Chapter 10 Comparisons Involving Means

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Chapter 10 Comparisons Involving Means

• Inferences about the difference between two
  population means ?1 and ?2 known
• Inferences about the difference between two
  population means ?1 and ?2 unknown
• Inferences about the difference between two
  population means Matched Samples

?1
?2
2
Inferences about the Difference Between Two
Population Means ?1 and ?2 Known

• Interval estimate of ?????
• Using Excel to construct a confidence interval
• Hypothesis test about ?????
• Using Excel to conduct a hypothesis test

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Point Estimator of the Difference between the
Means of Two Populations

• Let ?1 equal the mean of population 1 and ?2
  equal the mean of population 2.
• The difference between the two population means
  is ?1 - ?2.
• To estimate ?1 - ?2, we will select a simple
  random sample of size n1 from population 1 and a
  simple random sample of size n2 from population
  2.
• Let equal the mean of sample 1 and equal
  the mean of sample 2.
• The point estimator of the difference between the
  means of the populations 1 and 2 is .

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Sampling Distribution of

• Properties of the Sampling Distribution of
• Expected Value
• Standard Deviation
• where ?1 standard deviation of population
  1
• ?2 standard deviation of population
  2
• n1 sample size from population 1
• n2 sample size from population 2

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Interval Estimate of ?1 - ?2 ?1 and ?2 Known

• Interval Estimate with ?1 and ?2 Known
• where
• 1 - ? is the confidence coefficient

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Example Par, Inc.

• Par, Inc. is a manufacturer of golf equipment
  and has developed a new golf ball that has been
  designed to
• provide extra distance. In a test of driving
  distance
• using a mechanical driving device, a sample of
  Par golf
• balls was compared with a sample of golf balls
  made by
• Rap, Ltd., a competitor. ?1 15, ?2 20.
• Sample 1 Sample 2
• Par, Inc. Rap, Ltd.
• Sample Size n1 120 balls n2 80 balls
• Mean 235 yards 218
  yards
• Standard Deviation ?1 15 yards ?2 20
  yards

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Example Par, Inc.

• Point Estimate of the Difference Between Two
  Population Means
• ?1 mean distance for the population of
• Par, Inc. golf balls
• ?2 mean distance for the population of
• Rap, Ltd. golf balls
• Point estimate of ?1 - ?2 235 -
  218 17 yards.

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Example Par, Inc.

• 95 Confidence Interval Estimate of the
  Difference Between Two Population Means ?1 and
  ?2 known
• 17 5.14 or 11.86 yards to 22.14 yards.
• We are 95 confident that the difference between
  the mean driving distances of Par, Inc. balls and
  Rap, Ltd. balls lies in the interval of 11.86 to
  22.14 yards.

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Now You Try, pg. 403, 1a., b.
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Hypothesis Tests about the Difference between the
Means of Two Populations ?1 and ?2 Known

• Hypotheses
• H0 ?1 - ?2 lt 0 H0 ?1 - ?2 gt 0
  H0 ?1 - ?2 0
• Ha ?1 - ?2 gt 0 Ha ?1 - ?2 lt 0
  Ha ?1 - ?2 ? 0
• Test Statistic

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Hypothesis Tests about the Difference between the
Means of Two Populations ?1 and ?2 Known

• Hypotheses
• H0 ?1 - ?2 lt 0 H0 ?1 - ?2 gt 0
  H0 ?1 - ?2 0
• Ha ?1 - ?2 gt 0 Ha ?1 - ?2 lt 0
  Ha ?1 - ?2 ? 0
• Rejection Rule
• One-tail test
• Upper tail
• Lower Tail
• Two-tail test

• Reject H0 if z gt z???

• Reject H0 if -z lt -z?

• Reject H0 if z gt z??? or if -z lt -z???

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One-Tailed Test
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Two-Tailed Test
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Example Par, Inc.

• Par, Inc. is a manufacturer of golf equipment and
  has developed a new golf ball that has been
  designed to
• provide extra distance. In a test of driving
  distance
• using a mechanical driving device, a sample of
  Par golf
• balls was compared with a sample of golf balls
  made by
• Rap, Ltd., a competitor. ?1 15 yards, ?2 20
  yards.
• Sample 1 Sample 2
• Par, Inc. Rap, Ltd.
• Sample Size n1 120 balls n2 80 balls
• Mean 235 yards 218
  yards
• Standard Deviation ?1 15 yards ?2 20
  yards

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Example Par, Inc.

• Hypothesis Tests About the Difference Between the
  Means of Two Populations Large-Sample Case
• Can we conclude, using a .01 level of
  significance, that the mean driving distance of
  Par, Inc. golf balls is greater than the mean
  driving distance of Rap, Ltd. golf balls?
• ?1 mean distance for the population of Par,
  Inc.
• golf balls
• ?2 mean distance for the population of Rap,
  Ltd.
• golf balls
• Hypotheses H0 ?1 - ?2 lt 0
• Ha ?1 - ?2 gt 0

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Example Par, Inc.

• Hypothesis Tests About the Difference Between the
  Means of Two Populations Large-Sample Case
• Rejection Rule
• Conclusion
• Reject H0. We are at least 99 confident
  that the mean driving distance of Par, Inc.
  golf balls is greater than the mean driving
  distance of Rap, Ltd. golf balls.

Reject H0 if z gt 2.33
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Using Excel to Conduct a Hypothesis Test about m1
m2

• Excels z-Test Two Sample for Means Tool
• Step 1 Select the Tools pull-down menu
• Step 2 Choose the Data Analysis option
• Step 3 Choose z-Test Two Sample for Means
• from the list of Analysis Tools
• continued

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Using Excel to Conduct a Hypothesis Test about m1
m2

• Excels z-Test Two Sample for Means Tool
• Step 4 When the z-Test Two Sample for Means
• dialog box appears
• Enter A1A121 in the Variable 1 Range
  box
• Enter B1A81 in the Variable 2 Range
  box
• Enter 0 in the Hypothesized Mean
  Difference box
• Enter 225 in the Variable 1 Variance
  (known) box
• Enter 400 in the Variable 1 Variance
  (known) box
• continued

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Using Excel to Conduct a Hypothesis Test about m1
m2

• Excels z-Test Two Sample for Means Tool
• Step 4 (continued)
• Select Labels
• Enter .01 in the Alpha box
• Select Output Range
• Enter D4 in the Output Range box
• (Any upper left-hand corner cell
  indicating
• where the output is to begin may be
  entered)
• Select OK

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Using Excel to Conduct a Hypothesis Test about m1
m2

• Value Worksheet

Note Rows 16-121 are not shown.
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Using Excel to Conduct a Hypothesis Test about m1
m2
Example Case Problem 1
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Interval Estimate of ?1 - ?2 ?1 and ?2 Unknown

• Interval Estimate of the difference between two
  population means with ?1 and ?2 Unknown
• where
• 1 - ? is the confidence coefficient

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Interval Estimate of ?1 - ?2 ?1 and ?2 Unknown

• Interval Estimate of the difference between two
  population means with ?1 and ?2 Unknown
• where
• 1 - ? is the confidence coefficient

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Example Specific Motors

• 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 data is
  below.
• Sample 1 Sample 2
• M Cars J Cars
• Sample Size n1 12 cars n2 8 cars
• Mean 29.8 mpg 27.3 mpg
• Standard Deviation s1 2.56 mpg s2
  1.81 mpg

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Example Specific Motors

• Point Estimate of the Difference Between Two
  Population Means
• ?1 mean miles-per-gallon for the population of
  
• M cars
• ?2 mean miles-per-gallon for the population of
• J cars
• Point estimate of ?1 - ?2 29.8 -
  27.3 2.5 mpg.

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Example Specific Motors

• 95 Confidence Interval Estimate of the
  Difference Between Two Population Means ?1 and
  ?2 Unknown

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Example Specific Motors

• 95 Confidence Interval Estimate of the
  Difference Between Two Population Means ?1 and
  ?2 Unknown
• t.025 (17 d.f.) 2.11

• We are 95 confident that the difference between
  the mean mpg ratings of the two car types is from
  .484 mpg to 4.516 mpg.

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Now You Try

• The following results are for independent random
  samples taken from two populations

• Construct the 95 confidence interval for the
  difference between the two population means.

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Hypothesis Tests about the Difference between the
Means of Two Populations ?1 and ?2 unknown

• Hypotheses
• H0 ?1 - ?2 lt 0 H0 ?1 - ?2 gt 0
  H0 ?1 - ?2 0
• Ha ?1 - ?2 gt 0 Ha ?1 - ?2 lt 0
  Ha ?1 - ?2 ? 0
• Test Statistic

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Hypothesis Tests about the Difference between the
Means of Two Populations ?1 and ?2 unknown

• Hypotheses
• H0 ?1 - ?2 lt 0 H0 ?1 - ?2 gt 0
  H0 ?1 - ?2 0
• Ha ?1 - ?2 gt 0 Ha ?1 - ?2 lt 0
  Ha ?1 - ?2 ? 0
• Rejection Rule
• One-tail test
• Upper tail
• Lower Tail
• Two-tail test

• Reject H0 if t gt t???

• Reject H0 if -t lt -t?

• Reject H0 if t gt t??? or if -t lt -t???

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Hypothesis Tests about the Difference between the
Means of Two Populations Independent Samples,
?1 and ?2 unknown

• Degrees of Freedom

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Example Starting Salaries

• Samples of starting annual salaries for
  individuals entering the public accounting and
  financial planning professions were collected to
  determine if there was any difference between the
  population mean starting annual salaries for the
  two professions. The sample data is below.
• Sample 1 Sample 2
• Public Accountant Financial Planner
• Sample Size n1 12 n2 14
• Mean 30.5
  27.0
• Standard Deviation s1 3.35 s2 2.64

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Example Starting Salaries

• Hypothesis Tests About the Difference Between the
  Means of Two Populations ?1 and ?2 unknown
• Can we conclude, using a .05 level of
  significance, that there is any difference
  between the population mean starting annual
  salaries of public accountants and financial
  planners ?
• ?1 mean starting annual salaries of public
  accountants
• ?2 mean starting annual salaries of financial
  planners
• Hypotheses H0 ?1 - ?2 0
• Ha ?1 - ?2 ? 0

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Example Starting Salaries

• Hypothesis Tests About the Difference Between the
  Means of Two Populations ?1 and ?2 unknown
• Rejection Rule
• Test Statistic

Reject H0 if t gt t??? or if -t lt -t??? or p-value
lt .05
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Example Starting Salaries

• Hypothesis Tests About the Difference Between the
  Means of Two Populations ?1 and ?2 unknown
• Rejection Rule
• Degrees of Freedom

Reject H0 if t gt t??? or if -t lt -t??? or p-value
lt .05
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Example Starting Salaries

• Hypothesis Tests About the Difference Between the
  Means of Two Populations ?1 and ?2 unknown
• Rejection Rule
• Conclusion

Reject H0 if t gt t??? or if -t lt -t??? or p-value
lt .05

• 2.92 gt 2.086. Therefore reject H0
• We are 95 sure that there is a difference
  between the population mean starting annual
  salaries of public accountants and financial
  planners.

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Using Excel to Conduct a Hypothesis Test about m1
m2Small Sample Size

• Excels t-Test Two Sample Assuming Equal
  Variances Tool
• Step 1 Select the Tools pull-down menu
• Step 2 Choose the Data Analysis option
• Step 3 Choose t-Test Two Sample Assuming Equal
• Variances from the list of Analysis Tools
• continued

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Using Excel to Conduct a Hypothesis Test about m1
m2Small Sample Size

• Excels t-Test Two Sample Assuming Equal
  Variances Tool
• Step 4 When the t-Test Two Sample Assuming
• Equal Variances dialog box appears
• Enter A!A13 in the Variable 1 Range Box
• Enter B1B15 in the Variable 2 Range box
• Type 0 in the Hypothesized Mean
• Difference box
• Select Labels

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Using Excel to Conduct a Hypothesis Test about m1
m2Small Sample Size

• Excels t-Test Two Sample Assuming Equal
  Variances Tool
• Step 4 (continued)
• Enter .05 in the Alpha box
• Select Output Range
• Enter D1 in the Output Range box (to
• identify the upper left corner of the
• section of the worksheet where the
• output will appear)
• Click OK

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Using Excel to Conduct a Hypothesis Test about m1
m2Small Sample Size
Test Statistic
Critical Values
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Using Excel to Conduct a Hypothesis Test about m1
m2Small Sample Size
p-values
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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.
• The matched-sample design can be referred to as
  blocking.
• 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.
• H0 ?1??2 0
• Ha ?1??2 ? 0

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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.
• The matched-sample design can be referred to as
  blocking.
• 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.
• H0 ?d 0
• Ha ?d ???

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Example Express Deliveries

• Inference About the Difference Between the Means
  of Two Populations Matched Samples
• 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. In testing the delivery
  times of the two services, the firm sent two
  reports to a random sample of ten district
  offices with one report carried by UPX and the
  other report carried by INTEX.
• Do the data that follow indicate a difference
  in mean delivery times for the two services?

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Example Express Deliveries

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

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Example Express Deliveries

• Inference About the Difference Between the Means
  of Two Populations Matched Samples
• Let ?d the mean of the difference values
  for the two delivery services
  for the population of district
  offices
• Hypotheses H0 ?d 0, Ha ?d ???
• Rejection Rule
• Assuming the population of difference
  values is approximately normally
  distributed, the t distribution with n - 1
  degrees of freedom applies. With ? .05, t.025
  2.262 (9 degrees of freedom).
• Reject H0 if t lt -2.262 or if t gt 2.262

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Example Express Deliveries

• Inference About the Difference Between the Means
  of Two Populations Matched Samples
• Conclusion 2.94 gt 2.262 Reject H0.
• There is a significant difference between the
  mean delivery times for the two services.

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Using Excel to Conduct a Hypothesis Test about
m1 m2 Matched Samples

• Excels t-Test Paired Two Sample for Means
  Tool
• Step 1 Select the Tools pull-down menu
• Step 2 Choose the Data Analysis option
• Step 3 Choose t-Test Paired Two Sample for
  Means
• from the list of Analysis Tools
• continued

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Using Excel to Conduct a Hypothesis 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
• Enter 0 in the Hypothesized Mean
  Difference box
• Select Labels
• Enter .05 in the Alpha box
• Select Output Range
• Enter E2 (your choice) in the Output
  Range box
• Select OK

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Using Excel to Conduct a Hypothesis Test about
m1 m2 Matched Samples

• Value Worksheet