lect13 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction...

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Unformatted text preview: Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction: The one-sample problem for means. The Z interval. 8.2 Confidence interval for a proportion. 8.3 Comparing means of two independent samples. 8.4 Comparing two proportions. For lecture 13: 8.5 The one-sample problem for means with unknown variance and small samples. Student’s t-distribution, the T-interval. 8.6 Two independent samples, small sample sizes, variances unknown. The two-sample t statistic. 8.7 Paired samples. 8.8 Confidence intervals for variances. The chi-squared distribution. 8.9 Comparing two variances. The F-distribution. 1 8.5 THE ONE-SAMPLE PROBLEM FOR MEANS WITH UNKNOWN VARIANCE AND SMALL SAMPLES Ref: Devore 6e, Pages 299–303. Consider the Z-interval of Sec. 8.1. We can substitute S for σ , when σ is unknown, but if n is small (say less than 30) we should compensate for this extra uncertainty by using slightly wider confidence intervals. This is done by using the t –percentage points of Student’s t dis- tribution instead of standard normal z-percentage points. The pdf of Student’s t is bell-shaped like the normal, but somewhat flatter: f ( x ) = Γ ‡ r +1 2 · Γ ‡ r 2 · √ πr 1 ‡ 1 + x 2 r · ( r +1) / 2 Having written down this pdf , we will never have to use it, since the cdf is tabulated (e.g. Devore 6e Table A8, pages 746–7). Note that the distribution depends on a parameter r , called the degrees of freedom. Some facts about the t distribution: Mean 0 (its density is symmetric), Variance is r/ ( r- 2) > 1, ( r must be greater than 2). Also f ( x ) → 1 √ 2 π e- 1 2 x 2 as r → ∞ Percentage points of t : t ( α ; r ) : (for more complete listings, see Devore 6e Table A8, pages 746–7.) degrees of α = . 1 .05 .025 .01 freedom, r 5 1.476 2.015 2.571 3.365 15 1.341 1.753 2.131 2.602 30 1.310 1.697 2.042 2.457 ∞ (normal) 1.282 1.645 1.960 2.326 2 The remarkable result of William Gossett (pseudonym “Student”) is that the pivotal quan-...
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lect13 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction...

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