confidence interval. assignment # 2 briefly describe following with the help of examples: point...

Confidence Interval

Assignment # 2

Briefly describe following with the help of examples: Point Estimation of Parameters Confidence Intervals Students’ t Distribution Chi Square Distribution Testing of Hypothesis

Hand Written assignments be submitted by 20 Dec 2012.

Copying and Late Submissions will have appropriate penalty.

Estimation Process

Mean, , is unknown

Population Random SampleMean X = 50

Sample

I am 95% confident that is between 40 & 60.

Confidence Interval provides more information about a population characteristic than does a point estimate. It provides a confidence level for the estimate.

Point Estimate

Lower Confidence Limit

Width of

Confidence Interval

Upper

Confidence

Limit

Interval Estimates

Provides range of values Takes into consideration variation in

sample statistics from sample to sample Is based on observation from one sample Gives information about closeness to

unknown population parameters Is stated in terms of level of confidence Never 100% certain

Confidence Interval Estimates

Mean

2 Unknown

ConfidenceIntervals

Variance

2 Known

Confidence Interval for µ( б2 Known)

Assumptions Population standard deviation is

known Population is normally distributed If population is not normal, use

large sample Confidence interval estimate

n

ckwherekxkxConf

..)..(

Steps to Determine Confidence Interval

Choose a Confidence Level γ (95%, 90%, etc) Determine the corresponding c [ Z(D) ]

Compute the mean x of the sample x1, x2,…, xn

Confidence interval estimate

n

ckwherekxkxConf

..)..(

γ 0.90 0.95 0.99 0.999

C 1.645 1.960 2.576 3.291

Problem 1

Find a 95% Confidence Interval

for the mean µ of a normal

population with standard

deviation 4.00 from the sample

30, 42, 40, 34, 48, 50

Confidence Interval for µ( б2 Unknown)

Choose a Confidence Level γ (95%, 90%, etc) Determine the solution C of the equation

F(C) = ½ ( 1 + γ)

[Table: t distribution with n-1 degrees of freedom] Compute the mean x and variance s2 of the

sample x1, x2,…, xn


n

cskwherekxkxConf

..)..(

Student’s t Distribution

Zt

0

t (df = 5)

t (df = 13)Bell-ShapedSymmetric

‘Fatter’ Tails

Standard Normal

Degrees of Freedom (df )

Number of observations that are free to vary after sample mean has been calculated

df = n - 1 Example: Mean of 3 numbers is 2

degrees of freedom = n -1 = 3 -1= 2

Student’s t Distribution

Let X1, X2,…, Xn be independent random variables with the same mean µ and the same variance б2. Then, the random variable

has a t distribution with n-1 degrees of freedom

nS

xT

/

2

1

2 )(1

1

XXn

Sn

jj

Student’s t Table

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0 2.920t Values

Let: n = 3 df = n - 1 = 2 = .10 /2 =.05

/ 2 = .05

Problem 1


for the mean µ of a normal

population with standard

deviation 4.00 from the sample

30, 42, 40, 34, 48, 50

Problem 9


for the mean of a normal

population. The length of 20 bolts

with Sample Mean 20.2 cm and

Sample Variance 0.04 cm2.

Confidence Interval for the Variance

Choose a Confidence Level γ (95%, 90%, etc) Determine the solutions C1 and C2 of the equation

F(C1) = ½ ( 1 - γ) and F(C2) = ½ ( 1 + γ)

[Table A10: Chi Square distribution with n-1 degrees of freedom]

Compute (n-1)S2, where S2 is the variance of the sample x1, x2,…, xn

Confidence Interval for the Variance

Compute

k1 = (n-1)S2 / C1

k2 = (n-1)S2 / C2

Confidence Interval Estimate

)( 12

2 kkConf

Chi Square Distribution

Let X1, X2,…, Xn be independent random variables with the same mean µ and the same variance б2. Then, the random variable

With

2

2

)1(

Sn

2

1

2 )(1

1

XXn

Sn

jj

has a Chi Square distribution with n-1 degrees of freedom

Chi Square Distribution

CDF

PDF

Problem 15


for the variance of a normal

population. The Sample has 30

values with variance 0.0007.

Problem 2

Does the Interval in Problem 1

get longer or shorter, if we take γ

= 0.99 instead of 0.95? By what

factor?

Problem 3

By what factor does the length

of the Interval in Problem 1

change, if we double the Sample

Size?

Problem 5

What Sample Size would be

needed for obtaining a 95%

Confidence Interval (3) of length 2

б? Of length б?

Problem 7

What Sample Size is needed to

obtain a 99% Confidence Interval

of length 2.0 for the mean of a

normal population with variance

25?

Problem 9



population. The length of 20 bolts

with Sample Mean 20.2 cm and

Sample Variance 0.04 cm2.

Problem 11



population. The Copper Content

(%) of brass is

66, 66, 65, 64, 66, 67, 64, 65, 63,

64

Problem 13

Find a 95% Confidence Interval for the percentage of cars on a certain highway that have poorly adjusted brakes, using a random sample of 500 cars stopped at a road block on a highway, 87 of which had poorly adjusted brakes.

Problem 17



population. The Sample is the

Copper Content (%) of brass:

66, 66, 65, 64, 66, 67, 64, 65,

63, 64

Problem 19



population. The Sample has Mean

Energy (keV) of delayed neutron

group (Group 3, half life 6.2 sec)

for uranium U235 fission:

435, 451, 430, 444, 438

Problem 21

If X is normal with mean 27 and

variance 16, what distribution do –

X, 3X and 5X-2 have?

Problem 23

A machine fills boxes weighing Y lb with X lb of salt, where X and Y are normal with mean 100 lb and 5 lb and standard deviation 1 lb and 0.5 lb, respectively. What percent of filled boxes weighing between 104 lb and 106 lb are to be expected?

Confidence Interval for Variance

Choose a Confidence Level γ (95%, 90%, etc) Determine the solution C1 and C2 of the equation

F(C1) = ½ ( 1 - γ) and F(C2) = ½ ( 1 + γ)

[Table: Chi Square distribution with n-1 degrees of freedom]

Compute (n-1) s2, where s2 is variance of the sample x1, x2,…, xn

Confidence Interval for Variance

Compute k1 = (n-1) s2/C1

and k2 = (n-1) s2/C2


)( 12

2 kkConf

Elements of Confidence Interval

Estimation

Level of confidence Confidence in which the interval will

contain the unknown population parameter

Precision (range) Closeness to the unknown parameter

Cost Cost required to obtain a sample of

size n

Level of Confidence

Denoted by A relative frequency interpretation

In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter

A specific interval will either contain or not contain the parameter No probability involved in a specific interval

100 1 %

100 1 %

Chap 7-Confidence Intervals-

37

Interval and Level of Confidence

Confidence Intervals

Intervals extend from

to

of intervals constructed contain ;

do not.

_Sampling Distribution of the Mean

XX Z

X/ 2

/ 2

XX

1

XX Z

1 100%

100 %

/ 2 XZ / 2 XZ


38

Factors Affecting Interval Width (Precision)

Data variation Measured by

Sample size

Level of confidence

Intervals Extend from

© 1984-1994 T/Maker Co.

X - Z to X + Z xx

Xn

100 1 %


39

Determining Sample Size (Cost)

Too Big:

• Requires too many resources

Too small:

• Won’t do the job


40

Determining Sample Size for Mean

What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45.

Round Up

2 22 2

2 2

1.645 45219.2 220

Error 5

Zn


41

Determining Sample Size for Mean in PHStat

PHStat | sample size | determination for the mean …

Example in excel spreadsheet


42

Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample

Use student’s t distribution Confidence interval estimate

Confidence Interval for( Unknown)

/ 2, 1 / 2, 1n n

S SX t X t

n n


43

Example

/ 2, 1 / 2, 1

8 850 2.0639 50 2.0639

25 2546.69 53.30

n n

S SX t X t

n n

A random sample of 25 has 50 and 8.

Set up a 95% confidence interval estimate for

n X S


44

Confidence Interval Estimate for Proportion p

Assumptions Two outcomes (0;1) Number of ‘1’ in n trials follows B(n,p) Normal approximation can be used if

and Confidence interval estimate

5np 1 5n p

/ 2 / 2

1 1S S S SS S

p p p pp Z p p Z

n n


45

ExampleA random sample of 400 voters showed 32 preferred candidate A. Set up a 95% confidence interval estimate for p.

/ /

1 1

.08 1 .08 .08 1 .08.08 1.96 .08 1.96

400 400.053 .107

s s s ss s

p p p pp Z p p Z

n n

p

p

2 2

Normal Table


46

Confidence Interval Estimate for Proportion in PHStat

PHStat | confidence interval | estimate for the proportion …


Microsoft Excel Worksheet


47

Determining Sample Size for Proportion

Out of a population of 1,000, we randomly selected 100, of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence?

Round Up

2 2

2 2

1 1.645 0.3 0.7

Error 0.05227.3 228

Z p pn


48

Determining Sample Size for Proportion in PHStat

PHStat | sample size | determination for the proportion …


Microsoft Excel Worksheet


49

Confidence Interval for Population Total Amount

Point estimate


NX

/ 2, 1 1n

N nSNX N t

Nn


50

Confidence Interval for Population Total: Example

An auditor is faced with a population of 1000 vouchers and wants to estimate the total value of the population. A sample of 50 vouchers is selected with average voucher amount of $1076.39, standard deviation of $273.62. Set up the 95% confidence interval estimate of the total amount for the population of vouchers.


51

Example Solution

/ 2, 1

1000 50 $1076.39 $273.62

1

273.62 1000 501000 1076.39 1000 2.0096

1000 11001,076,390 75,830.85

n

N n X S

N nSNX N t

Nn

The 95% confidence interval for the population total amount of the vouchers is between 1,000,559.15, and 1,152,220.85


52

Confidence Interval for Total Difference in the Population

Point estimate Where is the sample

average difference


Where

ND 1

n

ii

DD

n

/ 2, 1 1

Dn

N nSND N t

Nn

2

1

1

n

ii

D

D DS

n


53

Estimation for Finite Population

Samples are selected without replacement Confidence interval for the mean

( unknown)

Confidence interval for proportion

/ 2, 1 1n

N nSX t

Nn

/ 2

1

1S S

S

p p N np Z

n N


54

Sample Size Determination for Finite Population

Samples are selected without replacement When estimating the mean

When estimating the proportion

2 2/ 2

0 2

Zn

e

2/ 2

0 2

1Z p pn

e


55

Ethical Considerations

Report confidence interval (reflect sampling error) along with the point estimate

Report the level of confidence Report the sample size Provide an interpretation

of the confidence interval estimate


56

Chapter Summary

Illustrated estimation process Discussed point estimates Addressed interval estimates Discussed confidence interval

estimation for the mean ( known) Addressed determining sample size Discussed confidence interval

estimation for the mean ( unknown)


57

Chapter Summary Discussed confidence interval

estimation for the proportion Addressed confidence interval

estimation for population total Discussed confidence interval

estimation for total difference in the population

Addressed estimation and sample size determination for finite population

Addressed confidence interval estimation and ethical issues

(continued)

confidence interval. assignment # 2 briefly describe following with the help of examples: point...

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