Download Confidence Intervals

Confidence
Intervals with
Means
What is the purpose of a
confidence interval?
To estimate an unknown
population parameter
Formula:
Standard
deviation of
Critical value
statistic
Confidence Interval :
  

x  z * 
 n
statistic
Margin of error
In a randomized comparative experiment on
the effects of calcium on blood pressure,
researchers divided 54 healthy, white males
at random into two groups, taking calcium
or placebo. The paper reports a mean
seated systolic blood pressure of 114.9 with
standard deviation of 9.3 for the placebo
group. Assume systolic blood pressure is
normally distributed.
Can you find a z-interval for this problem?
Why or why not?
Student’s t- distribution
• Continuous distribution
• Unimodal, symmetrical, bell-shaped
density curve
• Above the horizontal axis
• Area under the curve equals 1
• Based on degrees of freedom
df = n - 1
Formula:
Standard
deviation of
Standard error
–
Critical value
statistic
when you
substitute s for .
Confidence Interval :
 s 

x  t * 
 n
statistic
Margin of error
How to find t*
• Use
B for
t distributions
CanTable
also use
invT
on the calculator!
• Look up confidence level at bottom &
Need
t* value with 5% is above –
df onupper
the sides
• df = n – 1 so 95% is below
invT(p,df)
Find these t*
90% confidence when n = 5
95% confidence when n = 15
t* = 2.132
t* = 2.145
Steps for doing a confidence
interval:
1) Assumptions –
2) Calculate the interval
3) Write a statement about the interval
in the context of the problem.
Statement: (memorize!!)
We are ________% confident
that the true mean context is
between ______ and ______.
Assumptions for t-inference
• Have an SRS from population (or
randomly assigned treatments)
•  unknown
• Normal (or approx. normal) distribution
– Given
– Large sample size
– Check graph of data
Use only one of
these methods to
check normality
Ex. 1) Find a 95% confidence interval for the
true mean systolic blood pressure of the
placebo group.
Assumptions:
• Have randomly assigned males to treatment
• Systolic blood pressure is normally distributed
(given).
•  is unknown
 9.3 
114.9  2.056
  (111.22, 118.58)
 27 
We are 95% confident that the true mean systolic
blood pressure is between 111.22 and 118.58.
Find a sample size:
• If a certain margin of error is wanted,
then to find the sample size necessary
for that margin of error use:
  
m  z *

 n
Always round up to the nearest person!
Ex 4) The heights of PWSH male
students is normally distributed with
 = 2.5 inches. How large a sample
is necessary to be accurate within +
.75 inches with a 95% confidence
interval?
n = 43
Hypothesis Tests
One Sample Means
How canagency
I tell ifhas
they really
A government
are
underweight?
received numerous complaints
A hypothesis test
that will
a particular
restaurant
has
allow me to
been
selling
underweight
decide
if the
claim
is true or not!
hamburgers.
restaurant
Take The
a sample
& find x.
advertises that it’s patties are
“a quarter
(4if ounces).
But howpound”
do I know
this x is one that I
expect to happen or is it one that is
unlikely to happen?
Steps for doing a hypothesis
test
“Since the p-value < (>) a, I reject
1) Assumptions
(fail to reject) the H0. There is (is
not) sufficient evidence to suggest
thathypotheses
Ha (in context).”
2) Write
& define parameter
H0: m = 12 vs Ha: m (<, >, or ≠) 12
3) Calculate the test statistic & p-value
4) Write a statement in the context of the
problem.
Formulas:
 unknown:
statistic - parameter
test statistic 
standard deviation of statistic
t=
x m
s
n
Calculating p-values
• For z-test statistic –
– Use normalcdf(lb,rb)
– [using standard normal curve]
• For t-test statistic –
– Use tcdf(lb, rb, df)
Draw & shade a curve &
calculate the p-value:
1) right-tail test
t = 1.6; n = 20
P-value = .0630
2) two-tail test
t = 2.3; n = 25
P-value = (.0152)2 = .0304
Example 1: Bottles of a popular cola are
supposed to contain 300 mL of cola.
There is some variation from bottle to
bottle. An inspector, who suspects that
the bottler is under-filling, measures the
contents of six randomly selected bottles.
Is there sufficient evidence that the
bottler is under-filling the bottles?
Use a = .1
299.4 297.7 298.9 300.2 297 301
• I have an SRS of bottles
SRS?
Normal?
•Since the boxplot is approximately symmetrical with
no
outliers, the sampling distribution is approximatelyHow do you
know?
normally distributed
Do you
know ?
What are your
H0: m = 300 where m is the true mean amount
hypothesis
statements? Is
Ha: m < 300 of cola in bottles
there a key word?
299 .03  300
t 
 1.576 p-value =.0880
a = .1
1.503
Plug p-value
values to
Compare your
6
into decision
formula.
a & make
Since p-value < a, I reject the null hypothesis.
Write conclusion in
There is sufficient evidence to suggest
that
the true
context
in terms
of Ha.
mean cola in the bottles is less than 300 mL.
•  is unknown
Matched Pairs
Test
A special type of
t-inference
Matched Pairs – two forms
• Pair individuals by
certain
characteristics
• Randomly select
treatment for
individual A
• Individual B is
assigned to other
treatment
• Assignment of B is
dependent on
assignment of A
• Individual persons
or items receive
both treatments
• Order of
treatments are
randomly assigned
before & after
measurements are
taken
• The two measures
are dependent on
the individual
Is this an example of matched pairs?
1)A college wants to see if there’s a
difference in time it took last year’s
class to find a job after graduation and
the time it took the class from five years ago
to find work after graduation. Researchers
take a random sample from both classes and
measure the number of days between
graduation and first day of employment
No, there is no pairing of individuals, you
have two independent samples
Is this an example of matched pairs?
2) In a taste test, a researcher asks people
in a random sample to taste a certain brand
of spring water and rate it. Another
random sample of people is asked to
taste a different brand of water and rate it.
The researcher wants to compare these
samples
No, there is no pairing of individuals, you
have two independent samples – If you would
have the same people taste both brands in
random order, then it would be an example
of matched pairs.
Is this an example of matched pairs?
3) A pharmaceutical company wants to test
its new weight-loss drug. Before giving the
drug to a random sample, company
researchers take a weight measurement
on each person. After a month of using
the drug, each person’s weight is
measured again.
Yes, you have two measurements that are
dependent on each individual.
A whale-watching company noticed that many
customers wanted to know whether it was
better to book an excursion in the morning or
the afternoon.
To test
this question, the
You may subtract
either
company
thewhen
following data on 15
way – collected
just be careful
writing Hadays over the past
randomly selected
month. (Note: days were not
consecutive.)
Day
1
2
Morning
8 9
3
4
5
6
7
8
9
10
11 12 13 14 15
7 9 10 13 10
8
2
5
7 7 6 8 7
After8 10 9 8 9 11 8
noon
Since you have two values for
10
4 7 8 9 6 6 9
First, you must find
the differences for
each day.
each day, they are dependent
on the day – making this data
matched pairs
Day
1
2
3
Morning
8
9
7 9 10 13 10
Afternoon
8 10
4
5
9 8 9
6
7
8
9
10
11 12 13 14 15
8
2
5
7 7 6 8 7
11
8 10 4 7 8 9 6 6 9
I subtracted:
Differenc
0 -1 -2 1 1 Morning
2 2 – -2
-2 -2 -1 -2 0 2 -2
afternoon
es
You could subtract the other
way!
• Have an SRS of days for whale-watching
You need to state assumptions using the
•  unknown
differences!
Assumptions:
•Since the normal probability plot is approximately
linear, the distribution of difference is approximately
Notice the granularity in this
normal.
plot, it is still displays a nice
linear relationship!
Differences
0
-1
-2
1
1
2
2
-2
-2
-2
-1 -2
0
2
Is there sufficient evidence that more whales are
sighted in the afternoon?
H0: mD = 0
Ha: mD < 0
Be careful writing your Ha!
Think about
how you–
If you subtract
afternoon
subtracted: M-A
Hdifferences
mD>0should
Notice morning;
we
mthen
a:more
D foris
Ifused
afternoon
& it equals
since the nullbeshould
the0 differences
+ or -?
be that there
NOat
difference.
Don’t islook
numbers!!!!
Where mD is the true mean
difference in whale sightings
from morning minus afternoon
-2
Differences
0
-1
-2
1
1
2
2
-2
finishing the hypothesis test:
x m
.4  0
t 

 .945
s
1.639
n
15
p  .1803
df  14
a  .05
-2
-2
-1 -2
0
2
In your calculator,
perform
t-test
Notice athat
if
the
youusing
subtracted
differences
(L3)
A-M, then your
test statistic
t = + .945, but pvalue would be
the same
Since p-value > a, I fail to reject H0. There
is
How could
I
insufficient evidence to suggest that more
whales
increase
theare
sighted in the afternoon than in the morning.
power of this
test?
-2