Please Subscribe to this Channel for more solutions and lectures http://www.youtube.com/onlineteaching Chapter 7: Estimating Parameters and Determining Sample Sizes 7.1: Estimating a Population Proportion

1. Elementary Statistics
Chapter 7: Estimating
Parameters and
Determining Sample
Sizes
7.1 Estimating a
Population Proportion
1

2. Chapter 7:
Estimating Parameters and Determining Sample Sizes
7.1 Estimating a Population Proportion
7.2 Estimating a Population Mean
7.3 Estimating a Population Standard Deviation or Variance
7.4 Bootstrapping: Using Technology for Estimates
2
Objectives:
• Find the confidence interval for a proportion.
• Determine the minimum sample size for finding a confidence interval for a proportion.
• Find the confidence interval for the mean when  is known.
• Determine the minimum sample size for finding a confidence interval for the mean.
• Find the confidence interval for the mean when  is unknown.
• Find a confidence interval for a variance and a standard deviation.

3. Margin of Error & Confidence Interval for Estimating a Population Proportion p &
Determining Sample Size:
When data from a simple random sample are used to estimate a population proportion p, the
Margin of Error (Maximum Error of the Estimate) (Error Bound), denoted by E, is the
maximum likely difference (with probability 1 – α, such as 0.95) between the observed
(sample) proportion 𝑝 and the true value of the population proportion p.
3
𝑝 ± 𝐸
𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸 𝑝 − 𝑧𝛼
2
𝑝𝑞
𝑛
< 𝑝 < 𝑝 + 𝑧𝛼
2
𝑝𝑞
𝑛
7.1 Estimating a Population Proportion Summary
𝜎𝑝 =
𝑝𝑞
𝑛
→ 𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛
𝑛 =
(𝑍𝛼/2)2𝑝𝑞
𝐸2
𝑛 =
(𝑍𝛼/2)20.25
𝐸2
When no estimate of 𝒑 is
known: 𝒑 = 𝒒 =0.5
Point estimate of p: 𝑝 =
𝑈𝐶𝐿+𝐿𝐶𝐿
2
, UCL: Upper Confidence Limit
Margin of error: 𝐸 =
𝑈𝐶𝐿−𝐿𝐶𝐿
2
, LCL: Lower Confidence Limi

4. Key Concept: Use a Sample proportion to make an inference about the value of the corresponding
population proportion.
A point estimate is a specific numerical value estimate of a parameter. (A point estimate is a single value (or
point) used to approximate a population parameter. )
Properties of a good estimator:
1. 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.
2. 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.
3. 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.
• The sample proportion 𝑝 is the best point estimate of the population proportion p.
• p =
𝑋
𝑁
= population proportion 𝑝 =
𝑥
𝑛
(read p “hat”) = sample proportion, 𝑞 = 1 − 𝑝
x number of sample units that possess the characteristics of interest and n = sample size.
X number of population units that possess the characteristics of interest and N = population size.
• Confidence Interval: We can use a sample proportion to construct a confidence interval estimate of the true
value of a population proportion.
• Sample Size: We should know how to find the sample size necessary to estimate a population proportion.
7.1 Estimating a Population Proportion
4

5. A survey of 1500 adults indicated that 48% of them have a Facebook page.
Based on that result, find the best point estimate of the proportion of all adults
who have a Facebook page.
Example 1
Solution
Point estimate (PE) of p = 𝑝 = 0.48
Example 2: In a recent survey of 200 households, 60 had central air conditioning.
Find 𝑝 and 𝑞 , where 𝑝 is the proportion of households that have central air
conditioning. x = 60 and n = 200
𝑝 =
𝑥
𝑛
=
60
200
= 0.3 = 30%
𝑞 = 1 − 𝑝 = 1 − 0.3 = 0.7 = 70%
5
𝑝 =
𝑥
𝑛
, 𝑞 = 1 − 𝑝

6. Confidence Interval: A confidence interval (or interval estimate) is a range (or
an interval) of values used to estimate the true value of a population parameter. A
confidence interval is sometimes abbreviated as CI. This estimate may or may not
contain the value of the parameter being estimated.
Confidence Level: The confidence level is the probability 1 − α (such as 0.95, or
95%) that the confidence interval actually does contain the population parameter,
assuming that the estimation process is repeated a large number of times. (The
confidence level is also called the degree of confidence, or the confidence
coefficient.) The confidence level of 95% is the value used most often.
7.1 Estimating a Population Proportion
Most Common Confidence Levels Corresponding Values of α
90% (or 0.90) confidence level: α = 0.10
95% (or 0.95) confidence level: α = 0.05
99% (or 0.99) confidence level: α = 0.01
6

7. Interpreting a Confidence Interval
We must be careful to interpret confidence intervals
correctly. There is a correct interpretation and many
different and creative incorrect interpretations of the
confidence interval 0.405 < p < 0.455.
Correct: “We are 95% confident that the interval from
0.405 to 0.455 actually does contain the true value of the
population proportion p.”
Wrong: “There is a 95% chance that the true value of p
will fall between 0.405 and 0.455.”
Wrong: “95% of sample proportions will fall between
0.405 and 0.455.”
The Process Success Rate: A confidence level of
95% tells us that the process we are using should, in the
long run, result in confidence interval limits that contain
the true population proportion 95% of the time.
7.1 Estimating a Population Proportion
Confidence
Interval
from 20
Different
Samples
7
Confidence intervals can be used
informally to compare different
data sets, but the overlapping of
confidence intervals should not
be used for making formal and
final conclusions about equality
of proportions.

8. A critical value is the number on the borderline separating sample statistics that are
significantly high or low from those that are not significant. The number zα/2 is a
critical value that is a z score with the property that it separates an area of α/2 in the
right tail of the standard normal distribution.
A standard z score can be used to distinguish between sample statistics that are likely
to occur and those that are unlikely to occur. Such a z score is called a critical value.
Critical values are based on the following observations:
8
7.1 Estimating a Population Proportion, Critical Values
1. Under certain conditions, the sampling distribution of sample
proportions can be approximated by a normal distribution.
2. A z score associated with a sample proportion has a
probability of α/2 of falling in the right tail.
3. The z score separating the right-tail region is commonly
denoted by zα/2 and is referred to as a critical value because it
is on the borderline separating z scores from sample
proportions that are likely to occur from those that are
unlikely to occur.

9. Finding zα/2 for a 95% Confidence Level
Critical Values
This is the most
common critical value,
and it is listed with two
other common values in
the table that follows.

10. Margin of Error & Confidence Interval for Estimating a Population Proportion p:
When data from a simple random sample are used to estimate a population proportion p, the margin of error (also called the maximum
error of the estimate ), denoted by E, is the maximum likely difference (with probability 1 – α, such as 0.95) between the observed
(sample) proportion 𝑝 and the true value of the population proportion p.
𝑝 = sample proportion
n = number of sample values
zα/2 = Critical value, z score separating an area of α/2 in the right tail of the standard normal distribution.
Standard deviation of sample proportions: 𝜎𝑝 =
𝑝𝑞
𝑛
→ In practice, we don’t’ know the value of 𝒑, so we substitute 𝒑 → 𝜎𝑝 =
𝑝𝑞
𝑛
E = margin of error
1. The sample is a simple random sample.
2. The conditions for the binomial distribution are satisfied: There is a fixed number of trials, the trials are independent, there
are two categories of outcomes, and the probabilities remain constant for each trial.
3. There are at least 5 successes and at least 5 failures (np ≥ 5, and nq ≥ 5 ).
10
7.1 Estimating a Population Proportion
𝑝 ± 𝐸
𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸 𝑝 − 𝑧𝛼
2
𝑝𝑞
𝑛
< 𝑝 < 𝑝 + 𝑧𝛼
2
𝑝𝑞
𝑛
𝜎𝑝 =
𝑝𝑞
𝑛
→ 𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛

11. Assume that a poll of 1487 adults showed that 43% of the respondents have Facebook pages.
a. Find the margin of error E 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 fewer than 50% of adults have Facebook pages?
Assuming that you are a newspaper reporter, write a brief statement that accurately describes the
results and includes all of the relevant information.
Example 2
Given: Binomial Distribution(BD) n = 1487, 𝑝 = 0.43
11
𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛
𝑏. 𝑝 ± 𝐸 →
c. yes, because the interval of values is an interval that is completely below 0.50.
𝑎. 𝐸 = 1.96
0.43(0.57)
1487
= 0.0252
Based on this poll of 1487 randomly selected adults in the
United States, 43% of adults have Facebook pages.
In theory, in 95% of such polls, the percentage should differ by
no more than 2.52 percentage points in either direction from
the percentage that would be found by interviewing all adults.
𝑝 ± 𝐸 → 𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸
TI Calculator:
Confidence Interval:
proportion
1. Stat
2. Tests
3. 1-prop ZINT
4. Enter: x, n & CL
n𝑝 = (1487)(0.43) = 639 & n𝑞 = (1487)(0.57) = 848 ⇾ n𝑝 ≥ 5 & n𝑞 ≥ 5
0.4048 < p < 0.4552

12. A poll of 1007 randomly selected adults
showed that 85% of respondents know
what Twitter is.
a. Find the margin of error E 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. Assuming that you are a newspaper
reporter, write a brief statement that
accurately describes the results and
includes all of the relevant information.
Example 3
Given: BD: n = 1007, 𝑝 = 0.85
12
𝑎. 𝐸 = 1.96
0.85 0.15
1007 = 0.0220545
𝑝 ± 𝐸 →
b. 0.85 − 0.0220545 < 𝑝 < 0.85 + 0.0220545
0.8279 < 𝑝 < 0.8721
c. Yes, because the interval of values is an interval
that is completely above 0.75.
d. 85% of U.S. adults know what Twitter is. That percentage is
based on this poll of 1007 randomly selected adults.
In theory, in 95% of such polls, the percentage should differ by
no more than 2.21 percentage points in either direction from the
percentage that would be found by interviewing all adults in the
United States.
𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛
𝑝 ± 𝐸 → 𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸
n𝑝 = (1007)(0.85) = 855.951007 & n𝑞 =
(1007)(0.85) = 151.05 ⇾ n𝑝 ≥ 5 & n𝑞 ≥ 5

13. Finding the Point Estimate and E from a Confidence Interval
Point estimate of p: 𝑝 =
𝑈𝐶𝐿+𝐿𝐶𝐿
2
, UCL: Upper Confidence Limit
Margin of error: 𝐸 =
𝑈𝐶𝐿−𝐿𝐶𝐿
2
, LCL: Lower Confidence Limit
13
7.1 Estimating a Population Proportion
Assume 70% of the 71 subjects that had smoke therapy did not smoke after 8
weeks and resulted in a 95% confidence interval [CI] of 58% to 81%. Find the
point estimate p and the margin of error E.
Example 4
Solution: 95% CI: 0.58 < p < 0.81
𝑝 =
0.81 + 0.58
2
= 0.695
𝐸 =
0.81 − 0.58
2
= 0.115

14. Determining Sample Size: Finding the Sample Size Required to Estimate a Population
Proportion: Objective:
Suppose we want to collect sample data in order to estimate some population proportion.
The question is how many sample items must be obtained?
14
7.1 Estimating a Population Proportion
(solve for n by algebra)
When no estimate of 𝒑 is not
known: 𝒑 = 𝒒 =0.5
𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛
𝑛 =
(𝑍𝛼/2)2
𝑝𝑞
𝐸2
𝑛 =
(𝑍𝛼/2)2
0.25
𝐸2

15. 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 a recent result from the Census Bureau: 66% of adults buy clothing online.
b. Assume that we have no prior information suggesting a possible value of the proportion.
Example 5
Given: a. 𝑝 = 0.66, 𝐶𝐿 = 95%, 𝐸 = 0.03
𝑞 = 1 − 𝑝 = 1 − 0.66 = 0.34
15
𝐶𝐿 = 95% → 𝑧𝛼
2
= 1.96
=
(1.96)2(0.66)(0.34)
(0.03)2
= 957.8 → 𝑛 = 958
b. Assume 𝑝 = 𝑞 = 0.5, 𝐶𝐿 = 95%, 𝐸 = 0.03
=
(1.96)20.25
(0.03)2 = 1067.11 → 𝑛 = 1068
𝑛 =
(𝑍𝛼/2)2𝑝𝑞
𝐸2 , 𝑛 =
(𝑍𝛼/2)20.25
𝐸2
𝑛 =
(𝑍𝛼/2)2𝑝𝑞
𝐸2
𝑛 =
(𝑍𝛼/2)20.25
𝐸2

16. A researcher wishes to estimate, with 95% confidence, the proportion of people
who own a home computer. A previous study shows that 40% of those
interviewed had a computer at home. The researcher wishes to be accurate
within 2% of the true proportion. Find the minimum sample size necessary.
Example 6
Given: a. 𝑝 = 0.4, 𝐶𝐿 = 95%, 𝐸 = 0.02
𝑞 = 1 − 𝑝 = 1 − 0.4 = 0.6
16
𝐶𝐿 = 95% → 𝑧𝛼
2
= 1.96
=
(1.96)2
(0.4)(0.6)
(0.02)2
= 2304.96 → 𝑛 = 2305
= 2304.96
𝑛 =
(𝑍𝛼/2)2𝑝𝑞
𝐸2 , 𝑛 =
(𝑍𝛼/2)20.25
𝐸2
𝑛 =
(𝑍𝛼/2)2𝑝𝑞
𝐸2

17. A survey of 1404 respondents found that 323 students paid
for their education by student loans. Find the 90%
confidence of the true proportion of students who paid for
their education by student loans.
Given: Binomial Distribution(BD) n = 1404, 𝑥 = 323, 𝐶𝐿 = 90%
17
𝑝 =
323
1404
= 0.23
𝑝 ± 𝐸 →
0.23 − 0.019 < 𝑝 < 0.23 + 0.019
0.211 < 𝑝 < 0.249
𝐸 = 1.645
0.23 0.77
1404
𝑞 = 1 − 0.23 = 0.77 ⇾ n𝑝 ≥ 5 & n𝑞 ≥ 5
You can be 90%
confident that the
percentage of
students who pay for
their college
education by student
loans is between
21.1% and 24.9%.
Example 7
TI Calculator:
Confidence Interval:
proportion
1. Stat
2. Tests
3. 1-prop ZINT
4. Enter: x, n & CL
= 0.019
𝐸 = 𝑧𝛼
2
𝑝𝑞
𝑛
𝑝 ± 𝐸 → 𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸

18. A survey of 1721 people found that 15.9% of individuals
purchase religious books at a Christian bookstore. Find the
95% confidence interval of the true proportion of people who
purchase their religious books at a Christian bookstore.
18
𝑝 − 𝑧𝛼
2
𝑝𝑞
𝑛
< 𝑝 < 𝑝 + 𝑧𝛼
2
𝑝𝑞
𝑛
0.159 − 1.96
0.159 0.841
1721
< 𝑝 < 0.159 + 1.96
0.159 0.841
1721
0.142 < 𝑝 < 0.176
Given: Binomial Distribution(BD) n = 1721, 𝑝 = 0.159, 𝐶𝐿 = 95%
Example 8 𝐸 = 𝑍𝛼/2
𝑝𝑞
𝑛
𝑝 ± 𝐸 → 𝑝 − 𝐸 < 𝑝 < 𝑝 + 𝐸