copyright © 2011 pearson education, inc. comparison chapter 18

Copyright © 2011 Pearson Education, Inc.

Comparison

Chapter 18

18.1 Data for Comparisons

A fitness chain is considering licensing a proprietary diet at a cost of $200,000. Is it more effective than the conventional free government recommended food pyramid?

Use inferential statistics to test for differences between two populations

Test for the difference between two means


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Comparison of Two Diets

Frame as a test of the difference between the means of two populations (mean number of pounds lost on Atkins versus conventional diets)

Let µA denote the mean weight loss in the population if members go on the Atkins diet and µC denote the mean weight loss in the population if members go on the conventional diet.


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In order to be profitable for the fitness chain, the Atkins diet has to win by more than 2 pounds, on average.

State the hypotheses as: H0: µA - µC ≤ 2

HA: µA - µC > 2


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Data used to compare two groups typically arise in one of three ways:

1. Run an experiment that isolates a specific cause.2. Obtain random samples from two populations.3. Compare two sets of observations.


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Experiments

Experiment: procedure that uses randomization to produce data that reveal causation.

Factor: a variable manipulated to discover its effect on a second variable, the response.

Treatment: a level of a factor.


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Experiments

In the ideal experiment, the experimenter

1. Selects a random sample from a population.2. Assigns subjects at random to treatments defined

by the factor.3. Compares the response of subjects between

treatments.


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The factor in the comparison of diets is the diet offered.

There are two treatments: Atkins and conventional.

The response is the amount of weight lost (measured in pounds).


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Confounding

Confounding: mixing the effects of two or more factors when comparing treatments.

Randomization eliminates confounding.

If it is not possible to randomize, then sample independently from two populations.


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18.2 Two-Sample t - Test

Two-Sample t – Statistic

with approximate degrees of freedom calculated using software.


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)(

)(

21

021

xxse

Dxxt


Two-Sample t – Test Summary


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Two-Sample t – Test Checklist

No obvious lurking variables. SRS condition. Similar variances. While the test allows the

variances to be different, should notice if they are similar.

Sample size condition. Each sample must satisfy this condition.


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4M Example 18.1: COMPARING TWO DIETS

Motivation

Scientists at U Penn selected 63 subjects from the local population of obese adults. They randomly assigned 33 to the Atkins diet and 30 to the conventional diet. Do the results show that the Atkins diet is worth licensing?


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Method

Use the two-sample t-test with α = 0.05. The hypotheses are

H0: µA - µC ≤ 2 HA: µA - µC > 2.


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4M Example 18.1: COMPARING TWO DIETSMethod – Check Conditions

Since the interquartile ranges of the boxplots appear similar, we can assume similar variances.


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4M Example 18.1: COMPARING TWO DIETSMethod – Check Conditions

No obvious lurking variables because of randomization. SRS condition satisfied. Both samples meet the sample size condition.


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Mechanics

with 60.8255 df and p-value = 0.0308; reject H0


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91.1369.3

2)00.742.15(

t


Message

The experiment shows that the average weight loss of obese adults on the Atkins diet does exceed the average weight loss of obese adults on the conventional diet. Unless the fitness chain’s membership resembles this population (obese adults), these results may not apply.


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18.3 Confidence Interval for the Difference

95% Confidence Intervals for µA and µC


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95% Confidence Intervals for µA and µC

The confidence intervals overlap. If they were nonoverlapping, we could conclude a significant difference. However, this result is inconclusive.


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95% Confidence Interval for µ1 - µ2

The 100(1 – α)% two-sample confidence interval for the difference in means is


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)()( 212/21 xxsetxx


95% Confidence Interval for µA - µc

Since the 95% confidence interval for µA - µB does not include zero, the means are statistically significantly different (those on the Atkins diet lose on average between 1.7 and 15.2 pounds more than those on a conventional diet).


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4M Example 18.2: EVALUATING A PROMOTION

Motivation

To evaluate the effectiveness of a promotional offer, an overnight service pulled records for a random sample of 50 offices that received the promotion and a random sample of 75 that did not.


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Method

Use the two-sample t –interval. Let µyes denote the mean number of packages shipped by offices that received the promotion and µno denote the mean number of packages shipped by offices that did not.


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Method – Check Conditions

All conditions are satisfied with the exception of no obvious lurking variables. Since we don’t know how the overnight delivery service distributed the promotional offer, confounding is possible. For example, it could be the case that only larger offices received the promotion.


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Mechanics


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Message

The difference is statistically significant. Offices that received the promotion used the overnight service to ship from 4 to 21 more packages on average than those offices that did not receive the promotion. There is the possibility of a confounding effect.


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18.4 Other Comparisons

Comparisons Using Confidence Intervals

Other possible comparisons include comparing two proportions or comparing two means from paired data.

95% confidence intervals generally have the form

Estimated Difference ± 2 Estimated Standard Error of the Difference


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Comparing Proportions

The 100(1 – α)% confidence z-interval for p1- p2 is

.

Checklist: No obvious lurking variables.SRS condition.Sample size condition (for proportion).


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)ˆˆ()ˆˆ( 212/21 ppsezpp

4M Example 18.3: COLOR PREFERENCES

Motivation

A department store sampled customers from the east and west and each was shown designs for the coming fall season (one featuring red and the other violet). If customers in the two regions differ in their preferences, the buyer will have to do a special order for each district.


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Method

Data were collected on a random sample of 60 customers from the east and 72 from the west. Construct a 95% confidence interval for pE - pW.

SRS and sample size conditions are satisfied. However, can’t rule out a lurking variable (e.g., customers may be younger in the west compared to the east).


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Mechanics


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Mechanics

Based on the data,

and the 95% confidence interval is

0.1389 ±1.96 (0.08645) [-0.031 to 0.308]


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.1389.04444.05833.0ˆˆ WE pp


Message

There is no statistically significant difference between customers from the east and those from the west in their preferences for the two designs. The 95% confidence interval for the difference between proportions contains zero.


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Paired Comparisons

Paired comparison: a comparison of two treatments using dependent samples designed to be similar (e.g., the same individuals taste test Coke and Pepsi).

Pairing isolates the treatment effect by reducing random variation that can hide a difference.


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Paired Comparisons

Given paired data, we begin the analysis by forming the difference within each pair (i.e., di = xi – yi ).

A two-sample analysis becomes a one-sample analysis. Let denote the mean of the differences and sd their standard deviation.


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d


Paired Comparisons

The 100(1 - α)% confidence paired t- interval is

with n-1 df

Checklist: No obvious lurking variables.SRS condition.Sample size condition.


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n

std

d

n 1;2/

4M Example 18.4: SALES FORCE COMPARISON

Motivation

The merger of two pharmaceutical companies (A and B) allows senior management to eliminate one of the sales forces. Which one should the merged company eliminate?


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Method

Both sales forces market similar products and were organized into 20 comparable geographical districts. Use the differences obtained from subtracting sales for Division B from sales for Division A in each district to obtain a 95% confidence t-interval for µA - µB.


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Method – Check Conditions

Inspect histogram of differences:

All conditions are satisfied.


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Mechanics

The 95% t-interval for the mean differences does not include zero. There is a statistically significant difference.


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Message

On average, sales force B sells more per day than sales force A.

The high correlation (r = 0.97) of sales between Sales Force A and Sales Force B in these districts confirms the benefit of a paired comparison.


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Best Practices

Use experiments to discover causal relationships.

Plot your data.

Use a break-even analysis to formulate the null hypothesis.


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Best Practices (Continued)

Use one confidence interval for comparisons.

Compare the variances in the two samples.

Take advantage of paired comparisons.


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Pitfalls

Don’t forget confounding.

Do not assume that a confidence interval that includes zero means that the difference is zero.

Don’t confuse a two-sample comparison with a paired comparison.

Don’t think that equal sample sizes imply paired data.


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copyright © 2011 pearson education, inc. comparison chapter 18

Documents

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