Estimation

ESTIMATIONPrepared by:

Paolo Lorenzo Bautista

Estimation

We wish to estimate a characteristic of

the population, by using information from

the sample

Types of Estimates

Point Estimate – single value

Interval Estimate – interval or range

ESTIMATION OF PARAMETERS

One Population

One Mean

Case 1

Case 2

One Proportion

One Variance

Two Populations

Difference of Two

Means

Case 1 to 3

Paired Observations

Difference of Two

Proportions

Ratio of Two

Variances

Estimation for One Mean

Best Point Estimate:

Interval Estimation for μ

CASE 1: σ is known, OR n ≥ 30

The (1-α)100% confidence interval for μ is

given by

MARGIN OF ERROR

Example 1

The mean and standard deviation for the

sample point average of a random

sample of 36 college seniors are

calculated to be 2.6 and 0.3 respectively.

Find a 95% confidence interval for the

grade point average of the entire senior

class.

Determining the Sample Size

If the sample mean is used as an estimate

of µ, we can be (1-α)100% confident that

the error will not exceed E when the

sample size is

How large a sample do we need if we do

not wish our error to exceed 0.05?

Case 2: σ is unknown, AND n < 30

A (1-α)100% confidence interval for μ is

given by

The t-value has n-1 degrees of freedom

MARGIN OF ERROR

Example 2

A random sample of 16 mid-sized cars

tested for fuel consumption gave a mean

of 26.4 miles per gallon with a standard

deviation of 2.3 miles per gallon. Assuming

that the miles per gallon given by all mid-

sized cars have a normal distribution, find

a 99% confidence interval for the

population mean.

Exercise

In an effort to estimate the mean amount

spent per customer for dinner at a major

Atlantis restaurant, data were collected

for a sample of 49 customers.

Assume a population standard deviation of

$5, and a 95% confidence level. What is the

margin of error?

If the sample mean is $34.80, what is the

95% confidence interval for the population

mean?

Exercise

In the testing of a new production

method, 18 employees were selected

randomly and asked to try the new

method. The sample mean production

rate was 80 pph, and the sample s.d. was

10 pph. Provide 90% and 95% confidence

intervals for the population mean

production rate for the new method,

assuming the population has a normal

probability distribution.

Estimation for the Proportion of

Success

Assume a binomial experiment is being

conducted

Estimation for the Proportion of

Success

Best point estimate:

A (1-α)100% confidence interval for p is

given by

MARGIN OF ERROR

Example 3

In a random sample of n = 500 families

owning television sets in Quezon City, it

was found that x = 340 subscribed to

Destiny. Find a 95% confidence interval for

the actual proportion of families in this city

who subscribe to Destiny.

Determining the Sample Size

If we use the sample proportion as an

estimate of p, we can be (1 – α)100%

confident that the error will not exceed E

when the sample size is approximately

How large a sample would be needed if

we do not wish our error to exceed 0.02 at

95% confidence?

Exercise

A random sample of 75 college students is

selected and 16 are found to have cars on

campus.

Use a 95% confidence interval to estimate the

fraction of students who have cars on campus.

How large a sample is needed if we do not wish

our error to exceed 0.075?

In a random sample of 1000 homes in a

certain city, it is found that 628 are heated by

natural gas. Find the 98% confidence interval

for the fraction of homes in this city that are

heated by natural gas.

Estimation for the Variance

Recall:

Estimation for the Variance


A (1-α)100% confidence interval for σ2 is

given by

The chi-squared values have n-1 degrees

of freedom

Example 4

The following are the volumes, in

deciliters, of 10 cans of peaches

distributed by a certain company: 46.4,

46.1, 45.8, 47.0, 46.1, 45.9, 46.9, 45.8, 45.2,

and 46.0. Find a 95% confidence interval

for the variance of all such cans assuming

that volume is normally distributed.

ESTIMATION OF PARAMETERS

One Population

One Mean

Case 1

Case 2

One Proportion

One Variance

Two Populations

Difference of Two

Means

Case 1 to 3

Paired Observations

Difference of Two

Proportions

Ratio of Two

Variances

Estimation for the Difference of

Two Means


Interval Estimation:

Case 1:

A (1-α)100% confidence interval for μ1-μ2 is

given by

MARGIN OF ERROR

Example 5

A standardized chemistry test was given

to 50 girls and 75 boys. The girls made an

average grade of 76 with a standard

deviation of 6, while the boys made an

average grade of 82 with a standard

deviation of 8. Find a 96% confidence

interval for the difference of the two

population means.

Case 2: σ1=σ2 unknown, AND

n1<30 and n2<30


given by

The t-value has

Example 6 A course in statistics is taught to 12 students by the

transmissive teaching method. A second group of 10

students was given the same course by means of the

transformative teaching method. At the end of the

trimester the same examination was given to each

group. The 12 students who used the transmissive

method got an average grade of 85 with a standard

deviation of 4, while the 10 students under the

transformative method had an average grade of 81

with a standard deviation of 5. Find a 90%

confidence interval for the difference between

population means, assuming the populations are

approximately normally distributed with equal

variances.

Case 3: σ1≠σ2 unknown, AND

n1<30 and n2<30


given by

The t-value has degrees of freedom given by:

Example 7

Records for the past 15 years have shown the

average rainfall in a certain region of the

country for the month of May to be 4.93

centimeters, with a standard deviation of 1.14

centimeters. A second region of the country

has had an average rainfall in May of 2.64

centimeters with a standard deviation of 0.66

centimeters during the past 10 years. Find a

95% confidence interval for the difference of

the true average rainfalls in these two regions,

assuming that the observations come from

normal populations with different variances.

Paired Observations

Used when the samples are not selected

independently

The t-value has n-1 degrees of freedom

n represents the no. of pairs

Example 8 Twenty college freshmen were divided into 10

pairs, each member of the pair having approximately the same IQ. One of each pair was selected at random and assigned to a statistics section using a transformative method of teaching. The other member of each pair was assigned to a section in which the professor lectured. At the end of the trimester, each group was given the same examination and the following grades were recorded:

Example 8

Pair Transformative Lecture

1 76 81

2 60 52

3 85 87

4 58 70

5 91 86

6 75 77

7 82 90

8 64 63

9 79 85

10 88 83

Find a 98%

confidence

interval for the

true difference

in the two

learning

procedures.

Estimation for the Difference of

Two Proportions

Best point estimate:

A (1-α)100% confidence interval for p1-p2

is given by

Example 9

A poll is taken among the residents of a

city and the surrounding county to

determine the feasibility of a proposal to

construct a civic center. If 2400 of 5000

city residents favor the proposal and 1200

of 2000 county residents favor it, find a

90% confidence interval for the true

difference in the fractions favoring the

proposal to construct the civic center .

Estimation for the Ratio of Two

Variances

A (1-α)100% confidence interval for the

ratio of two variances is given by

Example 10 A standardized placement test in

mathematics was given to 25 boys and 16 girls. The boys made an average grade of 82, with a standard deviation of 8, while the girls made an average grade of 78 with a standard deviation of 7. Find a 98% confidence interval for σ1

2/σ22, where σ1

2 and σ2

2 are the variances of the populations of grades for all boys and girls, respectively, who at some time have taken or will take this exam. Assume the populations to be normally distributed.

Estimation

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