Download 9.1 Day 2

Section 9.1
CI for a Mean
Day 2
• Does “double-blind” study mean the
treatments are randomly assigned?
• Does “double-blind” study mean the
treatments are randomly assigned?
NO!
• “Double-blind” means that neither the
doctor nor the patient knows which
treatment the patient is given.
Experiment
• “Randomized” means treatments are
randomly assigned.
• “double-blind, randomized, clinical
trial.”
Do Computations
A confidence interval for the population
mean,  , is given by:
s
x  t* 
n
Can use calculator: STAT, TESTS
8: TInterval
Using Original Data
Using Summary Statistics
Give Interpretation in Context
Good interpretation is of this form: “I am
95% confident that the population mean,
 , is in this interval.”
Be specific about the confidence interval
and describe the population you are
talking about.
Give Interpretation in Context
Suppose we want to construct a 95% CI for
the mean number of hours students study
each night. The interval is (1.7, 3.5).
Give Interpretation in Context
Suppose we want to interpret a 95% CI for
the mean number of hours students study
each night. The interval is (1.7, 3.5).
Consider three different surveys:
1) random sample of 40 students from EHS
2) random sample of 120 students from IL
3) random sample of 500 students from US
Give Interpretation in Context
Suppose we want to interpret a 95% CI for
the mean number of hours students study
each night. The interval is (1.7, 3.5).
1) random sample of 40 students from
EHS
I’m 95% confident that the mean number of
hours students at EHS study each night
is in the interval (1.7, 3.5).
Give Interpretation in Context
Suppose we want to interpret a 95% CI for
the mean number of hours students study
each night. The interval is (1.7, 3.5).
2) random sample of 120 students from
IL
I’m 95% confident that the mean number of
hours students in IL study each night is
between 1.7 hours to 3.5 hours.
Give Interpretation in Context
Suppose we want to interpret a 95% CI for
the mean number of hours students study
a night. The interval is (1.7, 3.5).
3) random sample of 500 students from
US
I’m 95% confident that the mean number of
hours students in the US study each
night is in the interval (1.7, 3.5).
Page 574, P3 (a)
Page 574, P3 (a)
TInterval
Inpt: Data Stats
Page 574, P3 (a)
TInterval
Inpt: Data Stats
Page 574, P3 (a)
TInterval
Inpt: Data Stats
x: 27
sx: 12
n: 4
C-Level: .95
Calculate
Page 574, P3 (a)
TInterval
Inpt: Data Stats
x: 27
sx: 12
n: 4
C-Level: .95
Calculate
(7.9053, 46.095)
Page 576, E3
Should Jack and Jill readjust the machine?
Use a statistical argument to support your
advice.
Page 576, E3
Check conditions:
Page 576, E3
Check conditions:
1) told bottles are random sample from the
day’s production
Page 576, E3
Check conditions:
1) told bottles are random sample form the
day’s production
2) told the distribution of number of ounces
of water in the bottle is approx. normal
(Which distribution: sample or
population?)
Page 576, E3
Check conditions:
1) told bottles are random sample form the
day’s production
2) told the distribution of number of ounces
of water in the bottle is approx. normal
Population
Page 576, E3
Check conditions:
1) told bottles are random sample form the
day’s production
2) told the distribution of number of ounces
of water in the bottle is approx. normal
3) day’s production is most likely more than
100 bottles, which is 10 times the sample
size
Page 576, E3
TInterval
Inpt: Data Stats
Page 576, E3
TInterval
Inpt: Data Stats
List: L1
Freq: 1
C-Level: .95
Calculate
Page 576, E3
TInterval
Inpt: Data Stats
List: L1
Freq: 1
C-Level: .95
Calculate
95% CI is (15.916, 16.031)
Page 576, E3
I’m 95% confident that the mean weight of
the water bottles produced that day is
between 15.917 oz and 16.031 oz.
Page 576, E3
I’m 95% confident that the mean weight of
the water bottles produced that day is
between 15.917 oz and 16.031 oz.
Because 16 oz is one of the plausible values
for the population mean, there is no need
to adjust the machine.
Page 575, P8
Page 575, P8
a) False.
A CI is a statement about plausible values
for a population mean, not about the
individual values within a population.
Page 575, P8
b) False.
A CI is a statement about plausible values
for a population mean, not about the
values in the sample.
Page 575, P8
c) False.
The method has a 95% chance of success,
but after this particular interval is
calculated, either it contains the population
mean or it doesn’t.
Page 575, P8
d) False.
The sample mean is the center of the CI.
Page 575, P8
e) True.
95% of 200 is 190. So about 190 of the 200
CIs would include the population mean
and approximately 10 would not.
Page 579, E13
x ± t*●
s
n
x ± t*●
s
n
(a) standard deviation of sample increased?
margin of error _________
x ± t*●
s
n
(a) standard deviation of sample increased?
margin of error increases
x ± t*●
s
n
(b) sample size is increased?
margin of error _______
x ± t*●
s
n
(b) sample size is increased?
margin of error decreases
x ± t*●
s
n
(c) confidence level is increased?
margin of error _______
x ± t*●
s
n
(c) confidence level is increased?
margin of error increases
Page 577, E6
Page 577, E6
a) This is not a random sample of bags of
fries. Also, because all the bags came
from one restaurant, even with a random
sample, you could only infer about bags of
fries from this restaurant.
Page 577, E6
a) Distribution of number of fries is fairly
symmetric, so there is no indication that
the underlying population is not normal
(See Display 9.13, page 578)
Page 577, E6
a) This McDonald’s restaurant probably
sells more than 320 bags of fries, which
would be more than 10 times the sample
size.
So, random sample condition not met but
the other conditions are.
Page 577, E6
b) 95% CI is (44.888, 51.362)
Page 577, E6
b) 95% CI is (44.888, 51.362)
If this were a random sample,
Page 577, E6
b) 95% CI is (44.888, 51.362)
If this were a random sample, then I would
be 95% confident that the mean number of
fries in a small bag at this McDonald’s
restaurant is between about 45 and 51
fries.
Page 574, P7
For aldrin data in P4,
95% CI is (4.229, 5.809)
Page 574, P7
D. You are 95% confident that the mean
aldrin level of the Wolf River falls in the
confidence interval (4.229, 5.809)
Questions?
Page 564, D1
Page 564, D1
The larger sample size of 1000 will give
capture rates closer to 95% because the
skewness of the sampling distribution of s
decreases as n increases.
Page 573, P1
Page 573, P1
or (4.335, 5.703)
Page 573, P1
Page 565, D2
Page 565, D2
If an interval’s true capture rate is smaller
that you want it to be, you can get a larger
capture rate by using a wider interval.
Page 568, D3
Page 568, D3
Page 571, D5
Page 571, D5
It is never possible to be 100% confident
with less than 100% of the population in
hand.
Page 571, D5
It is never possible to be 100% confident
with less than 100% of the population in
hand.
For samples, high level of confidence
intervals produce wide intervals.
A level of 99.5% may produce an interval so
wide to be of little use.
Page 572, D7
Page 572, D7
Saying “I’m 95% confident that the
population mean is in the confidence
interval I just constructed” is actually
correct.
Page 572, D7
Saying “I’m 95% confident that the
population mean is in the confidence
interval I just constructed” is actually
correct.
It would be incorrect to say “there is a 95%
probability (95% chance) that the true
population mean falls in this interval.”
Why?
Page 572, D7
It would be incorrect to say “there is a 95%
probability (95% chance) that the true
population mean falls in this interval.”
Why?
Once you have constructed the CI, then the
population mean is either in the interval or
it is not
Meaning of “95% Confident”
Saying you are 95% confident means that if
you could take random samples
repeatedly from the population and
compute a confidence interval for each
sample, in the long run 95% of these
different intervals would contain (or
capture) the population mean.
Page 574, P4
Page 574, P4
Interval from P1 is (4.335, 5.704)
Page 574, P4
Interval from P1 is (4.335, 5.704)
Interval from P4 is (4.229, 5.809)
Why is the second interval wider?
Page 574, P4
Interval from P1 is (4.335, 5.704)
Interval from P4 is (4.229, 5.809)
Why is the second interval wider? Because
P1 used z* and P4 used t*