Chapter 10 Statistical Inference About Means and Proportions With Two Populations

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LESSON 4
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Chapter 10 Statistical Inference About Means and
Proportions With Two Populations

• Inferences About the Difference Between
• Two Population Means s 1 and s 2 Known

Inferences About the Difference Between
Two Population Means s 1 and s 2 Unknown
Inferences About the Difference Between
Two Population Means Matched Samples
Inferences About the Difference Between
Two Population Proportions
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Inferences About the Difference BetweenTwo
Population Means s 1 and s 2 Known

• Interval Estimation of m 1 m 2
• Hypothesis Tests About m 1 m 2

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Point Estimator of the Difference Betweenthe
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 (Standard Error)
• 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 Estimation of ?1 - ?2 s 1 and s 2 Known

• Interval Estimate

where 1 - ? is the confidence coefficient
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Example Par, Inc.

• Interval Estimate of ?1 - ?2
• 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.
• The sample statistics appear on the next slide.

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Example Par, Inc.

• Interval Estimate of ?1 - ?2
• Sample Statistics
• Sample 1
  Sample 2
• Par, Inc. Rap, Ltd.
• Sample Size n1 120 balls n2 80 balls
• Mean 235 yards 218
  yards
• Standard Dev. ?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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Point Estimator of the Difference Between the
Means of Two Populations
Population 1 Par, Inc. Golf Balls m1 mean
driving distance of Par golf balls
Population 2 Rap, Ltd. Golf Balls m2 mean
driving distance of Rap golf balls
m1 -m2 difference between the mean distances
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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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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• Hypotheses

Left-tailed
Right-tailed
Two-tailed

• Test Statistic

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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• Example Par, Inc.

Can we conclude, using a .01, that the
mean driving distance of Par, Inc. golf
balls is greater than the mean driving distance
of Rap, Ltd. golf balls?
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• p Value and Critical Value Approaches

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

• where
• ?1 mean distance for the population
• of Par, Inc. golf balls
• ?2 mean distance for the population
• of Rap, Ltd. golf balls

2. Specify the level of significance.
a .01
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known
Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• p Value and Critical Value Approaches

3. Compute the value of the test statistic.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• p Value Approach

4. Compute the pvalue.
For z 6.49, the p value lt .0001.
5. Determine whether to reject H0.
Because pvalue lt a .01, we reject H0.
At the .01 level of significance, the sample
evidence indicates the mean driving distance of
Par, Inc. golf balls is greater than the mean
driving distance of Rap, Ltd. golf balls.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known
Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• Critical Value Approach

4. Determine the critical value and rejection
rule.
For a .01, z.01 2.33
Reject H0 if z gt 2.33
5. Determine whether to reject H0.
Because z 6.49 gt 2.33, we reject H0.
The sample evidence indicates the mean
driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap,
Ltd. golf balls.
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Inferences About the Difference BetweenTwo
Population Means s 1 and s 2 Unknown

• Interval Estimation of m 1 m 2
• Hypothesis Tests About m 1 m 2

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Interval Estimation of ?1 - ?2s 1 and s 2
Unknown
When s 1 and s 2 are unknown, we will

• use the sample standard deviations s1 and s2
• as estimates of s 1 and s 2 , and

• replace za/2 with ta/2.

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Interval Estimation of ?1 - ?2s 1 and s 2
Unknown

• Interval Estimate

Where the degrees of freedom for ta/2 are
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Difference Between Two Population Means s 1 and
s 2 Unknown

• Example Specific Motors

• Specific Motors of Detroit
• has developed a new automobile
• known as the M car. 24 M cars
• and 28 J cars (from Japan) were road
• tested to compare miles-per-gallon (mpg)
  performance.
• The sample statistics are shown on the next slide.

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Difference Between Two Population Means s 1 and
s 2 Unknown

• Example Specific Motors

Sample 1 M Cars
Sample 2 J Cars
Sample Size
24 cars 28 cars
Sample Mean
29.8 mpg 27.3 mpg
Sample Std. Dev.
2.56 mpg 1.81 mpg
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Difference Between Two Population Means s 1 and
s 2 Unknown

• Example Specific Motors

Let us develop a 90 confidence interval
estimate of the difference between the mpg
performances of the two models of automobile.
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Point Estimate of m 1 - m 2
Point estimate of ?1 - ?2
29.8 - 27.3
2.5 mpg
where ?1 mean miles-per-gallon for the
population of M cars ?2 mean
miles-per-gallon for the population of J
cars
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Interval Estimation of m 1 - m 2s 1 and s 2
Unknown
The degrees of freedom for ta/2 are
With a/2 .05 and df 41, ta/2 1.683
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Interval Estimation of m 1 - m 2s 1 and s 2
Unknown
Interval Estimation of m 1 - m 2s 1 and s 2
Unknown
2.5 1.051 or 1.449 to 3.551 mpg
We are 90 confident that the difference
between the miles-per-gallon performances of M
cars and J cars is 1.449 to 3.551 mpg.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Interval Estimation of m 1 - m 2s 1 and s 2
Unknown

• Hypotheses

Left-tailed
Right-tailed
Two-tailed

• Test Statistic

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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• Example Specific Motors

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?
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• p Value and Critical Value Approaches

1. Develop 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

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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• p Value and Critical Value Approaches

2. Specify the level of significance.
a .05
3. Compute the value of the test statistic.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• p Value Approach

4. Compute the p value.
The degrees of freedom for ta are
Because t 4.003 gt t.005 2.701, the pvalue lt
.005.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• p Value Approach

5. Determine whether to reject H0.
Because pvalue lt a .05, we reject H0.
We are at least 95 confident that the
miles-per-gallon (mpg) performance of M cars is
greater than the miles-per-gallon performance of
J cars?.
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• Critical Value Approach

4. Determine the critical value and rejection
rule.
For a .05 and df 41, t.05 1.683
Reject H0 if t gt 1.683
5. Determine whether to reject H0.
Because 4.003 gt 1.683, we reject H0.
We are at least 95 confident that the
miles-per-gallon (mpg) performance of M cars is
greater than the miles-per-gallon performance of
J cars?.
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Large-Sample Case
Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown

• Using Excel
• Select Tools pull-down menu
• Choose the Data analysis option
• Choose t-test Two Sample Assuming Unequal
  Variances from the list of Analysis tools
• Enter 0 in the Hypothesized Mean Difference box.

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• Salary 31 F Vs 43 M

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• Excel Output

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Interpretation

• Because the P-value P(Zltz) two-tail
  0.000556901, is less than the level of
  significance, ?.05, we have sufficient
  statistical evidence to reject the null
  hypothesis and conclude that the mean salary for
  Male and Female is not equal.

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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.

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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?
• Hypotheses H0 ?d 0,
• Rejection Rule Ha ?d ???

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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 Reject H0.
• There is a significant difference between the
  mean delivery times for the two services.

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Confidence Interval

• Confidence Interval

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Excel

• Select tools pull-down menu
• Choose Data analysis option
• Choose t-Test paired Two Sample for Means
• Enter 0 in the hypothesized Mean Difference Box
• Enter .05 in the alpha box (level of
  significance)
• Enter Cell in the Variable 1 Range Box
• Enter Cell in the Variable 2 Range Box

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Excel Output

• Excel Output

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Interpretation

• Because the P-value P(Tltt) two-tail 0.02, is
  less than the level of significance, ?.05, we
  have sufficient statistical evidence to reject
  the null hypothesis and conclude that There is a
  significant difference between the mean delivery
  times for the two services.

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Inferences About the Difference Between the
Proportions of Two Populations

• Sampling Distribution of
• Interval Estimation of p1 - p2
• Hypothesis Tests about p1 - p2

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Sampling Distribution of

• Expected Value
• Standard Deviation

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Interval Estimation of p1 - p2

• Interval Estimate
• Point Estimator of

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Example MRA

• MRA (Market Research Associates) is conducting
  research to evaluate the effectiveness of a
  clients new advertising campaign. Before the
  new campaign began, a telephone survey of 150
  households in the test market area showed 60
  households aware of the clients product. The
  new campaign has been initiated with TV and
  newspaper advertisements running for three weeks.
  A survey conducted immediately after the new
  campaign showed 120 of 250 households aware of
  the clients product.
• Does the data support the position that the
  advertising campaign has provided an increased
  awareness of the clients product?

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Example MRA

• Point Estimator of the Difference Between the
  Proportions of Two Populations
• p1 proportion of the population of households
• aware of the product after the new campaign
• p2 proportion of the population of
  households
• aware of the product before the new campaign
  
• sample proportion of households aware
  of the
• product after the new campaign
• sample proportion of households aware
  of the
• product before the new campaign

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Example MRA

• Interval Estimate of p1 - p2 Large-Sample Case
• For ?? .05, z.025 1.96
• .08 1.96(.0510)
• .08 .10
• or -.02 to .18
• Conclusion
• At a 95 confidence level, the interval
  estimate of the difference between the proportion
  of households aware of the clients product
  before and after the new advertising campaign is
  -.02 to .18.

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Hypothesis Tests about p1 - p2

• Hypotheses
• H0 p1 - p2 lt 0
• Ha p1 - p2 gt 0
• Test statistic
• Point Estimator of where p1 p2
• where

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Example MRA

• Hypothesis Tests about p1 - p2
• Can we conclude, using a .05 level of
  significance, that the proportion of households
  aware of the clients product increased after the
  new advertising campaign?
• p1 proportion of the population of households
• aware of the product after the new campaign
• p2 proportion of the population of
  households
• aware of the product before the new campaign
  
• Hypotheses H0 p1 - p2 lt 0
• Ha p1 - p2 gt 0

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Example MRA

• Hypothesis Tests about p1 - p2
• Rejection Rule Reject H0 if z gt 1.645
• Test Statistic
• Conclusion Do not reject H0.