs 5 in 100 times (.05) or 1 in 100 times (.01)
Selecting a Level of Significance:
What is Unlikely?
• Goal is to determine how consistent or
inconsistent the sample data are with the
null hypothesis
• Usually select some small (conservative)
level of significance (.05, .01, .001)
• Level chosen depends on seriousness of the
consequences of one’s decision
Hypothetical Frequency Distribution of
1000 Samples
68%
Unlikely at .05
Unlikely at .05
Unlikely at .01
Unlikely at .01
95%
99%
97.75
98.5
99.25
100
population mean
100.75
101.5
102.25
One- and Two-Tail Test?
• One- and two-tail tests tell you which tail(s) in the
sampling distribution of means should be used to
determine “How likely?”
• Two-Tail Test: Willing to Entertain a Sample Mean in
Either Tail--H1 :Population Mean not = 100
• One-Tail Test: Willing to Specify the Direction of the
Sample Mean (Above or Below the Population Mean
Under the Null Hypothesis): H1 :Population Mean > 100
Two Tail
.025
One Tail
.025
.05
Critical Values for Case Ia:
Z-Test
Typ e of Alt erna t vi e Hypo theses
L eve l of
Signific a nce
Dire c itona l
"One-T a lie d"
Non-Dire cti onal
"T wo-T a lie d"
Alph a (or p) =.05
1.65
1.96
Alph a (or p) =.01
2.33
2.58
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
Z Distribution
Normal Curve
Two tailed
Test
-2se
Critical
Values
P=
-2.58se
-1se
-1.96se
.05 = outisde of 1.96 either end
u
+1se
+2se
+1.96se
.01 = outside of 2.58 either end
+2.58se
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
Z Distribution
Normal Curve
-2se
Critical
Values
P=
One tailed
test
-1se
u
+1se
+2se
+1..65se
.05
+2.33se
.01
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
Z Distribution
Normal Curve
One tailed
test
-2se
-1se
Critical
Values
-2.33se
-1..65se
P=
.01
.05
u
+1se
+2se
The Decision Regarding H0:
The Lingo
• Reject H0 : Take position that null hypothesis is probably
false
–
–
–
–
“H0 (the null hypothesis) was rejected”
“A statistically significant finding was obtained”
“A reliable difference was observed”
“p is less than X” (a small decimal value (p<.05,p< .01))
• Fail-to-reject H0: Take the position that there is not
enough evidence to reject the null hypothesis
– “H0 was tenable”
– “H0 was accepted”
– “No reliable differences were observed”
– “No significant differences were found” (ns)
– “p is greater than X” (a small decimal value (p>.05,p> .01))
Significance Testing vs Hypothesis
Testing
• Hypothesis Testing:
– Alpha level is preset
– Decision is “reject” or “do not reject”
– Don’t discuss impressive p-levels
• Significance Testing
– No alpha levels preset
– Data speak through p-levels
– Strength of significance discussed