Unformatted text preview: Chapter 10 Comparisons Involving Means
Estimation of the Difference Between the Means of Two Populations: Independent Samples s Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples
s Inferences about the Difference between the Means of Two Populations: Matched Samples
s =
µ1 ?
µ2 Slide 1 Estimation of the Difference Between the Means of Two Populations: Independent Samples
s s
s
s Point Estimator of the Difference between the Means of Two Populations
x1 − x2
Sampling Distribution of
Interval Estimate of µ1 −µ2 : LargeSample Case
LargeSample Case
Interval Estimate of µ1 −µ2 : SmallSample Case
SmallSample Case Slide 2 Point Estimator of the Difference Between
the Means of Two Populations
s s s s s Let µ1 equal the mean of population 1 and µ2 equal the mean of population 2.
The difference between the two population means is µ1 µ 2.
To estimate µ1 µ2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2.
x2
x1
Let equal the mean of sample 1 and equal the mean of sample 2.
The point estimator of the difference between the means x1 − x2
of the populations 1 and 2 is . Slide 3 Sampling Distribution of x1 − x2
s Properties of the Sampling Distribution of • Expected Value x1 − x2 E ( x1 − x2 ) = µ1 − µ 2 • Standard Deviation σ x1 − x2 2
σ1 σ 2
=
+2
n1 n2 where: σ1 = standard deviation of population 1 σ2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2 Slide 4 100(1 α) % Confidence Interval of µ 1 µ 2:
LargeSample Case (n1 > 30 and n2 > 30)
s Interval Estimate with σ1 and σ2 Known
where: s x1 − x2 ± zα / 2 σ x1 − x2 1 α is the confidence coefficient
Interval Estimate with σ1 and σ2 Unknown x1 − x2 ± zα / 2 sx1 − x2
where: sx1 − x2 2
2
s1 s2
=
+
n1 n2 Slide 5 Example: Par, Inc.
s Interval Estimate of µ1 µ2: LargeSample Case 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. Slide 6 Example: Par, Inc.
s Interval Estimate of µ1 µ2: LargeSample Case • Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd.
Sample Size
n1 = 120 balls n2 = 80 balls
Mean
Standard Dev. x x1
2 = 235 yards = 218 yards s1 = 15 yards s2 = 20 yards Slide 7 Example: Par, Inc.
s Point Estimate of the Difference Between Two Population Means
µ1 = mean distance for the population of Par, Inc. golf balls
µ2 = mean distance for the population of Rap, Ltd. golf balls
(a) What is the point estimate of µ1 µ2? Slide 8 Point Estimator of the Difference Between the Means of Two Populations
Population 1
Population 2
Population 1
Population 2
Par, Inc. Golf Balls
Rap, Ltd. Golf Balls
Par, Inc. Golf Balls
µ1 = mean driving
µ2 = mean driving
1 = mean driving
2 distance of Par distance of Rap golf balls
golf balls
golf balls
µ – µ2 = difference between
= difference between
1 the mean distances Simple random sample of n1 Par golf balls of n1 Par golf balls Simple random sample of n2 Rap golf balls n2 x1 = sample mean distance
x1 = sample mean distance
for sample of Par golf ball
for sample of Par golf ball x2 = sample mean distance
x2 = sample mean distance
for sample of Rap golf ball
for sample of Rap golf ball x1 x2 = Point Estimate of µ – µ2
1 Slide 9 Example: Par, Inc.
s (b) Compute the 95% Confidence Interval Estimate of the Difference Between Two Population Means: LargeSample Case, σ1 and σ2 Unknown 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 yards. Slide 10 100(1 α) % Confidence Interval of µ 1 µ 2:
SmallSample Case (n1 < 30 and/or n2 < 30)
s Assume both populations have normal distributions. s Interval Estimate with σ1 and σ2 Known x1 − x2 ± zα / 2 σ x1 − x2
where: σ x1 − x2 2
σ1 σ 2
=
+2
n1 n2 Slide 11 100(1 α) % Confidence Interval of µ 1 µ 2:
SmallSample Case (n1 < 30 and/or n2 < 30)
s If σ1 and σ2 Unknown, make following assumptions: •
• s s Both populations have normal distributions
2
2 σ
= σ2 = σ 2
Variances are equal:
1
11
Then σ x1 − x2 = σ 2 ( + )
n1 n2
Interval Estimate with σ1 and σ2 Unknown x1 − x2 ± tα / 2 sx1 − x2
where:
sx1 − x2 11
= s( + )
n1 n2
2 2
2
( n1 − 1) s1 + ( n2 − 1) s2
s2 =
n1 + n2 − 2 Slide 12 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 milesper
gallon (mpg) performance. The sample statistics are: Sample Size
Mean
Standard Deviation Sample #1 Sample #2 M Cars J Cars n1 = 12 cars n2 = 8 cars
x1
x2 = 29.8 mpg = 27.3 mpg s1 = 2.56 mpg s2 = 1.81 mpg Slide 13 Example: Specific Motors
s Point Estimate of the Difference Between Two Population Means µ1 = mean milespergallon for the population of M cars
µ2 = mean milespergallon for the population of J cars
(a) What is the point estimate of µ1 µ2? Slide 14 Example: Specific Motors
s 95% Confidence Interval Estimate of the Difference Between Two Population Means: SmallSample Case
We will make the following assumptions:
• The miles per gallon rating must be normally distributed for both the M car and the J car.
• The variance in the miles per gallon rating must be the same for both the M car and the J car.
Using the t distribution with n1 + n2 2 = 18 degrees
of freedom, the appropriate t value is t.025 = 2.101.
We will use a weighted average of the two sample
variances as the pooled estimator of σ 2. Slide 15 Example: Specific Motors
s (b) Construct the 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small
Sample Case We are 95% confident that the difference between the
mean mpg ratings of the two car types is from mpg . Slide 16 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples (LargeSample Case)
s Hypotheses
H0: µ1 µ2 < 0 H0: µ1 µ2 > 0 H0: µ1 µ2 = 0
Ha: µ1 µ2 > 0 Ha: µ1 µ2 < 0 Ha: µ1 µ2 ≠ 0 Test Statistic LargeSample Case (n1 > 30 and n2 > 30)
s σ1, σ2 known σ1, σ2 unknown
( x1 − x2 ) − ( µ1 − µ 2 )
( x1 − x2 ) − ( µ1 − µ 2 )
z=
z=
2
2
σ1 n1 + σ 2 n2
s12 n1 + s2 n2
2 Slide 17 LargeSample Case (n1 > 30 and n2 > 30)
s Rejection Rule Slide 18 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples (SmallSample Case)
s Test Statistic: SmallSample Case (n1 < 30 and/or n2 < 30) σ1, σ2 known σ1, σ2 unknown
( x1 − x2 ) − ( µ1 − µ 2 )
( x1 − x2 ) − ( µ1 − µ 2 )
z=
t=
2
2
σ1 n1 + σ 2 n2
s2 (1 n1 + 1 n2 ) where s 2
2
( n1 − 1) s1 + ( n2 − 1) s2
s2 =
n1 + n2 − 2 If σ1 and σ2 Unknown, make the assumptions: •
• Both populations have normal distributions
σ 12 = σ 22 = σ 2
Variances are equal: Slide 19 SmallSample Case (n1 < 30 and/or n2 < 30)
s Rejection Rule Slide 20 Example: Par, Inc.
s Hypothesis Tests About the Difference Between the Means of Two Populations: LargeSample Case 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. Slide 21 Example: Par, Inc.
s Hypothesis Tests About the Difference Between the Means of Two Populations: LargeSample Case
• Sample Statistics Sample #1 Par, Inc. Sample Size
n1 = 120 balls
Mean
Standard Dev. Sample #2 Rap, Ltd. n2 = 80 balls x1 = 235 yards s1 = 15 yards x2 = 218 yards s2 = 20 yards Slide 22 Example: Par, Inc.
Hypothesis Tests About the Difference Between the Means of Two Populations: LargeSample Case Q. 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?
µ1 = mean distance for the population of Par, Inc.
s golf balls
µ2 = mean distance for the population of Rap, Ltd.
golf balls
(a) What are the Hypotheses H0 and Ha? Slide 23 Example: Par, Inc.
s Hypothesis Tests About the Difference Between the Means of Two Populations: LargeSample Case
(b) Compute the Test Statistic (c) What is the conclusion? Slide 24 Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 12 M cars and 8 J cars
(fromJapan) were road tested to compare milesper
gallon (mpg) performance. The sample statistics are: Sample Size
Mean
Standard Deviation Sample #1 Sample #2 M Cars J Cars n1 = 12 cars n2 = 8 cars
x1
x2 = 29.8 mpg = 27.3 mpg s1 = 2.56 mpg s2 = 1.81 mpg Slide 25 Example: Specific Motors
Hypothesis Tests About the Difference Between the Means of Two Populations: SmallSample Case
Q. Can we conclude, using a .05 level of significance, that the milespergallon (mpg) performance of M cars is greater than the milespergallon performance of J cars?
µ1 = mean mpg for the population of M cars s µ2 = mean mpg for the population of J cars
(a) What are the Hypotheses H0 and Ha? Slide 26 Example: Specific Motors
s Hypothesis Tests About the Difference Between the Means of Two Populations: SmallSample Case (b) Find the Test Statistic (c) What is the conclusion? Slide 27 Inference About the Difference Between the Means of Two Populations: Matched Samples
s s
s With a matchedsample design each sampled item provides a pair of data values.
The matchedsample design can be referred to as blocking.
This design often leads to a smaller sampling error than the independentsample design because variation between sampled items is eliminated as a source of sampling error. Slide 28 Example: Express Deliveries
s Inference About the Difference Between the Means of Two Populations: Matched Samples A Chicagobased 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? Slide 29 Example: Express Deliveries
District Office Delivery Time (Hours)
UPX INTEX Difference Seattle
Los Angeles
Boston
Cleveland New York Houston Atlanta
St. Louis Milwaukee 7
Denver 32 25 30 24 19 15 16 15 15 13 18 15 14 15 10 8 9 16 11 Slide 30 Example: Express Deliveries
s Inference About the Difference Between the Means of Two Populations: Matched Samples Let µd = the mean of the difference values for the two delivery services for the population of district offices •
• Hypotheses H0: µd = 0, Ha: µd ≠ 0 Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n 1 degrees of freedom applies. With α = .05, t.025 = 2.262 (9 degrees of freedom). Reject H0 if t < 2.262 or if t > 2.262 Slide 31 Example: Express Deliveries
s Inference About the Difference Between the Means of Two Populations: Matched Samples
∑di
d=
n
sd =
t= • (di − d ) 2
∑
n −1 d − µd
sd
n What is the conclusion? Slide 32 ...
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