Confidence Interval

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.

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

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

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 Intervals Mean Variance 2 2 Known Unknown

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

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

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 Confidence interval estimate

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

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

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

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 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

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

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 cm2.

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 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 Confidence interval estimate

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

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

Factors Affecting Interval Width (Precision) Data variation Measured by Sample size Level of confidence Intervals Extend from X - Z to X + Z  x x © 1984-1994 T/Maker Co.

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

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

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

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

Example

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

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

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

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

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

Confidence Interval for Population Total Amount Point estimate Confidence interval estimate

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.

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

Confidence Interval for Total Difference in the Population Point estimate Where is the sample average difference Confidence interval estimate Where

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

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

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

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)

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