Estimation Problems: Point Estimates and Confidence Intervals, Slides of Data Analysis & Statistical Methods

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8: Estimation

Problems

EM 7: Engineering Data Analysis

Second Semester, 2019-

Pamantasan ng Lungsod ng Valenzuela

Introduction

• Previously, sampling properties are based on

the sample mean and variances. This is to

build a foundation that allows us to draw

conclusions about the population parameters

from experimental data.

• For example, the CLT provides information

about the distribution of the sample mean 𝑋

and the population mean 𝜇. Similar comments

apply to 𝑆

ଶ

and 𝜎

ଶ

(chi-squared distributions).

2

Introduction

• Statistical inference may be divided into two

major areas: estimation and tests of

hypotheses.

• Estimation requires to establish a knowledge of the

sampling distribution of a proportion.

• Test of hypotheses do not need such, there is no

need to estimate a parameter just need to try to

arrive at a correct decision about a pre-stated

hypothesis.

4

Classical Methods of

Estimation

5

Classical Methods of

Estimation

• An estimator is not expected to estimate the

population parameter without error, but we

certainly hope that the estimate is not far off.

• For example, we do not expect 𝑋

to estimate 𝜇

exactly, it is possible to obtain a closer

estimate of 𝜇 by using the sample median 𝑋

as

an estimator. However, not knowing the true

value of 𝜇, we must decide in advance whether

to use 𝑋

or 𝑋

as the estimator.

7

Unbiased Estimator

Let Θ

be an estimator whose value 𝜃

is a point

estimate of some unknown population

parameter 𝜃. Certainly, we would like the

sampling distribution of Θ

to have a mean equal

to the parameter estimated. An estimator

possessing this property is said to be unbiased.

8

DEFINITION: A statistic Θ ෡ is said to be an unbiased estimator of the

parameter 𝜃 if

𝜇 ஀ ෡ = 𝐸 Θ ෡ = 𝜃

10

Sampling distributions of different estimators of 𝜃.

Interval Estimation

Even the most efficient unbiased estimator is

unlikely to estimate the population parameter

exactly. In situations in which it is preferable to

determine an interval within which we would

expect to find the value of the parameter, the

interval estimate is used.

11

DEFINITION: An interval estimate of the population parameter 𝜃 is an

interval of the form,

𝜃

መ ௅ < 𝜃 < 𝜃

መ ௎

where 𝜃 መ ௅

and 𝜃 መ ௎

are bounds of the interval estimate that depends on the

value of the statistic Θ ෡ for a particular sample and also on the sampling

distribution of Θ

෡ .

Interval Estimation

The interval Θ

௅

௎

is called a 100

𝛼)% confidence interval, the fraction 1 − 𝛼 is

called the confidence coefficient or the degree

of confidence. The endpoints Θ

௅

and Θ

௎

are

called the lower and upper confidence limits,

respectively.

13

Interval Estimation

When 𝛼 = 0.05, we have a 95% confidence

interval. When 𝛼 = 0.01, we have a wider 99%

confidence interval. The wider the confidence

interval is, the more confident we are that the

interval contains the unknown parameter.

Ideally, we prefer a short interval with a high

degree of confidence. Sometimes, restrictions

on the size of our sample prevent us from

achieving short intervals without sacrificing some

degree of confidence.

14

Single Sample: Estimating the

Mean

The sampling distribution of 𝑋

is centered at 𝜇,

and in most applications the variance is smaller

than that of any other estimators of 𝜇. Thus, the

sample mean 𝑥̅ will be used as a point estimate

for the population mean 𝜇.

Considering the interval estimate of 𝜇. If our

sample is selected from a normal population or,

failing this, if 𝑛 is sufficiently large, we can

establish a confidence interval for 𝜇 by

considering the sampling distribution of 𝑋

16

Single Sample: Estimating the

Mean

According to CLT, we can expect the sampling

distribution of 𝑋

to be approximately normally

distributed with mean 𝜇 ௑

ത = 𝜇 and standard

deviation 𝜎 ௑

ത

17

𝑃 −𝑧 ఈ/ଶ <

𝑋 ത − 𝜇

𝜎/√𝑛

< 𝑧 ఈ/ଶ = 1 − 𝛼

Example

The average zinc concentration recovered from

a sample of measurements taken in 36 different

locations in a river is found to be 2.6 grams per

milliliter. Find the 95% and 99% confidence

intervals for the mean zinc concentration in the

river. Assume that the population standard

deviation is 0.3 gram per milliliter.

19

Point estimate of mu is x-bar = 2.60.

z-value with an area of 0.025 to the right (0.975 to the left) is z0.025 = 1.96 (z-table)

Hence, 95% CI is 2.6 – 1.96(0.3)/sqrt(36) < mu < 2.6 + 1.96(0.3)/sqrt(36)

which reduces to 2.50 < mu < 2.

For 99% CI, z0.005 = 2.575, hence 2.47 < mu < 2.

Single Sample: Estimating the

Mean

When solving for the sample size n, we round all

fractional values up to the next whole number.

This is to be sure that the degree of confidence

never falls below 100 1 − 𝛼 %.

20

THEOREM: If 𝑥̅ is used as an estimate of 𝜇, we can be 100 1 − 𝛼 %

confident that the error will not exceed 𝑧 ఈ/ଶ

ఙ

௡

.

THEOREM: If 𝑥̅ is used as an estimate of 𝜇, we can be 100 1 − 𝛼 %

confident that the error will not exceed a specified amount 𝑒 when the

sample size is,

𝑛 =

𝑧 ఈ/ଶ

𝜎

𝑒

ଶ