Download Confidence Intervals for two means

Section 10.2 – Confidence Intervals for 𝜇1 − 𝜇2
Conditions:
 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 𝑛1 ≤ 10 𝑁1 and 𝑛2 ≤ 10 𝑁2 .

Normal/Large Sample: Satisfied if:
o Both Populations are Normal OR
o Both Samples are ≥ 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 (σ) for each population, so we will be
using t – procedures.
̅𝟏 − 𝒙
̅𝟐
Standard Error of 𝒙
𝑆𝐸𝑥̅ 1 −𝑥̅2 = √
𝑠12 𝑠22
+
𝑛1 𝑛2
Two-Sample t Interval for a Difference Between Two Means
When conditions are met, an approximate C% confidence interval for 𝑥̅1 − 𝑥̅2 is
𝑠12
(𝑥̅1 − 𝑥̅2 ) ± 𝑡 ∗ √
𝑛1
𝑠2
+ 𝑛2
2
Where t* is the critical value with C% of its area between –t* and t* for the t distribution with degrees of
freedom technology or smaller than n1 – 1 and n2 – 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 “How do the sizes of longleaf pine trees in the
northern and southern halves of the forest compare?” 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 𝜇1 − 𝜇2 where 𝜇1 = the true mean DBH of all trees in the southern
half of the forest and 𝜇2 = 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 𝜇1 − 𝜇2
1. Random: The data come from independent random samples of 30 trees each from the northern and
southern halves of the forest.
1
1
 10%: 30 ≤ 10 (𝐴𝑙𝑙 𝑡𝑟𝑒𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑛𝑜𝑟𝑡ℎ𝑒𝑟𝑛 ℎ𝑎𝑙𝑓), 30 ≤ 10 (𝐴𝑙𝑙 𝑡𝑟𝑒𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑜𝑢𝑡ℎ𝑒𝑟𝑛 ℎ𝑎𝑙𝑓)
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
We’ll use d.f. = 29  t* = 1.699
𝑠12
(𝑥̅1 − 𝑥̅2 ) ± 𝑡 ∗ √
𝑛1
𝑠2
+ 𝑛2 = (34.53 − 23.70) ± 1.699√
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 
Put in 𝑥̅ 1: 34.53 Sx1: 14.26, n1: 30, 𝑥̅ 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.