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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 X t / 2,n 1( -5 -4 -3 -2 s s ) X t / 2,n 1( ) n n -1 0 1 2 3 4 5 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? Example (cont.) X = ______________ s = _______________ α = ________________ α/2 = _____________ t0.025,24 = _____________ __________________ < μ < ___________________ 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. Solution X = ______________ s = _______________ α = ________________ α/2 = _____________ (draw the picture) T___ , ____ = _____________ __________________ < μ < ___________________ Standard Error of a Point Estimate • Case 1: σ known – The standard deviation, or standard error of X is n • Case 2: σ unknown, sampling from a normal distribution – The standard deviation, or (usually) estimated standard error of X is ______ 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, x0 is 1 1 X z / 2 1 x0 X z / 2 1 n n if σ is known, and 1 1 X t / 2,n 1s 1 x0 X t / 2,n 1s 1 n n if σ is unknown 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. Summary • Confidence interval population mean μ • Prediction interval a new observation x0 • Tolerance interval a (1-α) proportion of the measurements can be estimated with a 100(1-γ)% confidence Estimating the Difference Between Two Means • Given two independent random samples, a point estimate the difference between μ1 and μ2 is given by the statistic x1 x 2 We can build a confidence interval for μ1 - μ2 (given σ12 and σ22 known) as follows: ( x 1 x 2 ) z / 2 12 n1 22 n2 1 2 ( x 1 x 2 ) z / 2 12 n1 22 n2 An example • Look at example 9.8, pg. 248 Differences Between Two Means: Variances Unknown • Case 1: σ12 and σ22 unknown but equal ( x1 x 2 ) t / 2,n1 n2 2Sp Where, 1 1 1 1 1 2 ( x 1 x 2 ) t / 2,n1 n2 2Sp n1 n2 n1 n2 (n1 1)S12 (n2 1)S22 S n1 n2 2 2 p Differences Between Two Means: Variances Unknown • Case 2: σ12 and σ22 unknown and not equal ( x 1 x 2 ) t / 2, Where, s12 s22 s12 s22 1 2 ( x 1 x 2 ) t / 2, n1 n2 n1 n2 (S12 / n1 S22 / n2 )2 S2 / n 2 S2 / n 2 1 1 2 2 n1 1 n2 1