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Math 011 – CHAPTER 7 Estimates and Sample Sizes
7-2 Estimating a Population Proportion
Notation:
GOALS:
DEFINITION
A point estimate
A Confidence Interval (CI)
A confidence level
Example 1: (Interpreting a Confidence Interval) – Based on the sample data of 1007 adults
polled, with 85% of them knowing what Twitter is, we can find a confidence interval:
The 95% confidence interval estimate of the population proportion p is 0.828 < p < 0.872
Correct Interpretation:
Incorrect Interpretation(s):
Note:
Example 2 (Finding a critical value zα/2) Find the critical value zα/2 corresponding to a 90%
confidence level.
Common Critical Values
Level
α
zα/2
DEFINITION
Confidence Interval for Estimating a Population Proportion p
Requirements:
1. The sample is a simple random sample.
2. The conditions for the binomial distribution are satisfied.
3. There are at least 5 successes and at least 5 failures.
Confidence Interval:
ˆ E  p  p
ˆ  E where
p
Also expressed as
or
E  z
2
ˆ ˆ
pq
n
ˆ
(
p
p̂
Round the confidence interval limits for p to three significant digits.
Example (Constructing a Confidence Interval) – In a poll of 1007 randomly selected US adults,
85% of the respondents know what Twitter is.
a. Find the margin of error that corresponds to a 95% confidence level.
b. Find the 95% confidence interval estimate of the population proportion p.
c. Based on the results, can we safely conclude that more than 75% of adults know what
Twitter is?
d. Write a brief summary that accurately describes the results and include all the relevant
information.
When analyzing polls:
Finding the Sample Size n Required to Estimate a Population Proportion
Requirement: The sample must be a simple random sample of independent sample units.
When an estimate is p known:
When no estimate is p known:
Rounding: If the computed sample size n is not a whole number, round the value UP to the
next larger whole number.
Example 4 (Computing a Sample Size) – Many companies are interested in knowing the
percentage of adults who buy clothing online. How many adults must be surveyed in order to
be 95% confident that the sample percentage is in error by no more than three percentage
points?
a. Use the recent result from the Census Bureau: 66% of adults buy clothing online.
b. Assume that we have no point estimate.
Caution:
Finding the Point Estimate P and Error E from a Confidence Interval
Example 5: Find the point estimate and margin of error of the statement: “95% confident that
the population proportion is from 60% to 85%”.
7-3 Estimating a Population Mean
Confidence Interval for Estimating a Population Mean µ with σ Not Known
Requirements:
1. The sample is a simple random sample.
2. Either or both of these conditions is satisfied.
a. The population must be normally distributed
b. n > 30
Confidence Interval:
x  E    x  E where
(Use df = n – 1)
where tα/2 = critical value separating an area of α/2 in the right tail of the student t
distribution, and df = the number of degrees of freedom is the sample size minus 1.
Also expressed as X ±𝐸 or
Round-off Rules:
Important Properties:
E
Example 1: (Finding a Critical t Value)
a. A sample of size 40 is a simple random sample obtained from a normally distributed
population. Find the critical value tα/2 corresponding to a 95% confidence level.
b. A sample size of 25 is a simple random sample obtained from a normally distributed
population. Find the critical value tα/2 corresponding to a 90% confidence level.
Example 2 (Constructing a Confidence Interval) – In a test of weight loss programs, 40 adults
used the Atkins weight loss program. After 12 months, their mean weight loss was found to be
2.1lbs, with a standard deviation of 4.8lbs. Construct a 90% confidence interval estimate of the
mean weight loss for all such subjects assuming the population is normally distributed.
Finding a Point Estimate and Margin of Error E from a Confidence Interval
Finding the Sample Size n Required to Estimate a Population Mean
Requirement: The sample must be a simple random sample.
Use Formula 7-4:
Rounding: If the computed sample size n is not a whole number, round the value UP to the
next larger whole number.
Confidence Interval for Estimating a Population Mean µ with σ Known
Requirements:
1. The sample is a simple random sample.
2. Either or both of these conditions is satisfied:
a.
b.
Confidence Interval:
x  E    x  E where E  z 2 

n
where zα/2 = critical z score separating an area of α/2 in the right tail of the student normal
distribution.
Also expressed as X ±𝐸 or
Example 3 (Confidence Interval Estimate of µ with known σ) – Twelve highway speeds were
measured from southbound traffic on 1-280 near Cupertino, California. The simple random
sample has mean of 60.7mi/h. Construct a 95% confidence interval estimate of the population
mean by assuming that σ is known to be 4.1.
Construct the confidence interval.
Section 7-4 Estimating a Population Standard deviation or Variance
n  1 s 2

 
2
2
Chi-Square Distribution:

Properties of the Chi-Square Distribution:
1.
2.
3.
Note: In table A-4, each critical value of χ2 in the body of the table corresponds to an area given
at the top row of the table, and each area is a cumulative area to the RIGHT of the critical value.
Example 1: (Finding Critical Values of χ2) – A simple random sample of 22 IQ scores is obtained.
Find the left and right critical values corresponding to a confidence level of 95%.
Confidence Interval for Estimating a Population Standard Deviation or Variance
Requirements:
1. The sample is a simple random sample.
2. The population must have normally distributed values (even if the sample is large).
n  1s
Confidence Interval for σ2:
2
2
R
n  1s
Confidence Interval for σ:
Round-Off Rules:

2
R
2
 2 
 
n  1s
2
 L2
n  1s

2
2
L
Example 2 (Constructing a Confidence Interval) – IQ scores for subject in three different lead
exposure groups were recorded. The 22 full IQ scores for the group with medium exposure to
lead have a standard deviation of 14.3. Consider the sample to be a simple random sample and
construct a 95% confidence interval estimate of σ.
Determining Sample Size: Refer to Table 7-2 to determine the sample size depending on the
desired margin of error and level of confidence.
σ2
σ
To be 95% confident
that s is within
1%
5%
10%
20%
30%
To be 99% confident
that s is within
1%
5%
10%
20%
30%
Of the value σ, the
sample size n should
be at last
19,205
768
192
48
21
Of the value σ, the
sample size n should
be at last
33,218
1,336
336
85
38
To be 95% confident
that s2 is within
1%
5%
10%
20%
30%
To be 99% confident
that s2 is within
1%
5%
10%
20%
30%
Of the value σ2, the
sample size n should
be at last
77,208
3,149
806
211
98
Of the value σ2, the
sample size n should
be at last
133,449
5,458
1,402
369
172
Example: We want to estimate the standard deviation of σ all IQ scores of people with exposure
to lead. How large should be sample if:
a. we want to be 95% confident that our estimate is within 10% of the true value of σ?
b. we want to be 95% confident that our estimate is within 5% of the true value of σ?
c. we want to be 99% confident that our estimate is within 1% of the true value of σ?
Example: Consider the following weights of post-1983 pennies:
3.1582, 3.0406, 3.0762, 3.0398, 3.1043, 3.1274, 3.0775,
3.1038, 3.0586, 3.0603, 3.0502, 3.1028, 3.0522
Assume the sample is a simple random sample obtained from a population with a normal
distribution. Construct a 98% confidence interval estimate of the standard deviation of the
weights of all post-1983 pennies.