Mean Variance

Sample

Population

n

xx

ni

i i

1

1

)(1

22

n

xxS

ni

i i

N

xNi

i i

1

22

)(

Size

n

N

IME 301

bz 1)( bzP

bz 025.0 975.0025.01)( bzP

)()( bZPb

975.0)96.1()96.1( ZP

b = is a random value = is probability

means

For example:

IME 301

Also:

For example

means

Then from standard normal table: b = 1.96

• Point estimator and Unbiased estimator

• Confidence Interval (CI) for an unknown parameteris an interval that contains a set of plausible values of the parameter. It is associated with a Confidence Level (usually 90% =<CL=< 99%) , which measuresthe probability that the confidence interval actuallycontains the unknown parameter value.

CI = – half width, + half width

An example of half width is:

• CI length increases as the CL increases. • CI length decreases as sample size, n, increases.• Significance level ( = 1 – CL)

IME 301

nZ 2/

_

X_

X

n

StX

n

StX nn )1(,2/)1(,2/

Confidence Interval for Population MeanTwo-sided, t-Interval

Assume a sample of size n is collected. Then sample mean, ,and sample standard deviation, S, is calculated.

The confidence interval is:

IME 301 (new Oct 06)

X

• Interval length is:

• Half-width length is:

• Critical Points are:

and

n

Stwidthhalf n )1(,2/

n

StL n )1(,2/*2

n

StX n )1(,2/

n

StX n )1(,2/

IME 301

n

StX n )1(,

X

Confidence Interval for Population MeanOne-sided, t-Interval

Assume a sample of size n is collected. Then sample mean, ,and sample standard deviation, S, is calculated.

The confidence interval is:

OR

IME 301 new Oct 06

n

StX n )1(,

Hypothesis: Statement about a parameter

Hypothesis testing: decision making procedure about the hypothesis

Null hypothesis: the main hypothesis H0

Alternative hypothesis: not H0 , H1 , HA

Two-sided alternative hypothesis, uses One-sided alternative hypothesis, uses > or <

IME 301

Hypothesis Testing Process:

1. Read statement of the problem carefully (*)

2. Decide on “hypothesis statement”, that is H0 and HA (**)

3. Check for situations such as: normal distribution, central limit theorem, variance known/unknown, …

4. Usually significance level is given (or confidence level)5. Calculate “test statistics” such as: Z0, t0 , ….

6. Calculate “critical limits” such as: 7. Compare “test statistics” with “critical limit”8. Conclude “accept or reject H0”

2/Z )1(,2/ nt

IME 301

IME 301

FACTH0 is true H0 is false

Accept no error Type II H0 error

DecisionReject Type I no error H0 error

=Prob(Type I error) = significance level = P(reject H0 | H0 is true) = Prob(Type II error) =P(accept H0 | H0 is false) (1 - ) = power of the test

The P-value is the smallest level of significance that would lead to rejection of the null hypothesis.

The application of P-values for decision making:

Use test-statistics from hypothesis testing to find P-value. Compare level of significance with P-value.

P-value < 0.01 generally leads to rejection of H0

P-value > 0.1 generally leads to acceptance of H0

0.01 < P-value < 0.1 need to have significance level to make a decision

IME 301 (new Oct 06)

Test of hypothesis on mean, two-sidedNo information on population distribution

Test statistic:

Reject H0 if

or P-value =

00 : H

01 : H

IME 301

s

Xnt

)( 00

)1(,2/00)1(,2/ .... nn ttortt

)(*201 ttP n

Test of hypothesis on mean, one-sidedNo information on population distribution

0: AH

IME 301

)1,( ntt

)( 1 ttP n

s

Xnt

)( 0

00 : H

0: AH

00 : H

)( 1 ttP n

)1,( ntt

Test statistic:

Reject Ho if P-value =

OR

Reject H0 if

Test of hypothesis on mean, two-sided, variance knownpopulation is normal or conditions for central limit theorem holds

Test statistic:

Reject H0 if or, p-value =

0

00 : H

)( 0

0

XnZ

0: AH

2/002/ .... ZZORZZ

IME 301

|)|(*2|)](|1[2 00 ZZ

Test of hypothesis on mean, one-sided, variance knownpopulation is normal or conditions for central limit theorem holds

0: AH

IME 301 and 312

Test statistic:

Reject Ho if P-value =

Or,

Reject H0 if

00 : H

0: AH

00 : H

)( 0

0

XnZ

ZZ 0 ZZ 0

))(1( 0Z )( 0Z

nZ

nZ 2/2/

Type II error, for mean with known variance

0

2

2/

ZZ

n

2

ZZ

n

)( Z

Where

Sample size, for mean with known variance

Where

Two-sided

One-sided

IME 301, Feb. 99