copyright © 2014, 2011 pearson education, inc. 1 chapter 17 comparison

Copyright © 2014, 2011 Pearson Education, Inc. 1

Chapter 17Comparison


17.1 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



Comparison of Two Diets

Frame as a test of the difference between the proportions of two populations.

Let pA denote the population proportion on the Atkins diet who renew membership and pC denote the proportion on the conventional diet who renew membership.




The difference pA- pC measures the extra proportion who renew if on the Atkins diet.

To be profitable this difference must be more than 4%.

H0: pA- pC ≤ 0.04HA: pA- pC > 0.04



Comparison of Two DietsData used to compare two groups arise from:

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

Method 3 usually not reliable.



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.



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.




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

It has two levels: Atkins and conventional.

The response is whether dieters renew membership in the fitness center.



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.


17.2 Two-Sample z-Test for Proportions

Two-Sample z – Statistic

using the estimated standard error of the difference between the sample proportions.

)ˆˆ(

)ˆˆ(

21

021

ppse

Dppz



Two-Sample z – Test Summary



Two-Sample z – Test Checklist

No obvious lurking variables. SRS condition. Sample size condition. Observe at least 10

“successes” and 10 “failures in each sample:

and 10)ˆ1(,10ˆ 1111 pnpn

10)ˆ1(,10ˆ 2222 pnpn




The p-value is 0.053. Cannot reject H0 at α = 0.05;

can reject at α = 0.01.

0493.0220

)60.01(60.0

150

)72.01(72.0)ˆˆ(

CA ppse

62.10493.0

04.060.072.0

z


17.3 Two Sample Confidence Interval for Proportions

Summary Statistics – Diet Comparison


17.3 Two-Sample Confidence Interval for Proportions


Two 95% confidence intervals (one for each sample in the diet comparison) overlap indicating no significant difference.



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

.

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

)ˆˆ()ˆˆ( 212/21 ppsezpp




The 95% confidence interval for the difference between the proportions who renew on the Atkins and conventional diets is between 0.023 and 0.217; it does not contain zero.

)ˆˆ()ˆˆ( 212/21 ppsezpp

)0493.0(96.1)60.072.0(



Interpreting the Confidence Interval

When the 95% confidence interval does not include zero, we say that the two proportions are statistically significantly different from each other.

Members on the Atkins diet renew memberships at a statistically significantly higher rate than those on the conventional diet.


4M Example 17.1: 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.


4M Example 17.1: COLOR PREFERENCESMethod

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).



Mechanics



Mechanics

Based on the data,

and the 95% confidence interval is

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

.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. Although the 95% confidence interval for the difference between proportions contains zero, before ordering the same lineup for both regions, consider gathering larger samples to narrow the confidence interval.


17.4 Two-Sample t - Test


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.




The null hypothesis specifies that the difference between population means is less than or equal to a predetermined constant D0.

To compare diets D0 = 5 pounds

H0: µA- µC ≤ 5HA: µA- µC > 5



Two-Sample t – Statistic

with approximate degrees of freedom calculated using software.

)(

)(

21

021

XXse

DXXt



Two-Sample t – Test Summary



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.


4M Example 17.2: 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 at α = 0.05 that the Atkins diet is worth the extra effort and produces 5 more pounds of weight loss?



Method

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

H0: µA - µC ≤ 5 HA: µA - µC > 5



Method – Check Conditions

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




No obvious lurking variables because of randomization.

SRS condition satisfied. Both samples meet the sample size

condition.



Mechanics

with 60.8255 df, p-value = 1572; cannot reject H0

015.1369.3

5)00.742.15(

t



Message

The experiment shows that the average weight loss of obese adults on the Atkins diet exceeds the average weight loss of obese adults on the conventional diet by 5 pounds. The difference is not statistically significant. Unless the fitness chain’s membership resembles this population (obese adults), these results may not apply.


17.5 Confidence Interval for the Difference between Means





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



The 100(1 – α)% confidence t-interval for µ1- µ2 is

.

Checklist: No obvious lurking variables.SRS condition.Similar variances.

Sample size condition.

)()( 212/21 XXsetXX



95% Confidence Interval for µA - µc

Since the 95% confidence interval for µA - µC 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).


4M Example 17.3: 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.



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.




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.



Mechanics



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.


17.6 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.

Randomization remains relevant.



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.

d



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

with n-1 df

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

n

std

d

n 1;2/


4M Example 17.4: 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?



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.




Inspect histogram of differences:

All conditions are satisfied.



Mechanics

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



Mechanics

The benefit of paired comparison; sales in these districts are highly correlated (r = 0.97).



Message

On average, sales force B sells more per day than sales force A. By comparing sales per representative, head to head in each district, a statistically significant difference in performance is detected.


Best Practices

Use experiments to discover causal relationships.

Plot your data.

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


Best Practices (Continued)

Use one confidence interval for comparisons.

Compare the variances in the two samples.

Take advantage of paired comparisons.


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.

copyright © 2014, 2011 pearson education, inc. 1 chapter 17 comparison

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