1. Confidence Interval

2. Assignment # 2 Point Estimation of Parameters Confidence Intervals
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.

3. Estimation Process Population Random Sample
I am 95% confident that  is between 40 & 60.
Mean
X = 50
Mean, , is unknown
Sample

4. Confidence Interval provides more information about a population characteristic than does a point estimate. It provides a confidence level for the estimate.
Upper
Confidence
Limit
Lower
Confidence
Limit
Point Estimate
Width of
Confidence Interval

5. 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

6. Confidence Interval Estimates
Intervals
Mean
Variance 2 2
Known
Unknown

7. 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

8. 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 γ
0.90
0.95
0.99
0.999 C
1.645
1.960
2.576
3.291

9. 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

10. 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
Confidence interval estimate

11. Student’s t Distribution
Standard Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5) Z t

12. 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

13. 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

14. Student’s t Table Let: n = 3 df = n - 1 = 2  = .10 /2 =.05 .05 2 t
Upper Tail Area df
.25
.10
.05 1
1.000
3.078
6.314
 / 2 = .05 2
0.817
1.886
2.920 3
0.765
1.638
2.353 t
2.920
t Values

15. 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

16. 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 cm2.

17. 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

18. Confidence Interval for the Variance
Compute
k1 = (n-1)S2 / C1
k2 = (n-1)S2 / C2
Confidence Interval Estimate

19. 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
has a Chi Square distribution with n-1 degrees of freedom

20. Chi Square Distribution
CDF
PDF

21. Problem 15
Find a 95% Confidence Interval for the variance of a normal population. The Sample has 30 values with variance

22. 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?

23. Problem 3
By what factor does the length of the Interval in Problem 1 change, if we double the Sample Size?

24. Problem 5
What Sample Size would be needed for obtaining a 95% Confidence Interval (3) of length 2 б? Of length б?

25. 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?

26. 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 cm2.

27. 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

28. 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.

29. 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

30. 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 U235 fission: 435, 451, 430, 444, 438

31. Problem 21
If X is normal with mean 27 and variance 16, what distribution do –X, 3X and 5X-2 have?

32. 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?

33. 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

34. Confidence Interval for Variance
Compute k1 = (n-1) s2/C1
and k2 = (n-1) s2/C2
Confidence interval estimate

35. 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

36. 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

37. Interval and Level of Confidence
Sampling Distribution of the Mean _
Intervals extend from to
of intervals constructed contain ;
do not.
Confidence Intervals

38. Factors Affecting Interval Width (Precision)
Data variation
Measured by
Sample size
Level of confidence
Intervals Extend from
X - Z to X + Z  x x
© T/Maker Co.

39. Determining Sample Size (Cost)
Too small:
Won’t do the job
Too Big:
Requires too many resources

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

41. Determining Sample Size for Mean in PHStat
PHStat | sample size | determination for the mean …
Example in excel spreadsheet

42. Confidence Interval for ( Unknown)
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

43. Example

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

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

46. Confidence Interval Estimate for Proportion in PHStat
PHStat | confidence interval | estimate for the proportion …
Example in excel spreadsheet

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

48. Determining Sample Size for Proportion in PHStat
PHStat | sample size | determination for the proportion …
Example in excel spreadsheet

49. Confidence Interval for Population Total Amount
Point estimate
Confidence interval estimate

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 $ , standard deviation of $ Set up the 95% confidence interval estimate of the total amount for the population of vouchers.

51. Example Solution
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
Confidence interval estimate
Where

53. Estimation for Finite Population
Samples are selected without replacement
Confidence interval for the mean ( unknown)
Confidence interval for proportion

54. Sample Size Determination for Finite Population
Samples are selected without replacement
When estimating the mean
When estimating the proportion

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
(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