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Chapter 9: One- and Two- Sample Estimation

This chapter discusses statistical inference, specifically the estimation of unknown parameters and hypothesis testing. It covers the calculation of confidence intervals and provides examples and explanations for both known and unknown variance scenarios.

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Chapter 9: One- and Two- Sample Estimation

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  1. Chapter 9: One- and Two- Sample Estimation • Statistical Inference • Estimation • Tests of hypotheses • Interval estimation: (1 – α) 100% confidence interval for the unknown parameter. • Example: if α = 0.01, we develop a 99% confidence interval. • Example: if α = 0.05, we develop a 95% confidence interval. JMB Spring 2009

  2. 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 JMB Spring 2009

  3. 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 known to be 4 seconds2. Based on this data, what is the 95% confidence interval (C.I.) around the mean stop time during a weekend day? JMB Spring 2009

  4. Example (cont.) X = ______________ σ = _______________ α = ________________ α/2 = _____________ Z0.025 = _____________ Z0.975 = ____________ _______________ < μ < _________________ z0.025 = -1.96 z0.975 = 1.96 13.2-(1.96)(2/sqrt(25)) = 12.416 13.2+(1.96)(2/sqrt(25)) = 13.984 JMB Spring 2009

  5. Your turn … • What is the 90% C.I.? What does it mean? 90% Z(.05) = + 1.645 12.542 < μ < 13.858 JMB Spring 2009

  6. What if σ2is unknown? • For example, what if the traffic engineer doesn’t know the variance of this population? • If n is sufficiently large (n > 30), then the large sample confidence intervalis calculated by using the sample standard deviation in place of sigma: • If σ2is unknown and n is not “large”, we must use the t-statistic … JMB Spring 2009

  7. 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: JMB Spring 2009

  8. 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, s2, is found to be 4 seconds2. Based on this data, what is the 95% confidence interval (C.I.) around the mean stop time during a weekend day? JMB Spring 2009

  9. Small Sample Example (cont.) X = ______________ s = _______________ α = ________________ α/2 = _____________ t0.025,24 = _____________ _______________ < μ < ________________ t 0.025,24 = 2.064 13.2-(2.064)(2/sqrt(25)) = 13.374 13.2+(2.064)(2/sqrt(25)) = 14.026 JMB Spring 2009

  10. 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. JMB Spring 2009

  11. Solution X = ______________ s = _______________ α = ________________ α/2 = _____________ (draw the picture) T___ , ____ = _____________ __________________ < μ < ___________________ X = 59.81 s = 4.94 α = .01 α/2 = .005 t (.005,14) = 2.977 Lower Bound 59.81 - (2.977)(4.94/sqrt(15)) = 56.01 Upper Bound 59.81 + (2.977)(4.94/sqrt(15)) = 63.61 JMB Spring 2009

  12. 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 JMB Spring 2009

  13. 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, x0is if σ is known, and if σ is unknown JMB Spring 2009

  14. 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 (page 761) gives values of k for (1-α) = 0.9, 0.95, or 0.99 and γ = 0.05 or 0.01 for selected values of n. JMB Spring 2009

  15. Example 9.8 (Page 284) • Find the 99% tolerance limits that will contain 95% of the metal pieces produced by the machine, given a sample mean of 1.0056 and a sample standard deviation of 0.0246. • Table A.7 (page 761) • (1-α) = 0.95 • ( 1-γ ) = 0.99 • n = 9. • k = 4.550 • x ± ks = 1.0056 ± (4.550) (0.0246) • We can assert with 99% confidence that the tolerance interval from 0.894 to 1.117 will contain 95% of the metal pieces produced by the machine JMB Spring 2009

  16. Summary JMB Spring 2009

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