Chapter%2010b

1
Chapter 10b

• Hypothesis Tests About the DifferenceBetween the
  Means of Two Populations Independent Samples,
  Small-Sample Case
• Using Excel to Conduct aHypothesis Test about µ1
  µ2 Small Sample
• Inference About the Difference between the Means
  of Two Populations Matched Samples

2
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Example Specific Motors
• Recall that Specific Motors of
• Detroit has developed a new
• automobile known as the M car.
• 12 M cars and 8 J cars (from Japan)
• were road tested to compare miles-per-gallon
  (mpg)
• performance. The sample statistics are shown on
  the
• next slide.

3
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Example Specific Motors

Sample 1 M Cars
Sample 2 J Cars
Sample Size
12 cars 8 cars
Sample Mean
29.8 mpg 27.3 mpg
Sample Std. Dev.
2.56 mpg 1.81 mpg
Can we conclude, using a .05 level of
significance, that the miles-per-gallon (mpg)
performance of M cars is greater than the
miles-per-gallon performance of J cars?
4
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the Test Statistic

1. Determine the hypotheses.
H0 ?1 - ?2 lt 0 ? Ha ?1 - ?2 gt 0

• where
• ?1 mean mpg for the population of M cars
• ?2 mean mpg for the population of J cars

5
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the Test Statistic

a .05
2. Specify the level of significance.
3. Select the test statistic.
where
4. State the rejection rule.
Reject H0 if t gt 1.734
(18 degrees of freedom)
6
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the Test Statistic

5. Compute the value of the test statistic.
Pooled Variance Estimator of s 2
7
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the Test Statistic

5. Compute the value of the test statistic.
(continued)
t Statistic
8
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the Test Statistic

6. Determine whether to reject H0.
t 2.384 gt t.05 1.734, so we reject H0.
At the .05 level of significance, the
sample evidence indicates that the mean mpg of M
cars is greater than the mean mpg of J cars.
9
Using Excel to Conduct aHypothesis Test about m1
m2 Small Sample

• Excels t-Test Two Sample Assuming Equal
  Variances Tool

Step 1 Select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose t-Test Two Sample Assuming
Equal Variances from the list of Analysis
Tools
10
Using Excel to Conduct aHypothesis Test about m1
m2 Small Sample

• Excels t-Test Two Sample Assuming Equal
  Variances Tool

Step 4 When the t-Test Two Sample Assuming
Equal Variances dialog box appears
Enter A1A13 in the Variable 1 Range box
Enter B1B9 in the Variable 2 Range box
Type 0 in the Hypothesized Mean Difference
box
11
Using Excel to Conduct aHypothesis Test about m1
m2 Small Sample

• Excels t-Test Two Sample Assuming Equal
  Variances Tool

Step 4 (continued)
Select Labels
Type .01 in the Alpha box
Select Output Range
Enter D1 in the Output Range box
(Any upper left-hand corner cell indicating where
the output is to begin may be entered)
Click OK
12
Using Excel to Conduct aHypothesis Test about m1
m2 Small Sample
13
Using Excel to Conduct aHypothesis Test about m1
m2 Small Sample

• Value Worksheet

14
Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples,
Small-Sample Case

• Using the p -Value

4. Compute the value of the test statistic.
The Excel worksheet shows t 2.369
5. Compute the pvalue.
The Excel worksheet shows p-value .0146
6. Determine whether to reject H0.
Because pvalue .0146 lt a .05, we reject H0.
15
Inference About the Difference between the Means
of Two Populations Matched Samples

• With a matched-sample design each sampled item
• provides a pair of data values.

• This design often leads to a smaller sampling
  error
• than the independent-sample design because
• variation between sampled items is
  eliminated as a
• source of sampling error.

16
Inference About the Difference between the Means
of Two Populations Matched Samples

• Example Express Deliveries
• A Chicago-based firm has
• documents that must be quickly
• distributed to district offices
• throughout the U.S. The firm
• must decide between two delivery
• services, UPX (United Parcel Express) and INTEX
• (International Express), to transport its
  documents.

17
Inference About the Difference between the Means
of Two Populations Matched Samples

• Example Express Deliveries
• In testing the delivery times
• of the two services, the firm sent
• two reports to a random sample
• of its district offices with one
• report carried by UPX and the
• other report carried by INTEX. Do the data on
  the
• next slide indicate a difference in mean
  delivery
• times for the two services? Use a .05 level of
  significance.

18
Inference About the Difference between the Means
of Two Populations Matched Samples
Delivery Time (Hours)
UPX
INTEX
Difference
District Office
32 30 19 16 15 18 14 10 7 16
25 24 15 15 13 15 15 8 9 11
7 6 4 1 2 3 -1 2 -2 5
Seattle Los Angeles Boston Cleveland New
York Houston Atlanta St. Louis Milwaukee Denver
19
Inference About the Difference between the Means
of Two Populations Matched Samples

• Using the Test Statistic

1. Determine the hypotheses.
H0 ?d 0 ? Ha ?d ???
Let ?d the mean of the difference values for
the two delivery services for the
population of district offices
20
Inference About the Difference between the Means
of Two Populations Matched Samples

• Using the Test Statistic

a .05
2. Specify the level of significance.
3. Select the test statistic.
and
where
4. State the rejection rule.
Reject H0 if t gt 2.262
(9 degrees of freedom)
21
Inference About the Difference between the Means
of Two Populations Matched Samples

• Using the Test Statistic

5. Compute the value of the test statistic.
22
Inference About the Difference between the Means
of Two Populations Matched Samples

• Using the Test Statistic

6. Determine whether to reject H0.
t 2.94 gt t.05/2 2.262, so we reject H0.
At the .05 level of significance, the sample
evidence indicates that there is a significant
difference between the mean delivery times for
the two services.
23
Using Excel to Conduct aHypothesis Test about m1
m2 Matched Samples

• Excels t-Test Paired Two Sample for Means
  Tool

Step 1 Select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose t-Test Paired Two Sample for
Means from the list of Analysis Tools
24
Using Excel to Conduct aHypothesis Test about m1
m2 Matched Samples

• Excels t-Test Paired Two Sample for Means
  Tool

Step 4 When the t-Test Paired Two Sample for
Means dialog box appears
Enter B1B11 in the Variable 1 Range box
Enter C1C11 in the Variable 2 Range box
Type 0 in the Hypothesized Mean Difference
box
Select Labels
Type .05 in the Alpha box
Select Output Range
Enter E2 (your choice) in the Output Range
box
Click OK
25
Using Excel to Conduct aHypothesis Test about m1
m2 Matched Samples
26
Using Excel to Conduct aHypothesis Test about m1
m2 Matched Samples

• Value Worksheet

27
Inference About the Difference between the Means
of Two Populations Matched Samples

• Using the p -Value

4. Compute the value of the test statistic.
The Excel worksheet shows t 2.9362
5. Compute the pvalue.
The Excel worksheet shows p-value .0166
6. Determine whether to reject H0.
Because pvalue .0166 lt a .05, we reject H0.