1. EGR 252 - Ch. 91 Chapter 9: One- and Two- Sample Estimation Statistical Inference – Estimation – Tests of hypotheses – “Even the most efficient unbiased estimator is unlikely to estimate the population parameter exactly.” (Walpole et al, pg. 233) Interval estimation: (1 – α)100% confidence interval for the unknown parameter. – Example: if α = 0.01, we develop a _______ confidence interval.

2. EGR 252 - Ch. 92 Single Sample: Estimating the Mean Given: –σ is known and X is the mean of a random sample of size n, Then, –the (1 – α)100% confidence interval for μ is given by

3. EGR 252 - Ch. 93 Example A traffic engineer is concerned about the delays at an intersection near a local school. The intersection is equipped with a fully actuated (“demand”) traffic light and there have been complaints that traffic on the main street is subject to unacceptable delays. To develop a benchmark, the traffic engineer randomly samples 25 stop times (in seconds) on a weekend day. The average of these times is found to be 13.2 seconds, and the variance is know to be 4 seconds 2. Based on this data, what is the 95% confidence interval (C.I.) around the mean stop time during a weekend day?

4. EGR 252 - Ch. 94 Example (cont.) X = ______________ σ = _______________ α = ________________ α/2 = _____________ Z 0.025 = _____________ Z 0.975 = ____________ __________________ < μ < ___________________

5. EGR 252 - Ch. 95 Your turn … What is the 90% C.I.? What does it mean?

6. EGR 252 - Ch. 96 What if σ 2 is unknown? For example, what if the traffic engineer doesn’t know the variance of this population? 1.If n is sufficiently large (> _______), then the large sample confidence interval is: 2.Otherwise, must use the t-statistic …

7. EGR 252 - Ch. 97 Single Sample: Estimating the Mean (σ unknown, n not large) Given: –σ is unknown and X is the mean of a random sample of size n (where n is not large), Then, –the (1 – α)100% confidence interval for μ is given by

8. EGR 252 - Ch. 98 Recall Our Example A traffic engineer is concerned about the delays at an intersection near a local school. The intersection is equipped with a fully actuated (“demand”) traffic light and there have been complaints that traffic on the main street is subject to unacceptable delays. To develop a benchmark, the traffic engineer randomly samples 25 stop times (in seconds) on a weekend day. The average of these times is found to be 13.2 seconds, and the sample variance, s 2, is found to be 4 seconds 2. Based on this data, what is the 95% confidence interval (C.I.) around the mean stop time during a weekend day?

9. EGR 252 - Ch. 99 Example (cont.) X = ______________ s = _______________ α = ________________ α/2 = _____________ t 0.025,24 = _____________ __________________ < μ < ___________________

10. EGR 252 - Ch. 910 Your turn A thermodynamics professor gave a physics pretest to a random sample of 15 students who enrolled in his course at a large state university. The sample mean was found to be 59.81 and the sample standard deviation was 4.94. Find a 99% confidence interval for the mean on this pretest.

11. EGR 252 - Ch. 911 Solution X = ______________ s = _______________ α = ________________ α/2 = _____________ (draw the picture) T ___, ____ = _____________ __________________ < μ < ___________________

12. EGR 252 - Ch. 912 Standard Error of a Point Estimate Case 1: σ known –The standard deviation, or standard error of X is Case 2: σ unknown, sampling from a normal distribution –The standard deviation, or (usually) estimated standard error of X is ______

13. EGR 252 - Ch. 913 9.6: Prediction Interval For a normal distribution of unknown mean μ, and standard deviation σ, a 100(1-α)% prediction interval of a future observation, x 0 is if σ is known, and if σ is unknown

14. EGR 252 - Ch. 914 9.7: Tolerance Limits For a normal distribution of unknown mean μ, and unknown standard deviation σ, tolerance limits are given by x + ks where k is determined so that one can assert with 100(1-γ)% confidence that the given limits contain at least the proportion 1-α of the measurements. Table A.7 gives values of k for (1-α) = 0.9, 0.95, 0.99; γ = 0.05, 0.01; and for selected values of n.

15. EGR 252 - Ch. 915 Summary Confidence interval  population mean μ Prediction interval  a new observation x 0 Tolerance interval  a (1-α) proportion of the measurements can be estimated with a 100(1-γ)% confidence

16. EGR 252 - Ch. 916 Estimating the Difference Between Two Means Given two independent random samples, a point estimate the difference between μ 1 and μ 2 is given by the statistic We can build a confidence interval for μ 1 - μ 2 (given σ 1 2 and σ 2 2 known) as follows:

17. EGR 252 - Ch. 917 An example Look at example 9.8, pg. 248

18. EGR 252 - Ch. 918 Differences Between Two Means: Variances Unknown Case 1: σ 1 2 and σ 2 2 unknown but equal Where,

19. EGR 252 - Ch. 919 Differences Between Two Means: Variances Unknown Case 2: σ 1 2 and σ 2 2 unknown and not equal Where,

20. EGR 252 - Ch. 920 Estimating μ 1 – μ 2 Example ( σ 1 2 and σ 2 2 known) : A farm equipment manufacturer wants to compare the average daily downtime of two sheet-metal stamping machines located in two different factories. Investigation of company records for 100 randomly selected days on each of the two machines gave the following results: x 1 = 12 minutesx 2 = 10 minutes s 1 2 = 12s 2 2 = 8 n 1 = n 2 = 100 Construct a 95% C.I. for μ 1 – μ 2

21. EGR 252 - Ch. 921 Solution α/2 = _____________ z _____ = ____________ __________________ < μ 1 – μ 2 < _________________ Interpretation: Picture

22. EGR 252 - Ch. 922 μ 1 – μ 2 : σ i 2 Unknown Example (σ 1 2 and σ 2 2 unknown but equal): Suppose the farm equipment manufacturer was unable to gather data for 100 days. Using the data they were able to gather, they would still like to compare the downtime for the two machines. The data they gathered is as follows: x 1 = 12 minutesx 2 = 10 minutes s 1 2 = 12s 2 2 = 8 n 1 = 18n 2 = 14 Construct a 95% C.I. for μ 1 – μ 2

23. EGR 252 - Ch. 923 Solution α/2 = _____________ t ____, ________ = ____________ __________________ < μ 1 – μ 2 < _________________ Interpretation: Picture

24. EGR 252 - Ch. 924 Paired Observations Suppose we are evaluating observations that are not independent … For example, suppose a teacher wants to compare results of a pretest and posttest administered to the same group of students. Paired-observation or Paired-sample test … Example: murder rates in two consecutive years for several US cities (see attached.) Construct a 90% confidence interval around the difference in consecutive years.

25. EGR 252 - Ch. 925 Solution D = ____________ t α/2, n-1 = _____________ a (1- α)100% CI for μ 1 – μ 2 is: __________________ < μ 1 – μ 2 < _________________ Interpretation: Picture

26. EGR 252 - Ch. 926 C.I. for Proportions The proportion, P, in a binomial experiment may be estimated by where X is the number of successes in n trials. For a sample, the point estimate of the parameter is The mean for the sample proportion is and the sample variance

27. EGR 252 - Ch. 927 C.I. for Proportions An approximate (1-α)100% confidence interval for p is: Large-sample C.I. for p 1 – p 2 is: Interpretation: _______________________________

28. EGR 252 - Ch. 928 Example 9.15 1.C.I. = (-0.0017, 0.0217), therefore no reason to believe there is a significant decrease in the proportion defectives using the new process. 2.What if the interval were (+0.0017, 0.0217)? 3.What if the interval were (-0.9, -0.7)?