Transcript PowerPoint

STATISTICS 200
Lecture #20
Thursday, October 27, 2016
Textbook: Sections 9.6, 11.1, 11.2
Objectives:
• Apply sampling distribution for one sample mean to
confidence intervals and hypothesis tests.
• Identify situations in which t-multipliers and t-tests should be
used instead of z-multipliers and z-tests.
We have begun a strong focus on
Inference
Means
Proportions
One
population
proportion
Two
population
proportions
This week
One
population
mean
Difference
between
Means
Mean
difference
Categorical
data
(2 categories)
Quantitative
data
parameter:
parameter:
statistic:
statistic:
Clicker Question:
Consider the following
three survey questions:
1. Do you plan to vote in
the upcoming
presidential election?
2. How old are you?
3. Which candidate do you
most support?
How many of these
questions will
produce
Quantitative data?
A. 0
B. 1
C. 2
D. 3
Example 1: The population
is normally distributed.
52
68
84
100
116
132
148
X sampling distribution
52
68
84
100
116
132
148
IQ
52
68
84
100
116
132
X sampling distribution
148
Clicker Question
Which statement(s) are false, when comparing the
original distribution to the two sampling distributions
a.
All three distributions have the same value for the
mean
b.
As the sample size increases, the standard deviation
for the sampling distribution decreases
c.
The original distribution will always have a smaller
standard deviation than what is found with either of
the two sampling distributions
d.
The sampling distributions suggest possible values for
the population mean.
Example 1: The population
is normally distributed.
52
68
84
100
116
132
148
X sampling distribution
52
68
84
100
116
132
148
IQ
52
68
84
100
116
132
X sampling distribution
148
Example 1: The population
is normally distributed.
52
68
84
100
IQ
116
132
148
Sampling Distribution:
Standard Deviation & Standard Error
Statistic Standard
Deviation
p̂
x
p(1  p)
s.d.(p̂) 
n
σ
s.d.(x) 
n
We generally do
not know p.
Thus, we don’t
know s.d.(p-hat).
Similarly: We
generally do not
know σ.
Thus, we don’t
know s.d.(x-bar).
Sampling Distribution:
Standard Deviation & Standard Error
Statistic Standard
Deviation
Standard Error:
Estimated St Dev
if p is unknown, use:
p̂
x
p(1  p)
s.d.(p̂) 
n
σ
s.d.(x) 
n
p̂(1  p̂)
s.e.(p̂) 
n
if σ is unknown, use:
s
s.e.(x) 
n
Here, s is
the sample
standard
deviation.
Example 1: The population
is normally distributed.
Substitute s
for σ:
52
68
84
100
IQ
116
132
148
Standard normal (dotted red) vs. t (solid black)
Degrees of freedom
for t distribution:
1, 5, and 20
(as d.f. increases, the
t looks more like the
standard normal.)
-3
-2
-1
0
1
2
3
Standard normal (dotted red) vs. t (solid black)
Degrees of freedom
for t distribution:
1, 5, and 20
We worry a lot about
teaching t vs. z, but
the difference is tiny
for degrees of
freedom usually seen
in practice.
-3
-2
-1
0
1
2
3
Confidence Interval Formula
Generic
Formula:
Specific for
Population
Mean: µ
sample estimate ± (margin of error)
sample estimate ± (multiplier × standard
error)
x t
*
s
n
Here, t* depends on confidence level and df = (n – 1).
Multipliers:
from the t table (not a complete list, obviously)
Conf. level: 0.90
1 df
6.31
0.95
12.71
0.98
31.82
0.99
63.66
2 df
3 df
9 df
20 df
2.92
2.35
1.83
1.72
4.30
3.18
2.26
2.09
6.96
4.54
2.82
2.53
9.92
5.84
3.25
2.85
30 df
Infinite df
1.70
1.645
2.04
1.96
2.46
2.326
2.75
2.576
Example 2:
We ask each of 31 students
“how many regular ‘text’
friends do you have?”
Survey results:
n = 31 X-bar = 6 friends
Clicker Question:
What kind of variable
is this?
A.Categorical
B.Quantitative
s = 2.0 friends
Calculate a 95% Confidence Interval:
How can we estimate the population mean number of regular
“text” friends for all STAT 200 students using these data?
Confidence Interval Formula
Generic
Formula:
sample estimate ± (margin of error)
sample estimate ± (multiplier × standard
error)
Survey results:
n = 31 X-bar = 6 friends
Thus, the 95% CI is
s = 2.0 friends
Confidence Interval Interpretation
We are 95% confident that the…
Calculated Interval:
6.0 ± 0.7 friends
(5.3 to 6.7 friends)
a.
b.
c.
d.
e.
sample mean
sample proportion
population mean
population proportion
range of values for the
…number of regular “text” friends for
STAT 200 students is between 5.3
and 6.7 friends.
Confidence Interval Conclusion
95% C.I.: 5.3 to 6.7 friends
In the population, we may conclude, with 95% confidence,
that on average, STAT 200 students have
A.
B.
C.
D.
more than 6 friends.
more than 4 friends.
fewer than 5 friends.
fewer than 6 friends.
15000
Example 3: The population is
NOT normally distributed.
Histogram of 100,000 samples (n=25)
0
5000
1/6
1
2
3
4
5
6
X
10000
Histogram of 100,000 samples (n=100)
4
5
6
6000
3
2000
2
0
1
1
2
3
4
X
5
6
No
Are all sampling distributions normal? _____
When do we have to be cautious?
small sample sizes
1.
with _____
2.
where the original population is not normal
______ in shape
One-Sample t procedure is valid
if one of the conditions for
normality is met:
Sample data suggest
a normal shape
or
We have a large sample
size (n ≥ 30
__)
Sampling distribution will look normal in shape
If you understand today’s lecture…
9.61, 9.62, 9.64, 9.65, 11.25, 11.30, 11.32,
11.33
Objectives:
• Apply sampling distribution for one sample mean to
confidence intervals and hypothesis tests.
• Identify situations in which t-multipliers and t-tests should be
used instead of z-multipliers and z-tests.