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Chapter 7
Section 7 - 1 Introduction
Inferential statistics is the process of making judgments about a population based on properties
of a sample from the population.
One aspect of inferential statistics is estimation.
The best point estimate of the population mean  is the sample mean x .
The best point estimate of the population proportion p is the sample proportion p̂ .
An important question in estimating the population mean and proportion is that of sample size.
How large should the sample be in order to make an accurate estimate?
The procedures for estimating the population mean, estimating the population proportion, and
estimating a sample size will be explained. The other type of inference is called hypothesis
testing, which is a decision-making process for evaluating claims about a population. The
procedures for hypothesis testing for the population mean and proportion will be explained in
Chapter 8.
Section 7 - 2 Estimating a population mean (Large-Sample Case)
I.
Level of Confidence 1 - 
Even the best point estimate of the population mean is the sample mean, for the most
part, the sample mean x will be different from the population mean  due to sampling
error. For this reason, statisticians prefer an interval estimate. The confidence level of an
interval estimate of a population mean  is the probability that the interval estimate will
contain  .
Example 1:
Find the critical values for each.
(a)
z / 2
for the 99% confidence interval
(b)
z / 2
for the 95% confidence interval
(c)
z / 2
for the 90% confidence interval
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 x from a large sample, the maximum error of the estimate, with
level of confidence 1- , is
E  z / 2 

n
When  is unknown, we can estimate it by s ; as long as n  30    s .
E  z / 2 

n
z / 2 
s
n
Example 1: Find the maximum error for  based on x =128.3, n = 64, s = 32.4, and
confidence level of 98%.
III.
Confidence Interval for 
xE
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 decimal places as given
for the mean.
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 of 23.4 minutes and a standard
deviation of 7.1 minutes. Find a 95% confidence interval for  .
Example 2:
(Ref: General Statistics by Chase/Bown, 4th Ed.)
A union official wanted to estimate the mean hourly wage  of its
members. A random sample of 100 members gave x = $18.30 and
s = $3.25 per hour.
(a)
Find an 80% confidence interval for  .
(b)
Find a 95% confidence interval for  .
(c)
If you were to construct a 90% confidence interval for 
(do not construct it), would the interval be longer or shorter
than the 80% confidence interval? Longer or shorter than the
95% confidence interval?
Example 3:
(Ref: General Statistics by Chase/Bown, 4th Ed.)
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):
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.
IV.
Determining the Sample Size for 
Maximum Error of Estimate for 
E  z / 2 
Solve for n

n

n  z / 2 
E
2
 z  
n    /2 
 E 
Round the answer up to obtain a whole number.
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.
Example 2:
(Ref: General Statistics by Chase/Bown, 4th Ed.)
Consider a population with unknown mean  and population standard
deviation  = 15.
(a) How large a sample size is needed to estimate  to within five units
with 95% confidence?
(b) Suppose you wanted to estimate  to within five units with 90%
confidence. Without calculating, would the sample size required
be larger or smaller than the one found in part (a)?
(c) Suppose you wanted to estimate  to within six units with 95%
confidence. Without calculating, would the sample size required
be larger or smaller than the one found in part (a)?
Section 7-3
I.
Confidence interval for  when  is unknown and n is small
When estimating  by x from a small sample, the maximum error of the estimate, with
level of confidence 1- , is
Maximum Error for 
E  t / 2 
Confidence Interval for
xE
s
n
t - table
-- bell shape with thick tails
-- t value depends on the degree of freedom ( df = n -1)
n=5
n=10
Example 1:
Find t / 2 with the following information.
(a) Level of confidence is 98% with n = 19
(b) Level of confidence is 90% with n = 25
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.
Example 3:
A random sample of 20 parking meters in a large municipality
showed the following incomes for a day.
$2.60
$2.00
$2.10
$1.05
$2.40
$1.75
$2.45
$2.35
$1.00
$2.90
$2.40
$2.75
$1.30
$1.95
$1.80
$3.10
$2.80
$1.95
$2.35
$2.50
Assume the population is approximately normal. Find the 95%
confidence interval of the true mean.
Section 7 - 4 Inference Concerning a Population Proportion
I.
Confidence Interval for p
When estimating p by pˆ 
x
, the maximum error of the estimate with confidence 1- is
n
E  z / 2 
ˆˆ
pq
n
where qˆ  1  pˆ
n will be sufficiently large if both x and n–x are at least 5.
Confidence Interval for p
p̂  E
II.
Determining the Sample Size for p
2
z 
n    / 2   pˆ  qˆ
 E 
Round the answer up to obtain a whole number.
Since the sample has not yet been obtained, we do not know the value of p̂ and q̂ .
However, it can be shown that regardless of the values of p̂ and q̂ , the value of
p̂  q̂ will never be more than ¼. Therefore, to be on the safe side, we should take
the sample size to be at least
2
2
z 
z  1
n    / 2   =   / 2   (0.25)
 E  4
 E 
Round the answer up to obtain a whole number.
Example 1:
(Ref: General Statistics by Chase/Bown, 4th Ed.)
A city council commissioned a statistician to estimate to proportion
p 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 p ?
(b) What is the maximum error of estimate for p ?
(c) Find a 90% confidence interval for p .
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.
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?
Example 4: How large a sample is necessary to estimate the true proportion of adults
who are overweight to within 2 % with 95% confidence?
Skip Section 7 – 5.