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Section 10.2 β Confidence Intervals for π1 β π2 Conditions: ο· Random: Data come from two independent random samples or from two groups in a randomized experiment. 1 1 o 10%: When sampling without replacement, check that π1 β€ 10 π1 and π2 β€ 10 π2 . ο· Normal/Large Sample: Satisfied if: o Both Populations are Normal OR o Both Samples are β₯ 30 OR o Graphs of the data show NO outliers or strong skewness (t-procedures) The Two-Sample t Statistic It is very unlikely that we will be given the population standard deviation (Ο) for each population, so we will be using t β procedures. Μ π β π Μ π Standard Error of π ππΈπ₯Μ 1 βπ₯Μ 2 = β π 12 π 22 + π1 π2 Two-Sample t Interval for a Difference Between Two Means When conditions are met, an approximate C% confidence interval for π₯Μ 1 β π₯Μ 2 is π 12 (π₯Μ 1 β π₯Μ 2 ) ± π‘ β β π1 π 2 + π2 2 Where t* is the critical value with C% of its area between βt* and t* for the t distribution with degrees of freedom technology or smaller than n1 β 1 and n2 β 1. ***Using the table with degrees of freedom, we will use the smaller d.f. from the samples. Example, pg. 641 Big Trees, Small Trees, Short Trees, Tall Trees The Wade Tract Preserve in Georgia is an old-growth forest of longleaf pines that has survived in a relatively undisturbed state for hundreds of years. One question of interest to foresters who study the area is βHow do the sizes of longleaf pine trees in the northern and southern halves of the forest compare?β To find out, researchers took random samples of 30 trees from each half and measured the diameter at breast height (DBH) in centimeters. The summary statistics are as follows: Descriptive Statistics: Variable North South North, South N Mean 30 23.70 30 34.53 St Dev 17.50 14.26 Construct and interpret a 90% confidence interval for the difference in the mean DBH of longleaf pines in the northern and southern halves of the Wade Tract Preserve. State: We want to estimate the difference π1 β π2 where π1 = the true mean DBH of all trees in the southern half of the forest and π2 = the true mean DBH of all trees in the northern half of the forest at a 90% confidence level. Plan: Two-sample t interval for π1 β π2 1. Random: The data come from independent random samples of 30 trees each from the northern and southern halves of the forest. 1 1 ο· 10%: 30 β€ 10 (π΄ππ π‘ππππ ππ π‘βπ ππππ‘βπππ βπππ), 30 β€ 10 (π΄ππ π‘ππππ ππ π‘βπ π ππ’π‘βπππ βπππ) Fairly safe to assume there are more than 300 trees in each half of the forest. 2. Normal/Large Sample: Because both sample sizes are 30, it is safe to use t-procedures by the Central Limit Theorem. Do: Method 1: Using a table Weβll use d.f. = 29 ο t* = 1.699 π 12 (π₯Μ 1 β π₯Μ 2 ) ± π‘ β β π1 π 2 + π2 = (34.53 β 23.70) ± 1.699β 2 14.262 30 + 17.52 30 = 10.83 ± 7.00 = (3.83, 17.83) Method 2: Using the calculator **CALCULATOR: Use 2-SampTInt: Stats ο Put in π₯Μ 1: 34.53 Sx1: 14.26, n1: 30, π₯Μ 2: 23.7 Sx2: 17.5, n2: 30 and C-Level: 90 Pooled: NO Calculate YOU WRITE: df = 55.728 (3.9362, 17.724) Conclude: We are 90% confident that the interval from 3.9362 to 17.724 centimeters captures the true difference in the actual mean DBH of the southern trees and the actual mean DBH of the northern trees.