week 10 1

Test on Pairs of Means – Case I

• Suppose are iid independent of that

are iid .

• Further, suppose that n1 and n2 are large or that are known.

• We are interested in testing H0: μx = μy versus a one sided or a two sided alternative…

• Then,…

1...,,1 nXX 2, xxN

2...,,1 nYY

2, yyN

2y

2 and x

week 10 2

Test on Pairs of Means – Case II

• Suppose are iid independent of that are

iid .

• Further, suppose that n1 and n2 are small and that are

unknown but we assume they are equal to σ2.

• We are interested in testing H0: μx - μy = δ versus a one sided or a two sided alternative…

• Then,…

1...,,1 nXX 2, xxN

2...,,1 nYY

2, yyN

2y

2 and x

week 10 3

Example

• The strength of concrete depends, to some extent, on the method used for drying it. Two drying methods were tested on independently specimens yielding the following results…

• We can assume that the strength of concrete using each of these methods follows a normal distribution with the same variance.

• Do the methods appear to produce concrete with different mean strength? Use α = 0.05.

week 10 4

Test on Pairs of Means – Case III

• Suppose are iid with E(Xi ) = µx and Var(Xi) = σx

independent of that are iid with E(Yi ) = µy and Var(Yi) = σy

• Further, suppose that n1 and n2 are both large.

• We are interested in testing H0: μx - μy = δ versus a one sided or a two sided alternative…

• Then,…

1...,,1 nXX

2...,,1 nYY

week 10 5

Test on Two Proportions

• Suppose are iid Bernoulli(θ1) independent of

that are iid Bernoulli(θ2).

• Further, suppose that n1 and n2 are large.

• We are interested in testing H0: θ1 = θ2 versus a one sided or a two sided alternative…

• Then,…

1...,,1 nXX

2...,,1 nYY

week 10 6

Example

week 10 7

Paired Observations • In a matched pairs study, subjects are matched in pairs and the

outcomes are compared within each matched pair. The experimenter can toss a coin to assign two treatment to the two subjects in each pair. One situation calling for match pairs is when observations are taken on the same subjects, under different conditions.

• A match pairs analysis is needed when there are two measurements or observations on each individual and we want to examine the difference. This corresponds to the case where the samples are not independent.

• For each individual (pair), we find the difference d between the

measurements from that pair. Then we treat the di as one sample and use the one sample t test and confidence interval to estimate and test the difference between the treatments effect.

week 10 8

Example

• Seneca College offers summer courses in English. A group of 20 students were given the TOFEL test before the course and after the course. The results are summarized in the next slide.

• Find a 95% CI for the average improvement in the TOFEL score.

• Test whether attending the course improve the performances on the TOFEL.

week 10 9

Data Display

Row Student Pretest Posttest improvement 1 1 30 29 -1 2 2 28 30 2 3 3 31 32 1 4 4 26 30 4 5 5 20 16 -4 6 6 30 25 -5 7 7 34 31 -3 8 8 15 18 3 9 9 28 33 5 10 10 20 25 5 11 11 30 32 2 12 12 29 28 -1 13 13 31 34 3 14 14 29 32 3 15 15 34 32 -2 16 16 20 27 7 17 17 26 28 2 18 18 25 29 4 19 19 31 32 1 20 20 29 32 3

week 10 10

• One sample t confidence interval for the improvement

T-Test of the Mean

Test of mu = 0.000 vs mu > 0.000 Variable N Mean StDev SE Mean T P improvemt 20 1.450 3.203 0.716 2.02 0.029

• MINITAB commands for the paired t-test Stat > Basic Statistics > Paired t

Paired T-Test and Confidence Interval

Paired T for Posttest – Pretest N Mean StDev SE Mean Posttest 20 28.75 4.74 1.06 Pretest 20 27.30 5.04 1.13 Difference 20 1.450 3.203 0.716 95% CI for mean difference: (-0.049, 2.949) T-Test of mean difference=0 (vs > 0): T-Value = 2.02 P-Value = 0.029

week 10 11

Character Stem-and-Leaf Display

Stem-and-leaf of improvement N = 20Leaf Unit = 1.0 2 -0 54 4 -0 32 6 -0 11 8 0 11 (7) 0 2223333 5 0 4455 1 0 7

86420-2-4

6

5

4

3

2

1

0

improvement

Fre

quency

week 10 12

Test for a Single Variance

• Suppose X1, …, Xn is a random sample from a N(μ, σ2) distribution.

• We are interested in testing versus a one sided or a

two sided alternative…

• Then…

20

20 : H

week 10 13

Test on Pairs of Variances

• Suppose are iid independent of that

are iid .

• We are interested in testing versus a one sided or a two

sided alternative…

• Then…

1...,,1 nXX 2, xxN 2

...,,1 nYY

2, yyN

220 : yxH

week 10 14

Example