lect13 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. Students 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 299303. Consider the Z-interval of Sec. 8.1. We can substitute S for , when 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 Students t dis- tribution instead of standard normal z-percentage points. The pdf of Students t is bell-shaped like the normal, but somewhat flatter: f ( x ) = r +1 2 r 2 r 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 7467). 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 e- 1 2 x 2 as r Percentage points of t : t ( ; r ) : (for more complete listings, see Devore 6e Table A8, pages 7467.) degrees of = . 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-...
View Full Document

Page1 / 8

lect13 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online