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.

Population

Mean,



, is unknown Sample

Estimation Process

Random Sample

Mean X = 50 I am 95% confident that



is between 40 & 60.

 Confidence information Interval about a provides more population characteristic than does a point estimate.

It provides a confidence level for the estimate.

Lower Confidence Limit Point Estimate Width of Confidence Interval Upper Confidence Limit

Interval Estimates

 Provides range of values  Takes into consideration variation sample statistics from sample to sample in  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

Confidence Intervals Mean Variance

 2

Known

 2

Unknown

Confidence Interval for

µ

( б

2

Known)

  Assumptions Population standard deviation is known    Population is normally distributed If population is not normal, use large sample Confidence interval estimate

Conf

 ( 

x



k

   

x



k

)..

where

..

k



c



n

Steps to Determine Confidence Interval

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

γ 0.90

0.95

0.99

0.999

C 1.645

1.960

2.576

3.291

 Compute the mean x of the sample x 1 , x 2 ,…, x n  Confidence interval estimate

Conf

 ( 

x



k

   

x



k

)..

where

..

k



c



n

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

( б

2

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 s 2 sample x 1 , x 2 ,…, x n  Confidence interval estimate

Conf

 ( 

x



k

   

x



k

)..

where

..

k



cs n

of the

Student’s

t

Distribution

Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5)

Z t

0

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 X 1 variance б , X 2 2 ,…, X n be independent random variables with the same mean µ and the same . Then, the random variable

T



S



x

 / 

n

has a t distribution with n-1 degrees of freedom

S

2 

n

1  1

j n

  1 (

X j

 

X

) 2

Student’s

t

Table

df Upper Tail Area

.25

.10

.05

1 1.000 3.078 6.314

2

0.817 1.886

2.920

3 0.765 1.638 2.353

Let: n = 3 df =

n

 - 1 = 2 = .10

 /2 =.05



/ 2 = .05

0 2.920

t

t

Values

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

Problem 9

Find a 99% Confidence Interval for the mean of a normal population. The length of 20 bolts with Sample Mean 20.2 cm and Sample Variance 0.04

cm 2 .

Confidence Interval for the Variance

 Choose a Confidence Level γ (95%, 90%, etc)  Determine the solutions C 1 and C 2 of the equation F(C 1 ) = ½ ( 1 - γ) and F(C 2 ) = ½ ( 1 + γ) [Table A10: Chi Square distribution with n-1 degrees of freedom]  Compute (n-1)S 2 , where S 2 sample x 1 , x 2 ,…, x n is the variance of the

Confidence Interval for the Variance

 Compute k 1 = (n-1)S 2 / C 1 k 2 = (n-1)S 2 / C 2 

Confidence Interval Estimate

Conf

 (

k

2   2 

k

1 )

Chi Square Distribution

Let X 1 variance б , X 2 2 ,…, X n be independent random variables with the same mean µ and the same . Then, the random variable   (

n

 1 )

S

 2 2 With

S

2 

n

1  1

j n

  1 (

X j

 

X

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

Chi Square Distribution

CDF PDF

Problem 15

Find a 95% Confidence Interval 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

Find a 99% Confidence Interval for the mean of a normal population. The length of 20 bolts with Sample Mean 20.2 cm and Sample Variance 0.04

cm 2 .

Problem 11

Find a 99% Confidence Interval for the mean of a normal 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

Find a 95% Confidence Interval for the variance of a normal population.

The Sample is the Copper Content (%) of brass: 66, 66, 65, 64, 66, 67, 64, 65, 63, 64

Problem 19

Find a 95% Confidence Interval for the variance of a normal population.

The Sample has Mean Energy (keV) of delayed neutron group (Group 3, half life 6.2 sec) for uranium U 235 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 C 1 equation and C 2 of the F(C 1 ) = ½ ( 1 - γ) and F(C 2 ) = ½ ( 1 + γ) [Table: Chi Square distribution with n-1 degrees of freedom]  Compute (n-1) s 2 , where s 2 sample x 1 , x 2 , …, x n is variance of the

Confidence Interval for Variance

 Compute k 1 = (n-1) s 2 /C 1 and k 2 = (n-1) s 2 /C 2  Confidence interval estimate

Conf

 (

k

2   2 

k

1 )

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  100 1    % 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

Interval and Level of Confidence  

Z

 / 2 

X

Intervals extend from  to  

X X

 / 2 1   

X

 / 2  

Z

 / 2 

X



X

 

X

 1    100% of intervals constructed  Confidence Intervals not.

 do Chap 7-Confidence Intervals-37

Factors Affecting Interval Width (Precision)  Data variation  Measured by     

X



n

Level of confidence     %

Intervals Extend from X - Z



x to X + Z



x

© 1984-1994 T/Maker Co.

Chap 7-Confidence Intervals-38

Determining Sample Size (Cost)

Too Big:

• Requires too many resources

Too small:

• Won’t do the job Chap 7-Confidence Intervals-39

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.

Z

2  2

n

 Error 2  1.645

2 5 2  219.2

 220 Round Up Chap 7-Confidence Intervals-40

Determining Sample Size for Mean in PHStat   PHStat | sample size | determination for the mean … Example in excel spreadsheet Chap 7-Confidence Intervals-41

Confidence Interval for      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 

S S X



t

 / 2,

n

 1

n X



t

 / 2,

n

 1

n

Chap 7-Confidence Intervals-42

Example A random sample of

n

 25 has

X

 50 and

S

 8.

  / 2,

n

 1

S

8

n

25 46.69

 

t

 / 2,

n

 1

S n

8 25 53.30

Chap 7-Confidence Intervals-43

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

np

 5 and

n

 1 

p

  Confidence interval estimate 5 

p S



Z

 / 2

p S

 1 

p S



n p S



Z

 / 2

p S

 1 

p S



n

Chap 7-Confidence Intervals-44

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

p s



Z

 / 2

p s

 1 

p s



n

  400 .053

p s



Z

 / 2 .107

p s

 1 

p s

 

n

 400 Normal Table Chap 7-Confidence Intervals-45

Confidence Interval Estimate for Proportion in PHStat   PHStat | confidence interval | estimate for the proportion … Example in excel spreadsheet Chap 7-Confidence Intervals-46

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?

n

  2

Z p



1 

p

Error 2 227.3

 228



 2

  

0.05

2 Round Up Chap 7-Confidence Intervals-47

Determining Sample Size for Proportion in PHStat   PHStat | sample size | determination for the proportion … Example in excel spreadsheet Chap 7-Confidence Intervals-48

Confidence Interval for Population Total Amount  Point estimate  

NX

Confidence interval estimate 

NX

   / 2,

n

 1 

S n

 

N N

 

n

1   Chap 7-Confidence Intervals-49

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.

Chap 7-Confidence Intervals-50

Example Solution

N

 1000

n

 50

X NX

   / 2,

n

 1 

S n

 $1076.39

S

 $273.62

 

N N

 

n

1           100 The 95% confidence interval for the population total amount of the vouchers is between 1,000,559.15, and 1,152,220.85

Chap 7-Confidence Intervals-51

Confidence Interval for Total Difference in the Population  Point estimate 

ND

difference

D



i n

  1

D i n

 Confidence interval estimate 

ND

   / 2,

n

 1 

S D n

 

N N

 

n

1    Where

S D



i n

  1 

D i n

  1

D

 2 Chap 7-Confidence Intervals-52

Estimation for Finite Population  Samples are selected without replacement  

X



t

 / 2,

n

 1

S n

 

N N

 

n

1    Confidence interval for proportion 

p S



Z

 / 2

p S

 1 

n p S

  

N N

 

n

1   Chap 7-Confidence Intervals-53

Sample Size Determination for Finite Population  Samples are selected without replacement  When estimating the mean 

Z

2  / 2  2 

n

0 

e

2 When estimating the proportion 

n

0 

Z

 2 / 2

p e

 2 1 

p

 Chap 7-Confidence Intervals-54

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 Chap 7-Confidence Intervals-55

Chapter Summary       Illustrated estimation process Discussed point estimates Addressed interval estimates Discussed confidence interval estimation  Addressed determining sample size Discussed confidence interval estimation  Chap 7-Confidence Intervals-56

     Chapter Summary (continued) 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 Chap 7-Confidence Intervals-57