Estimation

1
Chapter 8

• Estimation

2
Table of Contents
8.1 Estimating µ When s Is Known
8.2 Estimating µ When s Is Unknown
8.3 Estimating p in the Binomial Distribution
8.4 Estimating µ1-µ2 and p1-p2
3
8.1 Estimating µ When s Is Known
Assumptions about the random variable x
1. We have a simple random sample of size
n drawn from a population of x values.
2. The value of s, the population
standard deviation of x, is known.
3. If the x distribution is normal, then
our methods work for any sample size n.
4. If x has an unknown distribution, then
we require a sample size n 30. However,
if the x distribution is distinctly skewed
and definitely not mound-shaped, a sample
of size 50 or even 100 or higher may
be necessary.
4
Definitions
8.1 Estimating µ when s Is Known
5
Definition
For a confidence level c, the critical value zc
is the number such that the area under the
standard normal curve between zc and zc equals c.
In mathematical notation, zc is the value that
satisfies P ( zc lt z lt zc ) c
8.1 Estimating µ when s Is Known
6
Note
We have gone over this concept in section 6.3
when we discussed the invNorm() function with the
center case. As a refresher, refer to example 9
and figure 6-28 on page 263. With the center
case, we used invNorm((1A)/2).
8.1 Estimating µ when s Is Known
7
8.1 Estimating µ when s Is Known
8
Definition
8.1 Estimating µ when s Is Known
9
Area c
8.1 Estimating µ when s Is Known
10
8.1 Estimating µ when s Is Known
11
8.1 Estimating µ when s Is Known
12
Definition
A c confidence interval for µ is an interval
computed from sample data in such a way that c is
the probability of generating an interval
containing the actual value of µ.
8.1 Estimating µ when s Is Known
13
How to find a confidence interval for µ when s is
known
If you can assume that x has a normal
distribution, then any sample size n will work.
If you cannot assume this, then use a sample size
of n 30.
8.1 Estimating µ when s Is Known
14
How to find a confidence interval for µ when s is
known
Confidence interval for µ when s is known
c confidence level (0 lt c lt 1)
zc critical value for confidence level c
8.1 Estimating µ when s Is Known
15
Note
Two consequences of these formulas are
1. As c increases, so does E, and
the confidence interval widens.
2. As n increases, E decreases, so
the confidence interval narrows.
8.1 Estimating µ when s Is Known
16
Calculator function
The TI-83/84 calculator has a function to compute
the confidence interval when s is known.
STATTESTSZInterval
8.1 Estimating µ when s Is Known
17
Example 2
s 1.80 min
n 90
c 95
We are 95 confident that 15.23 lt µ lt 15.97.
8.1 Estimating µ when s Is Known
18
Figure 8-4
0.90 Confidence Intervals for Samples of the Same
Size
At 90, we have a 1/10 chance that our interval
does not contain µ.
8.1 Estimating µ when s Is Known
19
Sample size for estimating the mean µ
Sometimes we will encounter situations where we
dont want the margin of error E to exceed a
certain value, given some confidence level.
Remember from earlier that the confidence
interval narrows as n increases. The goal is to
find the sample size n so that the margin of
error E doesnt exceed a certain value given some
confidence level c.
8.1 Estimating µ when s Is Known
20
How to find the sample size n for estimating µ
when s is known
If n is not a whole number, increase n to the
next higher whole number. Note that n is the
minimal sample size for a specified confidence
level and maximal error of estimate E.
8.1 Estimating µ when s Is Known
21
How to find the sample size n for estimating µ
when s is known
For this equation, either s is known, or n 30
and we can use s as an estimate for s.
8.1 Estimating µ when s Is Known
22
Example 3
We need to find n big enough to reduce E down to
0.20 lb at the 99 confidence level.
We also need a value for s. Since n 50 30 in
the preliminary study, we can use s s 2.15 lb.
8.1 Estimating µ when s Is Known
23
Example 3
769.2
n 770
We need to find n big enough to reduce E down to
0.20 lb at the 99 confidence level.
We also need a value for s. Since n 50 30 in
the preliminary study, we can use s s 2.15 lb.
8.1 Estimating µ when s Is Known
24
8.2 Estimating µ When s Is Unknown
In 8.1, we knew s before trying to determine a
confidence interval. Then, if x were normal, any
sample size n would work, and if x were not
normal, then we would need a sample size of at
least 30.
In 8.2, we wont know s before we try to
determine a confidence interval. In this case,
we require x to be normal, or at least
approximately normal, or that we have n 30.
25
Definition
has a Students t distribution with degrees of
freedom d.f. n 1.
8.2 Estimating µ when s Is Unknown
26
Normal
d.f. 20
d.f. 8
d.f. 4
d.f. 1
Thicker tails than normal
4
3
2
1
0
1
2
3
4
8.2 Estimating µ when s Is Unknown
27
Figure 8-6
Area c
tc
0
tc
The critical value tc for a c confidence level is
found in a fashion similar to finding zc in that
the area above the t-interval tc,tc is c.
8.2 Estimating µ when s Is Unknown
28
Figure 8-6
Area c
tc
0
tc
In mathematical notation, tc is the value that
satisfies P ( tc lt t lt tc ) c
8.2 Estimating µ when s Is Unknown
29
Degrees of freedom for the t distribution
For confidence intervals involving µ, the degrees
of freedom that determine the t distribution is
one less than the sample size. In other words,
d.f. n - 1
8.2 Estimating µ when s Is Unknown
30
Example 4
d.f. n 1 5 1 4
c 0.99
tc 4.604
8.2 Estimating µ when s Is Unknown
31
How to find a confidence interval for µ when s is
unknown
If you can assume that x has a normal
distribution or simply a mound-shaped symmetric
distribution, then any sample size n will work.
If you cannot assume this, then use a sample size
of n 30.
8.2 Estimating µ when s Is Unknown
32
How to find a confidence interval for µ when s is
unknown
Confidence interval for µ when s is unknown
c confidence level (0 lt c lt 1)
tc critical value for confidence level c
and degrees of freedom d.f. n 1
8.2 Estimating µ when s Is Unknown
33
Calculator function
The TI-83Plus calculator has a function to
compute the confidence interval when s is unknown.
STATTESTSTInterval
8.2 Estimating µ when s Is Unknown
34
Example 4
If we have raw data to start with, we can enter
the data into a list, and then choose Data for
the Input type. Lets enter data into L1.
Freq will always be 1
We are 99 confident that 44.5 lt µ lt 47.8.
8.2 Estimating µ when s Is Unknown
35

Which distribution should you use for x?
Examine problem statement
s is known
s is unknown
Use ZInterval
Use TInterval
8.2 Estimating µ when s Is Unknown
36
8.3 Estimating p in the Binomial Distribution
In order to estimate p from a sample, we want to
have np gt 5 and nq gt 5, just as with section 6.4.
Definition
The point estimates for p and q are
q 1 p
where n number of trials and r number of
successes.
37
Calculator function
The TI-83/84 calculator has a function to compute
the confidence interval for a proportion p.
STATTESTS1-PropZInt
This returns a confidence interval (low p, high
p). E, if needed, can be back-calculated by
finding p low p.
8.3 Estimating p in the Binomial Distribution
38
Example 6
r 600
n 800
(c) Successes 600 gt 5, failures 200 gt 5.
Yes, a normal approximation is justified.
1-PropZInt x600 n800 C-Level.99 Calculate
(d)
r becomes x in the TI-83/84
We are 99 confident that 0.71 lt p lt 0.79.
If we were asked to find E, it would be E p
low p 0.75 0.7106 0.0394.
8.3 Estimating p in the Binomial Distribution
39
General interpretation of poll results
1. When a poll states the results of a
survey, the proportion reported to respond in
the designated manner is p, the
sample estimate of the population proportion.
2. The margin of error is the maximal error
E of a 95 confidence interval for p.
3. A 95 confidence interval for
the population proportion p is poll report p
margin of error E lt p lt poll report p
margin of error E
8.3 Estimating p in the Binomial Distribution
40
Definition
For a binomial distribution
If we have a preliminary estimate for p, then
If we have no preliminary estimate for p, then
In both cases, if n is not a whole number, then
always round up.
8.3 Estimating p in the Binomial Distribution
41
Example 7
E 0.01
(a)
9604
(b) p 0.86
4625.2864
n 4626
8.3 Estimating p in the Binomial Distribution
42
8.4 Estimating µ1 - µ2 and p1 - p2
Definitions
Two samples are independent if sample data drawn
from one population are completely unrelated to
the selection of sample data from the other
population.
Two samples are dependent if each data value in
one sample can be paired with a corresponding
data value in the other sample.
43
Note
8.4 deals exclusively with data from independent
samples.
8.5 Estimating µ1-µ2 and p1-p2
44
Meaning of figure 8-7
Back in section 7.1, we studied an example of
trout lengths from a pond.
8.5 Estimating µ1-µ2 and p1-p2
45
Meaning of figure 8-7
8.5 Estimating µ1-µ2 and p1-p2
46
Meaning of figure 8-7
Now suppose there is another set of data to
examine, lets say from a nearby pond. To
distinguish these data, lets call one of them x1
and the other x2.
8.5 Estimating µ1-µ2 and p1-p2
47
Meaning of figure 8-7
10.6
12.3
1.7
8.5 Estimating µ1-µ2 and p1-p2
48
Meaning of figure 8-7
We can now make an x1 x2 distribution.


10.8
13.2
2.4

10.4
11.9
1.5

1.7
8.5 Estimating µ1-µ2 and p1-p2
49
Theorem 8.1
1. a normal distribution
2. mean µ1 µ2
3. standard deviation
8.5 Estimating µ1-µ2 and p1-p2
50
Calculator function
The TI-83/84 calculator has a function to compute
the confidence interval for µ1 µ2 when both s1
and s2 are known.
STATTESTS2-SampZInt
8.5 Estimating µ1-µ2 and p1-p2
51
Possible conclusions
When µ1 µ2 lies between two negative numbers,
then µ1 µ2 must be negative, so µ1 lt µ2 at the
given confidence level c.
When µ1 µ2 lies between two positive numbers,
then µ1 µ2 must be positive, so µ1 gt µ2 at the
given confidence level c.
When µ1 µ2 ranges between a negative value and
a positive value, then we cant say that either
of the means µ1 or µ2 is greater than the other
at the given confidence level c.
8.5 Estimating µ1-µ2 and p1-p2
52
Example 8
We are 95 confident that 2.10 lt µ1 µ2 lt 1.10.
8.5 Estimating µ1-µ2 and p1-p2
53
Example 8
Since µ1 µ2 lies between two negative numbers,
we can be 95 confident that µ1 lt µ2.
8.5 Estimating µ1-µ2 and p1-p2
54
Calculator function
The TI-83/84 calculator has a function to compute
the confidence interval for µ1 - µ2 when both s1
and s2 are unknown.
STATTESTS2-SampTInt
In this function, we will always set the Pooled
option to No.
8.5 Estimating µ1-µ2 and p1-p2
55
Example 9
We can enter our data into two lists and let the
calculator do the computing for us. Lets put
Group 1 into L1 and Group 2 into L2.
We are 90 confident that 11.9 lt µ1 µ2 lt 14.3.
8.5 Estimating µ1-µ2 and p1-p2
56
Example 9
We can enter our data into two lists and let the
calculator do the computing for us. Lets put
Group 1 into L1 and Group 2 into L2.
We are 90 confident that µ1 gt µ2.
8.5 Estimating µ1-µ2 and p1-p2
57
Something you should know
The TI-83/84 uses a different formula for
computing degrees of freedom (just) in the
two-sample t-interval calculations. Your answers
will only be very (very very) slightly different
from the books answers because of this. On the
exam, I will give full credit regardless of
whether you use the 2-SampTInt function or
whether you use the t-table.
8.5 Estimating µ1-µ2 and p1-p2
58
For those of you who absolutely must know how the
calculator determines degrees of freedom when
dealing with differences, it uses a much more
complicated delta formula ? rounded down to the
nearest integer. The following formula for ? is
given below purely for those who are curious. It
is actually NOT part of this course. We will not
be using it.
8.5 Estimating µ1-µ2 and p1-p2
59
Which distribution should you use for x1 x2?


Examine problem statement
s1 and s2 are known
s1 and s2 are unknown
Use 2-SampZInt
Use 2-SampTInt
8.5 Estimating µ1-µ2 and p1-p2
60
Calculator function
The TI-83/84 calculator has a function to compute
the confidence interval for p1 p2.
STATTESTS2-PropZInt
8.5 Estimating µ1-µ2 and p1-p2
61
Example 10
2-PropZInt x149 n1175 x263 n2180
C-Level.95 Calculate
We are 95 sure that 0.166 lt p1 p2 lt 0.026.
Since p1 p2 straddles negative to positive, we
cant be sure that one populations p is greater
than the other at the 95 level of confidence.
8.5 Estimating µ1-µ2 and p1-p2