Means Robb T. Independent Samples: Comparing Means

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction

Independentvs.DependentSamples

The Sum ofTwoDistributions

Assignment

Independent Samples:Comparing Means

Lecture 37Sections 11.1, 11.2, 11.4

Robb T. Koether

Hampden-Sydney College

Wed, Nov 5, 2008

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Outline

1 Homework Review

2 Introduction

3 Independent vs. Dependent Samples

4 The Sum of Two Distributions

5 Assignment

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Homework Review

Exercise 10.21, page 648.

The value reported as lost for a random sample ofn = 20 pickpocket offenses occurring in a city is givenhere.

883 447 207 627 214313 844 253 587 217768 1064 549 833 277805 443 649 554 570

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Homework Review

Exercise 10.21, page 648.

(a) Use the data to construct a 95% confidence interval forthe mean value lost in all pickpocket offenses for thiscity.

(b) What is the margin of error for the interval estimate inpart (a)?

(c) Give an interpretation of the interval and of theconfidence level.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Homework Review

Solution(a) Use the TI-83 to find the mean and standard deviation

of the sample.

x = 555.2

s = 256.9.

Use the t table to find t0.025,19 = 2.093. The confidenceinterval is

x± t(

s√n

)= 555.2± 2.093

(256.9√

20

)= 555.2± 120.2.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Homework Review

Solution(b) The margin of error is 120.2.(c) The interpretation is that if we repeated this procedure

many times, about 95% of our confidence intervalswould contain the true value of µ.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Introduction

In this chapter we will consider problems that comparetwo populations.You may have seen much of this last night:

How did white Catholics vote compared to Protestants?How did union members vote compared to non-unionmembers?And so on.

The bad news: Things will get a bit more complicated.The good news: You will not need to memorize anymore formulas.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Dependent Samples

Definition (Bivariate data, paired data)Bivariate data are data in which each datum is a pair ofobservations. These are also called paired data. Typicallythe two values are called x and y.

Definition (Paired samples, dependent samples)If the data are paired, then the sample of x values and thesample of y values are called paired samples or dependentsamples.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Dependent Samples

Paired data are often “before” and “after” observations.By comparing the mean before treatment to the meanafter treatment, we can determine whether thetreatment had an effect.To make direct comparisons of the two samples, theymust be measuring the same sort of thing.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Independent Samples

On the other hand, with independent samples, wesimply take one sample from one population andanother sample from another population.There is no logical way to pair the data.Furthermore, the independent samples could be ofdifferent sizes.Clearly, paired samples must be of the same size.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Sum of Two Distributions

1 Suppose x and y are random variables, each with itsown distribution.

2 What is the distribution of their sum, x + y?

µx+y = µx + µy

σ2x+y = σ2

x + σ2y

σx+y =√σ2

x + σ2y

3 Furthermore, if x and y are both normal x + y is alsonormal.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Central Limit Theorem

1 This is the basis for the Central Limit Theorem.2 When we gather a sample of size n, we are getting n

values of identical random variables x1, x2, . . . , xn, eachwith mean µ and standard deviation σ.

3 The mean and standard deviation of their sum are

µx1+···+xn = µ+ · · ·+ µ = nµ.

σ2x1+···+xn

= σ2 + · · ·+ σ2 = nσ2.

σx1+···+xn =√

nσ2 =√

nσ.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Central Limit Theorem

1 Now divide x1 + · · ·+ xn by n:

x =1n

(x1 + · · ·+ xn) .

2 Its mean and standard deviation are

µx =nµn

= µ.

σ2x =

nσ2

n2 =σ2

n.

σx =

√σ2

n=

σ√n.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Sum of Two Distributions

1 That was interesting, but we need to know about thedifference between variables.

2 For the difference x− y, the situation is very similar tothe sum.

3 For any two random variables x and y

µx−y = µx − µy

σ2x−y = σ2

x + σ2y

σx−y =√σ2

x + σ2y

4 If x and y are both normal x− y is also normal.

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Distribution of x

10090 110 12080706050400 10 20 30

N(100, 6)

N(80, 8)

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Distribution of y

10090 110 12080706050400 10 20 30

N(100, 6)

N(80, 8)

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

The Distribution of x− y

10090 110 12080706050400 10 20 30

N(100, 6)

N(80, 8)N(20, 10)

IndependentSamples:

ComparingMeans

Robb T.Koether

HomeworkReview

Introduction



Assignment

Assignment

HomeworkRead Sections 11.1, 11.2, 11.4, pages 669 - 678, 694 -695.Let’s Do It! 11.1, 11.2, 11.3.Exercises 1 - 8, page 676.

Means Robb T. Independent Samples: Comparing Means

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