Download 95% confidence interval.

Lecture Notes
Chapter 7: Inferences Based on a Single Sample: Estimation with
Confidence Intervals
A survey by the Roper Organization found that
45% of the people who were offended by a
television program would change the channel,
while 15% would turn off their television sets.
The survey further stated that the margin of error
is 3 percentage points and 4000 adults were
introduced.
1. How do these estimates compare with the true
population percentages?
2. Is the sample of 4000 large enough to
represent the population of all adults who
watch television in the United States.

1.
2.
3.
4.
5.
Find the confidence interval for the mean
when σ is known or n  30.
Determine the minimum sample size for
finding a confidence interval of the mean
Find the confidence interval for the mean
when σ is unknown and n< 30.
Find the confidence interval for a proportion
Determine the minimum sample size for
finding a confidence interval of the
proportion

Target Parameter
Key words associated with parameters:
Parameter
Key Words or Phrases

Mean; average
p
Proportion; percentage; fraction; rate

Recall that a statistic such as the sample
mean is a point estimator of the population
mean .
The goal is to determine how to estimate the
population mean and assess the estimate’s
reliability.
 Example: Find
such that
.
Solution:





According to the Central Limit Theorem, the
distribution of
will be approximately a normal and
95% of all
from a sample of size n lie within
of the mean.
There is a probability of .95 that  will lie in the
interval
.
The interval
is called a large-sample 95%
confidence interval for the population mean .
Large sample means the sample must be large
enough so that the Central Limit Theorem can be
applied. (Rule of thumb, n ≥ 30)




Calculating
knowing ,
. We usually don’t
know , but with a large sample s is a good estimator of .
The interval
being called the 95% confidence
interval for the population mean  means that if a large
number of samples were taken and this interval calculated
each time, 95% would contain .
The probability, .95, that measures the confidence we can
place in the interval estimate is called a confidence
coefficient. The percentage, 95%, is called the confidence
level for the interval estimate.
The Margin of error=
(This is the maximum
error of estimate for a 95% confidence interval)


Confidence coefficient – probability that a
randomly selected confidence interval
encloses the population parameter
Confidence level – Confidence coefficient
expressed as a percentage

The confidence coefficient is equal to 1- , where 
is called the significance level and it is the amount
of area assigned to the tails of the sampling
distribution, and is split between the two tails of
the distribution.

John says he is 90% confident that the
population mean is contained within the
interval I when the values of the population
are normally distributed.
A publishing company has just published has just a
new textbook. Before the company decides the price
at which to sell this textbook, it wants to know the
average price of all such textbooks in the market. The
research department at the company took a sample
of 36 comparable textbooks and collected
information on their prices. This information
produces a mean of $70.50 for this sample. It is
known that the standard deviation of the prices of all
such textbooks is $4.50.
A) What is the point estimate of the mean price of all
such college textbooks?
B) What is the margin error of this estimate?
C) Construct the 90% confidence interval for the mean
price of all such college textbooks.
Conditions required for a Valid Large-Sample Confidence Interval for 
1.
A random sample is selected from the target population.
2.
The sample size n is large, n  30.
 p. 306 # 11
A random sample of 100 observations from a
normally distributed population possesses a mean
equal to 83.2 and standard deviation equal to 6.4.
a. Find a 95% confidence interval for μ.
b. What do you mean when you say that a confidence
coefficient is .95?
c. Find a 99% confidence interval for μ.
d. What happens to the width of a confidence interval
as the value of the confidence coefficient is
increased while the sample size is held fixed?
e. Would your confidence intervals of parts a and c be
valid if the distribution of the original population
were not normal? Explain.


Many times, inferences must be made from small
samples. But The Central Limit Theorem does
not guarantee that sampling distribution of will
be normal for small sample sizes. The sampling
distribution of
will be normal if the population
is normal.
Also, the population standard deviation may not
be known and the sample standard deviation s
may not provide a good approximation for .
Instead of using the statistic
, which
requires a good approximation of , the statistic
is used.


The t-statistic is very much like the zstatistic. It is mound shaped, symmetric, and
has mean 0. The t-statistic is different in
that it has two random quantities ( and s)
while the z-statistic only has one ( ).
The variability of t depends on the sample
size n. Variability is expressed as (n-1)
degrees of freedom (df). As df gets smaller,
variability increases.


Table for t-distribution contains t-value for various combinations of
degrees of freedom and tα. tα is the point where the upper tail of
the t-distribution contains an area of .
The last row, where df = , contains the standard normal z-values.
Conditions required for a Valid Large-Sample Confidence Interval for 
1.
A random sample is selected from the target population.
2.
The population has a relative frequency distribution that
is approximately normal.

The data below represent a sample of the
number of homes fires started by candles for
the past several years. Find the 99%
confidence interval for the mean of homes
started by candle each year.
5460
5900
6090
6310
7160
8440
9930
 p. 317 # 33ab,39
The following random sample was selected from a normal
distribution: 4, 6, 3, 5, 9, 3.
a. Construct a 90% confidence interval for the population
mean μ.
b. Construct a 95% confidence interval for the population
mean μ.
c. Construct a 99% confidence interval for the population
mean μ.
d. Assume that the sample mean and sample standard
deviation s remain exactly the same as those you just
calculated, but that they are based on a sample of n = 25
observations rather than n = 6 observations. Repeat
parts a-c . What is the effect of increasing the sample
size on the width of the confidence intervals?
Example: p. 318 #39
Radioactive lichen. Refer to the Lichen
Radionuclide Baseline Research project at the
University of Alaska, presented in Exercise 2.36 (p.
47). Recall that the researchers collected 9 lichen
specimens and measured the amount (in
microcuries per milliliter) of the radioactive
element cesium-137 for each. (The natural
logarithms of the data values are saved in the
LICHEN file.) A MINITAB printout with summary
statistics for the actual data is shown below.
Variable
N
Mean
StDev
SE Mean
CESIUM
9
0.009027
0.004854
0.001618
95% CI
(0.005296, 0.012759)
a. Give a point estimate for the mean amount of
cesium in lichen specimens collected in Alaska.
b. Give the t-value used in a small-sample 95%
confidence interval for the true mean amount of
cesium in Alaskan lichen specimens.
c. Use the result you obtained in part b and the
values of and s shown on the MINITAB printout
to form a 95% confidence interval for the true
mean amount of cesium in Alaskan lichen
specimens.
d. Check the interval you found in part c with the
95% confidence interval shown on the MINITAB
printout.
e. Give a practical interpretation for the interval
you obtained in part c .
 Confidence
intervals around a proportion are
confidence intervals around the probability of
success in a binomial experiment.
 Sample statistic of interest is , where .
Conditions required for a Valid Large-Sample Confidence Interval of p
1.
2.
A random sample is selected from the target population.
The sample size n is large. (A sample size is considered
large if both
and
.
and
are the number
of success and the number of failures in the sample.)

A sample of 500 nursing applications
included 60 from men. Find the 90%
confidence interval of the true proportion of
men who applied to the nursing program.
 p. 325 # 51,55
Example: p. 325#51
A random sample of size n = 196 yielded = .64.
a. Is the sample size large enough to use the
methods of this section to construct a
confidence interval for p? Explain.
b. Construct a 95% confidence interval for p.
c. Interpret the 95% confidence interval.
d. Explain what is meant by the phrase “95%
confidence interval.”
Example: p. 325#55
a.
b.
c.
d.
Is Starbucks coffee overpriced? The Minneapolis
Star Tribune (August 12, 2008) reported that 73%
of Americans say that Starbucks coffee is
overpriced. The source of this information was a
national telephone survey of 1,000 American
adults conducted by Rasmussen Reports.
Identify the population of interest in this study.
Identify the sample for the study.
Identify the parameter of interest in the study.
Find and interpret a 95% confidence interval for
the parameter of interest.
A. Large-Sample Confidence Interval for a Population Mean
For random samples of size  30, the confidence interval is expressed as
B.
Small-Sample Confidence Interval for a Population Mean
The small sample confidence interval will be
Where
is based on (n – 1) degrees of freedom.
C. Large-Sample Confidence Interval for a Population Proportion
Large-Sample Confidence Interval for p
Where

and
.
A sample size is considered large if both
and
.
A. Estimating a Population Mean
 The width of a confidence interval depends on the sample size:


As the sample size increases, the width of the interval decreases for any
given confidence coefficient.
When we want to estimate  to within a given number of units with a
(1- ) level of confidence, we can calculate the sample size needed by
solving the equation
= interval width for n.
The reliability associated with a confidence interval for the population
mean is expressed using the sampling error within which we want to
estimate  with 100(1- )% confidence.
Example: p. 332#71
If you wish to estimate a population mean to
within .2 with a 95% confidence interval and you
know from previous sampling that σ2 is
approximately equal to 5.4, how many
observations would you have to include in your
sample?

The Sampling Error (SE) is half the width of
the confidence interval.
#71 p.
#78, 81 p.
Example: p. 332#81
Scanning errors at Wal-Mart. Refer to the National
Institute for Standards and Technology (NIST)
study of the accuracy of checkout scanners at WalMart stores in California, presented in Exercise
3.52 (p. 132). NIST sets standards so that no
more than 2 of every 100 items scanned through
an electronic checkout scanner can have an
inaccurate price. Recall that in a sample of 60
Wal-Mart stores, 52 violated the NIST scanner
accuracy standard (Tampa Tribune, Nov. 22,
2005). Suppose you want to estimate the true
proportion of Wal-Mart stores in California that
violate the NIST standard.
a. Explain why the large-sample methodology of
Section 7.4 is inappropriate for this study.
b. Determine the number of Wal-Mart stores that
must be sampled in order to estimate the true
proportion to within .05 with 90% confidence,
using the large-sample method.