# Confidence intervals for μ l hypothesis tests for μ

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Confidence intervals for μ l Hypothesis tests for μ l But have assumed known population variance σ 2 l This is unrealistic l Do have s 2 - a consistent & unbiased estimator of σ 2 l Why not use s 2 in place of σ 2?

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10 Practical considerations… l When sample size is large l CLT  sampling distribution of sample mean is approximately normal irrespective of population distribution l When σ is unknown & is replaced by s our standardized test statistic remains approximately (asymptotically) normally distributed l Why? Because in large samples s will be close to σ with high probability (it is a consistent estimator of σ 29 l So, for large n nothing has changed
Practical considerations... l But what about small n ? l Now need to consider sampling from a normal population l Let sample be iid from N( μ , σ 2) l Recall linear combinations of normal rv’s are also normal l Hence the sample mean and its standardized version (using σ ) will also be normally distributed l What happens when we standardize using s ? 11

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12 t distribution freedom of degrees 1 with on distributi Student a has ratio) ( statistic The ~ ? of on distributi the is What ) 1 , 0 ( ~ and /n) , ( then ) , ( If 1 2 2 n- t t t t n s X t n s X N n X Z ~N X X~N n - - = - - = μ μ σ μ σ μ σ μ
13 Properties of t distribution l Symmetric, unimodal distribution l Looks very similar to normal but has “fatter” tails l Is characterized by degrees of freedom ν l For large ν the distribution tends to a normal distribution l Keller Table 4 & BES t tables only provide critical values l P ( t > t α , ν ) = α l What are t. 05,9 , t. 01,15 , t. 05,120 , z. 05?

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14 Table 4: Upper-tail Critical Values of t -Distribution: t(α, ν ) α ν 0.100 0.050 0.025 0.010 0.005 1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 60 1.296 1.671 2.000 2.390 2.660 120 1.289 1.658 1.980 2.358 2.617 1.282 1.645 1.960 2.326 2.576 0 t
15 Inference using the t distribution l Strategies for constructing confidence intervals & conducting tests remain intact l What has changed is the distribution of our test statistic l What is a (100- α )% CI for the sample mean when X ~ N ( μ , σ 2 ) but σ 2 is unknown?

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