Mr. Mark Anthony Garcia, M.S.
Mathematics Department
De La Salle University
Recall: Parameter
A parameter is a numerical value
associated to a population.
 Some examples are the population
mean 𝜇, population standard deviation
𝜎 , population variance 𝜎 2
and
population proportion 𝑝.

Recall: Statistic
A statistic is a numerical value
associated to a sample.
 Some examples are the sample mean 𝑥,
sample standard deviation 𝑠 , sample
variance 𝑠 2 and sample proportion 𝑝.

Estimation
In this chapter, we are concerned in
estimating the value of the population
parameter using the value of the sample
statistic.
 In particular, we estimate the value of
the population proportion 𝑝 using the
value of the sample proportion 𝑝.

Estimation
Estimation is a procedure by which a
numerical value or values are assigned to
a population parameter based on the
information collected from a sample.
Examples of using Estimation
1.
2.
A political organization may want to
estimate the proportion of voters that
will vote for their presidential candidate.
A TV network may want to estimate the
proportion of TV viewers who watch a
particular primetime show.
Situation: Estimation
Suppose that a political analyst wants to
estimate the percentage or proportion of
voters that will vote for senatorial
candidate A in city X for the upcoming
senatorial elections.
 The population of study of the political
analyst is the set of all voters in city X.
Suppose that the population size is 𝑁 =
1000 voters.

Situation: Estimation
The political analyst used simple
random sampling to obtain a sample of
𝑛 = 50 voters and asked these voters if
they will vote for senatorial candidate A
or not.
 The survey result showed that 34 out of
the 50 voters will vote for senatorial
candidate A.

Situation: Estimation
The result showed that the sample
34
proportion 𝑝 = = 0.68 or 68% of the
50
sample will vote for senatorial candidate
A.
 However, the result is obtained from a
sample of 𝑛 = 50 voters. The sample
that the political analyst obtained is only
one possible sample of 50 voters out of
many different sample of 50 voters.

Situation: Estimation

If the political analyst takes another
sample of 50 voters (common voters
from the previous sample may be
included in the new sample), he will get
a different sample proportion or
percentage.
Situation: Estimation
The problem of the political analyst is to
estimate the value of the population
proportion 𝑝 using the different sample
proportions obtained from different
samples.
 The question is, what percentage of the
𝑁 = 1000 voters in city X will vote for
senatorial candidate A for the upcoming
elections?

Situation: Estimation
To answer the problem of the political
analyst, we use interval estimation of
parameters.
 But before we reveal this topic, let us
differentiate point estimation from
interval estimation.

Point Estimation
Point estimation is the procedure of
assigning a single value to the
population parameter.
 Normally, we use the value of the
sample statistic as point estimate for the
population parameter.
 Disadvantage: We can have a lot of
point estimates since different samples
may be obtained from the population.

Example: Point Estimation
In the situation of the political analyst, we
34
can use the sample proportion 𝑝 = =
50
0.68 or 68% as a point estimate for the
population proportion 𝑝.
Interval Estimation
Interval estimation is the procedure of
assigning multiple values to a population
parameter.
 It is a procedure of obtaining an interval
where
the
population
parameter
coincides.

Interval Estimation
The interval is dependent on the point
estimate, standard error and maximum
error.
 Each interval is constructed with regard
to a given confidence level and is called
a confidence interval.

Confidence Level
The confidence level attached to a
confidence interval is viewed as the
probability that the population parameter
lies inside the confidence interval.
 It is of the form 1 − 𝛼 100% .
 If the confidence level is 95%, then it is
of
the
form
1 − 0.05 100% =
0.95 100% = 95%. This means that
𝛼 = 0.05.

Confidence Level
The value of 𝛼 is viewed as the
probability that the population parameter
is not inside the confidence interval.
 Now, let us interpret the 95% confidence
level using the political analyst situation.

Confidence Level

Suppose that the political analyst
obtained 100 samples of 50 voters each.
Then 95 out of the 100 samples will give
us a sample proportion that is inside the
confidence interval estimate.
Interval Estimation of Proportion
𝑥
𝑛
The sample proportion is given by 𝑝 =
where 𝑥 is the number of successes in
the sample and 𝑛 is the sample size.
 The standard error is given by the

formula 𝑆𝐸 =
𝑝𝑞
𝑛
where 𝑞 = 1 − 𝑝. This
can be obtained from the PhStat output.
Interval Estimation of Proportion
The formula for the confidence interval
estimate for the population proportion 𝑝
is sample proportion ± maximum error.
 We use minus for the lower bound and
plus for the upper bound.

Interval Estimation of Proportion
Interval Estimation of Proportion

Thus, we are 95% confident that the
proportion of voters that will vote for
senatorial candidate A is between the
interval [0.5507, 0.8093] or between
55.07% and 80.93%.
Example 1: Interval Estimation
In a random sample of 500 people eating
lunch at a hospital cafeteria on various
Fridays, it was found that 160 preferred
seafood.
A. Find a 95% confidence interval for the
actual proportion of people who prefer
seafood on Fridays at this cafeteria.
B.
How large a sample is required if we
want to be 95% confident that our
estimate is within 0.02?
Example 1: Interval Estimation
A.

Find a 95% confidence interval for the
actual proportion of people who prefer
seafood on Fridays at this cafeteria.
Given: 𝑝 =
is 95%.
160
500
= 0.32, confidence level
Example 1: Interval Estimation
Data
Sample Size
500
Number of Successes
160
Confidence Level
95%
|𝒛𝜶 | = 𝟏. 𝟗𝟔
𝟐
Intermediate Calculations
Sample Proportion
Z Value
0.32
-1.959963985
Standard Error of the Proportion
0.020861448
Interval Half Width
0.040887686
Confidence Interval
Maximum error
Interval Lower Limit
0.279112314
Interval Upper Limit
0.360887686
Example 1: Interval Estimation
We are 95% confident that the proportion
of people eating lunch at a hospital
cafeteria on various Fridays who prefer
seafood is between [0.2791,0.3609] or
between 27.91% and 36.09%
Sample Size Determination
If the sample proportion 𝑝 is used as an
estimate for 𝑝, then we can be (1-α)100%
confident that the error will not exceed a
specified maximum error 𝑒 when the
sample size is
𝑛=
𝑧𝛼
2
𝑒
𝑝𝑞
2
Example 1: Interval Estimation
B.
o
o
How large a sample is required if we
want to be 95% confident that our
estimate is within 0.02?
The maximum error 𝑒 = 0.02 is given.
The problem is interpreted as the
number of samples to be considered
so that the error will not exceed 0.02.
Example 1: Interval Estimation
Data
Estimate of True Proportion
0.32
Sampling Error
0.02
Confidence Level
95%
Intermediate Calculations
Z Value
-1.95996398
Calculated Sample Size
2089.753598
Result
Sample Size Needed
2090
Example 2: Interval Estimation
In a random sample of 1000 homes in a
certain city it is found that 228 are heated
by oil. Find a 99% confidence interval for
the proportion of homes in this city that are
heated by oil.
Example 2: Interval Estimation
Data
Sample Size
1000
Number of Successes
228
Confidence Level
99%
Intermediate Calculations
Sample Proportion
Z Value
0.228
-2.5758293
Standard Error of the Proportion
0.013267102
Interval Half Width
0.034173791
Example 2: Interval Estimation
The lower and upper bounds are not
given.
 Thus, we use the formula sample
proportion ± maximum error.
 In the output, the maximum error is the
interval half width.
 Hence, we have sample proportion ±
interval half width.

Example 2: Interval Estimation

Thus, we get 0.228 ± 0.034173791.
 Therefore,
we are 99% confident that
the proportion of all homes in that city
that are heated by oil is between
[0.1938,0.2622] or between 19.38%
or 26.22%