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Math 227 Elementary Statistics
Bluman 5th edition
2
CHAPTER 7
Confidence Intervals
and Sample Size
3
Objectives
• Find the confidence interval for the mean
when is known or n > 30.
• Determine the minimum sample size for
finding a confidence interval for the mean.
• Find the confidence interval for the mean
when is unknown and n < 30.
4
Objectives (cont.)
• Find the confidence interval for a proportion.
• Determine the minimum sample size for finding a confidence interval for a proportion.
• Find a confidence interval for a variance and a standard deviation.
5
Introduction
• Estimation is the process of estimating the
value of a parameter from information
obtained from a sample.
• The procedures for estimating the
population mean, estimating the population
proportion, and estimating a sample size
will be explained.
6
7.1 Confidence Intervals for the Mean
When Known and Sample Size
Three Properties of a Good Estimator • The estimator should be an unbiased estimator. That is,
the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.
• The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.
• The estimator should be a relatively efficient estimator; that is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.
7
Point and Interval Estimates
• A point estimate is a specific numerical value
of a parameter. The best point estimate of the
population mean m is the sample mean .
• An interval estimate of a parameter is an
interval or a range of values used to estimate
the parameter. This estimate may or may not
contain the value of the parameter being
estimated.
X
8
I. Confidence Intervals
• Even though the best point estimate of the
population mean is the sample mean, for the
most part, the sample mean will be different
from the population mean m due to sampling
error. For this reason, statisticians prefer an
interval estimate. The confidence level of an
interval estimate of a population mean m is the
probability that the interval estimate will contain m.
X
9
Confidence Level and
Confidence Interval
• The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.
• A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.
10
Formula
• Formula for the Confidence Interval of the Mean for a Specific a
For a 90% confidence interval,
for a 95% confidence interval,
and for a 99% confidence interval,
α 2z =1.96
α 2z = 2.58
nzX
nzX
m
aa 2/2/
2 1.65za
11
• Greek letter (alpha) represents the area in both tails of the standard normal distribution curve, and represents the area of each one of the tails.
• The relationship between and the confidence level is that the stated confidence level is the percentage equivalent to the decimal value of , and vice versa.
Ex) For 95% confidence interval,
For 90% confidence interval,
a
2a
a
1 a0.05a 0.10a
12
Example 1 : Find the critical values for each.
(a) for the 99% confidence interval
22 58a / .z
%99
2/az
0 995.0 005.
13
(b) for the 95% confidence interval
21 96a / .z
95%
2/az
0 975.0 025.
14
(c) for the 90% confidence interval
21 65a / .z
90%
2/az
0 95.0 05.
15
II. Maximum Error of the Estimate
The maximum error of estimate is the maximum
difference between the point estimate of a parameter and
the actual value of the parameter.
Definition :
When estimating by from a large sample, the maximum error of the
estimate, with level of confidence , is
When is unknown, we can estimate it by , as long as →
16
95% Confidence Interval
For =0.05, 95% of the sample means will fall within the
error value on either side of the population mean.
a
17
Example 1 :
Find the maximum error for based on = 128.3, n = 64 , s = 32.4 and
confidence level of 98%.
First , figure out what formula to use :
is unknown, but →
%98
22 33a / .z
0 99.0 01.
18
III. Rounding Rule for a Confidence
Interval for a Mean
• When you are computing a confidence interval
for a population mean by using raw data, round
off to one more decimal place than the number
of decimal places in the original data.
• When you are computing a confidence interval
for a population mean by using a sample mean
and a standard deviation, round off to the same
number of a decimal places as given for the
mean.
19
Example 1 : (Ref General Statistics by Chase/Bown, 4th Ed.)
A physician wanted to estimate the mean length of time that a patient had to
wait to see him after arriving at the office. A random sample of 50 patients
showed a mean waiting time 23.4 minutes and a standard deviation of 7.1
minutes. Find a 95% confidence interval for .
is unknown, but →
0.975
Now
Formula to use : , but first find .
0 025.
20
A union official wanted to estimate the mean hourly wage of its members. A
random sample of 100 members gave = $18.30 and = $3.25 per hour.
Example 2 : (Ref General Statistics by Chase/Bown, 4th Ed.)
(a) Find an 80% confidence interval for .
is unknown, but → Now
Formula to use : , but first find .
0.900 10.
21
(b) Find an 95% confidence interval for .
Now
0.9750 025.
22
(c) If you were to construct a 90% confidence interval for (do not construct it),
would the interval be longer or shorter than 80% confidence interval? Longer
or shorter than the 95% confidence interval?
%80
%90
%90
%95
Interval of 90% is longer than interval
of 80%.
Interval of 90% is shorter than interval
of 95%.
23
A restaurant owner believed that customer spending was below normal at tables
manned by one of the waiters. The owner sampled 36 checks from the waiter’s
tables and got the following amounts (rounded to the nearest dollar) :
Example 3 : (Ref General Statistics by Chase/Bown, 4th Ed.)
47 46 56 70 52 58 48 57 49 61 52 40 60 22 74 59 60 30
61 44 62 41 53 57 50 52 57 59 69 51 58 56 44 36 47 51
Find a 95% confidence interval for the true mean amount of money spent at the
waiter’s tables.
Using calculator,
0.9750 025.
24
IV. Determining the Sample Size for
Maximum Error of the estimate for
Solve for n
Round the answer up to obtain a whole number
Note : If level of confidence is not given, then use 95%.
25
Example 1 :
To estimate , what sample size is required so that the maximum error of the
estimate is only 8 square feet? Assume is 42 square feet. (Use 95%
confidence level.)
0.9750 025.
26
Example 2 : (Ref General Statistics by Chase/Bown, 4th Ed.)
Consider a population with unknown mean and population standard deviation
(a) How large a sample size is needed to estimate to within five units with
95% confidence?
= 15.
(b) Suppose you wanted to estimate to within five units with 90% confidence.
Without calculating, would the sampling size required be larger or smaller than the
found in part (a) ?
90% confidence means smaller → n is smaller.
27
(c) Suppose you wanted to estimate to within six units with 95% confidence.
Without calculating, would the sampling size required be larger or smaller the found
in part (a) ?
n is smaller because when you divide by a bigger E.
28
7.2 Confidence Intervals for the Mean
( Unknown and )
• When the population sample size is less
than 30, and the standard deviation is
unknown, the t distribution must be used.
29
Characteristics of the t
Distribution
• The t distribution is similar to the standard
normal distribution in the following ways:
1. It is bell shaped.
2. It is symmetrical about the mean.
3. The mean, median, and mode are equal to 0
and are located at the center of the
distribution.
4. The curve never touches the x axis.
30
Characteristics of the t
Distribution (cont.)
• The t distribution differs from the standard normal distribution in the following ways.
1. The variance is greater than 1.
2. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size.
3. As the sample size increases, the t distribution approaches the standard normal distribution.
31
t Distribution
• The degrees of freedom are the number of
values that are free to vary after a sample
statistic has been computed.
• The degrees of freedom for the confidence
interval for the mean are found by
subtracting 1 from the sample size. That
is,
1.. nfd
32
Formula
• Formula for the Confidence Interval of the Mean When is Unknown and
/ 2 / 2
s sX t X t
n na am
Maximum Error for :
. . 1d f n Degree of Freedom :
33
Example 1 : Find with the following information.
(a) Level of confidence is 98% with n = 19
%98
%12/ a
Look up on t – table
(b) Level of confidence is 90% with n = 25
%90
%52/ a
Look up on t – table
34
Example 2 :
A sample of 25 two-year-old chickens shows that they lay an average of 21 eggs
per month. The standard deviation of the sample was 2 eggs. Assume the
population is approximately normal. Construct a 99% confidence interval for the
true mean.
%99
%5.02/ a
Look up on t – table
35
Example 3 :
A random sample of 20 parking meters in a large municipality showed the following
incomes for a day.
$2.60 $1.05 $2.45 $2.90 $1.30 $3.10 $2.35
$2.00 $2.40 $2.35 $2.40 $1.95 $2.80 $2.50
$2.10 $1.75 $1.00 $2.75 $1.80 $1.95
Assume the population is approximately normal. Find the 95% confidence interval
of the true mean.
%95
%5.22/ a
Look up on t – table
36
When to Use the Z or t Distribution
37
7.3 Confidence Intervals and
Sample Size for Proportions
I. Confidence Intervals for Proportions
• Symbols Used in Proportion Notation
p = symbol for the population proportion
(read p “hat”) = symbol for the sample proportion
For a sample proportion,
where X = number of sample units that possess the
characteristics of interest and n = sample size.
and or 1ˆ ˆ ˆX n X
p q pn n
p̂
38
Formula
• Formula for a specific confidence interval for
a proportion
when np and nq are each greater than or
equal to 5.
n
qpzpp
n
qpzp aa
ˆˆˆ
ˆˆˆ
2/2/
39
II. Determining the Sample Size for p
Round the answer up to obtain a whole number.
Since the sample has not yet been obtained, we do not know the value of
and . However, it can be shown that regardless of the values of and ,
the value of will never be more than ¼ . Therefore, to be on the safe
side, we should take the sample size to be at least
Round the answer up to obtain a whole number.
40
Example 1 : (Ref: General Statistics by Chase/Bown, 4th Ed.)
A city council commissioned a statistician to estimate the proportion of voters in
favor of a proposal to build a new library. The statistician obtained a random
sample of 200 voters, with 112 indicating approval of the proposal.
(a) What is a point estimate for ?
(b) What is the maximum error of estimate for with
a 90% confidence level?
(c) Find a 90% confidence interval for .
Example 2 :
A Roper poll of 2,000 American adults showed that 1,440 thought that chemical
dumps are among the most serious environmental problems. Estimate with a 98%
confidence interval the proportion of population who consider chemical dumps
among the most serious environmental problem.
41
22 33a / .z
0 99.0 01.
42
Example 3 :
A recent study indicated that 29% of the 100 women over age 55 in the study were
widows.
(a) How large a sample must one take to be 90% confident that the estimate is
within 0.05 of the true proportion of women over 55 who are widows?
(b) If no estimate of the sample proportion is available, how large should the
sample be?
43
Example 4 :
How large a sample is necessary to estimate the true proportion of adults who are
overweight to within 2% with a 95% confidence?
0.9750 025.
44
Summary
• A good estimator must be unbiased, consistent, and relatively efficient.
• There are two types of estimates of a parameter: point estimates and interval estimates.
• A point estimate is a single value. The problem with point estimates is that the accuracy of the estimate cannot be determined, so the interval estimate is preferred.
45
Summary (cont.)
• By calculating a 95% or 99% confidence interval
about the sample value, statisticians can be 95%
or 99% confident that their estimate contains the
true parameter.
• Once the confidence interval of the mean is
calculated, the z or t values are used depending
on the sample size and whether the standard
deviation is known.
46
Summary (cont.)
• The following information is needed to
determine the minimum sample size
necessary to make an estimate of the
mean:
1. The degree of confidence must be stated.
2. The population standard deviation must be
known or be able to be estimated.
3. The maximum error of the estimate must be
stated.
47
Conclusions
Estimation is an important aspect of
inferential statistics.
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