1. Part B Business Statistics: Communicating with Numbers
By Sanjiv Jaggia and Alison Kelly
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Chapter 10 Learning Objectives (LOs)
LO 10.1(Part A): Make inferences about the difference between two population means based on independent sampling. µ1 - µ2 LO 10.2 (Part B): Make inferences about the mean difference based on matched-pairs sampling. µd LO 10.3 (Part B): Make inferences about the difference between two population proportions based on independent sampling. P1 – P2

3. Chapter Case - Effectiveness of Mandatory Caloric Postings
In March 2010, federal health-care law required chain restaurants with 20 locations or more to post caloric information on their menus.
This would make it easier for consumers to select healthier food options.
Nutritionist Molly Hosler would like to study the effects of a recent local menu ordinance requiring caloric postings in San Mateo, California.
Molly obtains transaction data for 40 Starbucks cardholders around the time that San Mateo implemented the ordinance.

4. Effectiveness of Mandatory Caloric Postings
Here is some of the sample data.
Molly wants to use the sample information to:
Determine whether average calories of purchased drinks declined after the passage of the ordinance. µd
Determine whether average calories of purchased food declined after the passage of the ordinance. µd
Assess the implications of caloric postings for Starbucks and other chains.

5. 10.1 Inference Concerning the Difference Between Two Means
LO 10.1 Make inferences about the difference between two population means based on independent sampling.
Independent Random Samples
Two (or more) random samples are considered independent if the process that generates one sample is completely separate from the process that generates the other sample.
The samples are clearly delineated.
m1 is the mean of the first population. m2 is the mean of the second population.

6. 10.2 Inference Concerning Mean Differences
LO 10.2 Make inferences about the mean difference based on matched-pairs sampling.
Matched-Pairs Sampling
Parameter of interest is the mean difference D where D = X1  X2 , and the random variables X1 and X2 are matched in a pair.
Both X1 and X2 are normally distributed or n > 30.
For example, assess the benefits of a new medical treatment by evaluating the same patients before (X1) and after (X2) the treatment.
Often more reliable data can be collected this way!

7. 10.2 Inference Concerning Mean Differences
LO 10.2
Recognizing a Matched-Pairs Experiment
“Before” and “after” studies characterized by a measurement, some type of intervention, and another measurement, all on the same subject.
Example: Measuring the weight of clients before and after a diet plan.
A pairing of observations, where it is not on the same subject that gets sampled twice.
Example: Matching 20 adjacent plots of land using a nonorganic fertilizer on one half of the plot and an organic fertilizer on the other.
This artificial process of create and collect data is called experiment.

8. 10.2 Inference Concerning Mean Differences
LO 10.2
Hypothesis Test for mD
When conducting hypothesis tests concerning mD, the competing hypotheses will take one of the following forms: where d0 typically is equal to 0.

9. 10.2 Inference Concerning Mean Differences
LO 10.2
Test Statistic for Hypothesis Tests About mD
The test statistic for hypothesis tests about mD is assumed to follow the tdf distribution with df = n  1, and its value is where and sD are the mean and standard deviation, respectively, of the n sample differences, and d0 is a given hypothesized mean difference.

10. 10.2 Inference Concerning Mean Differences
LO 10.2
Chapter Case
Here are data for 40 Starbucks cardholders.
At the 5% significance level, does the posting of caloric information reduce the intake of average food calories (mD > 0)?
This is a matched-pairs experiment where D = X1 – X2.
Let X1 denote drink calories before the ordinance and X2 denote drink calories after the ordinance.
The data file name is Food_Calories in S:drive.
The way to find differences and their mean and standard deviation is demonstrated in D2L

11. 10.2 Inference Concerning Mean Differences
LO 10.2
4 Steps of HT
State hypotheses (H0 and HA)
Find test statistic (t in this case)
Find critical or p-value
Conclusion and interpretation

12. 10.2 Inference Concerning Mean Differences
LO 10.2
4 Steps of HT
1. State hypotheses (H0 and HA)
H0: µD ≤ 0 HA: µD > 0
2. Find t test statistic
= 6.78
As shown in Excel demo,
Df= 39 d = d0 = 0 SD = 8.1 n = 40

13. Reduced calories 10.2 Inference Concerning Mean Differences
4 Steps of HT
3 & 4 Find critical and p-value
Right-tail test, level of significance (α) = 0.05
Df = 39
Critical value = 1.685
T test statistic falls in the tail – reject H0
p-value < 0.005,
smaller than α = 0.05 – reject H0
Reduced calories

14. 10.2 Inference Concerning Mean Differences
LO 10.2
Using Excel to test a hypothesis about mD.
Open the Food Calories data file.
In Excel, choose Data > Data Analysis > t-Test: Paired Two Sample for Means > OK.
In the resulting dialog, specify the required information as shown.
Click OK.
This is demonstrated in D2L

15. 10.2 Inference Concerning Mean Differences
LO 10.2
Using Excel to test a hypothesis about mD.
The Excel output gives both the p-values and critical values for a one-tail and two-tail test.
The relevant p-value for H0: mD < 0, HA: mD > 0, is 2.15E-08 or 0.
2.15E-08 < a = thus, we reject H0.
Conclude that the average food caloric intake has declined after the ordinance.

16. Practice
You can do the same test with Drink_calories data both manually and with Excel. It is done manually in Example 10.7 on p Differences and their mean and standard deviation were calculated with Excel.
Statistics and Data

17. 10.3 Inference Concerning the Difference Between Two Proportions
LO 10.3
Hypothesis Test for p1  p2
The null and alternative hypotheses for testing the difference between two population proportions under independent sampling will take one of the following forms:
Where d0 is a hypothesized difference (typically 0) between p1 and p2.

18. 10.2 Inference Concerning Mean Differences
LO 10.2
4 Steps of HT
State hypotheses (H0 and HA)
Find test statistic (t in this case)
Find critical or p-value
Conclusion and interpretation

19. 10.3 Inference Concerning the Difference Between Two Proportions
LO 10.3
The Test Statistic for Testing p1  p2
The test statistic is assumed to follow the z distribution.
If the hypothesized difference d0 is zero, then the value of the test statistic is where
More often the case

20. 10.3 Inference Concerning the Difference Between Two Proportions
LO 10.3
The Test Statistic for Testing p1  p2
The test statistic is assumed to follow the z distribution.
If the hypothesized difference d0 is not zero, then the value of the test statistic is

21. 10.3 Inference Concerning the Difference Between Two Proportions
LO 10.3
Example 10.9: Hypothesis test about p1  p2.
Three months ago, out of 120 registered voters, 55 said that they would vote for Candidate A.
Today, 41 registered voters in a sample of 80 said that they would vote for Candidate A.
Is today different from 3 months ago
Determine: H0: p1  p2 = 0, HA: p1  p2 ≠ 0
Where p1 is the population proportion of the electorate who support the candidate today and p2 is the proportion from three months ago.

22. 10.3 Inference Concerning the Difference Between Two Proportions
LO 10.3
H0: p1 − p2 = 0
HA: p1 − p2 ≠ 0
Step 1
Test statistic

23. 10.3 Inference Concerning the Difference Between Two Proportions
Z test statistic = 0.52
Steps 3 & 4
Critical value = ± 1.96 with α = 0.05
Z test stat falls in non-rejection area.
P-value = 2*P (Z > 0.52) = 0.603
P-value > α
Do not reject H0, hence no evidence of difference.
Step 2
Statistics and Data

24. 10.3 Inference Concerning the Difference Between Two Proportions
EXAMPLE 10.11
Recent research by analysts and retailers claims significant gender differences when it comes to online shopping (The Wall Street Journal, March 13, 2008). A survey revealed that 5,400 of 6,000 men said they “regularly” or “occasionally” make purchases online, compared with 8,600 of 10,000 women surveyed. At the 5% significance level, test whether the proportion of all men who regularly or occasionally make purchases online is greater than the proportion of all women.
Statistics and Data

25. 10.3 Inference Concerning the Difference Between Two Proportions
H0: p1 − p2 ≤ 0
HA: p1 − p2 > 0
Since the hypothesized difference is zero, or d0 = 0, we compute the value of the test statistic as
We first compute the sample proportions
= x1/n1 = 5,400/6,000 = 0.90
= x2/n2 = 8,600/10,000 = 0.86
Step 1
Statistics and Data

26. 10.3 Inference Concerning the Difference Between Two Proportions
Test stat
Step 2
For a right-tailed test with α = 0.05, the appropriate critical value is zα = z0.05 = The decision rule is to reject H0 if z > Since 7.41 > 1.645, we reject H0.
Also p-value = P(Z>7.41) = 0, which is smaller than α = Hence reject H0.
So there is evidence that the proportion of all men who regularly or occasionally make purchases online is greater than the proportion of all women.
Steps
3&4
Statistics and Data