Unformatted text preview: Chapter 10: Hypothesis Testing II Comparison of Two Population Means
Overview s Independent Samples • Interval estimation of the difference between the means of two populations • Hypothesis tests about the difference between the means of two populations s Matched Samples • Inferences about the difference between the means of two populations ? µ = µ Slide 1 Difference Between the Means of Two Populations: Independent Samples
Population 1 µ 1, σ 1 Population 2 µ 2, σ 2 µ1 − µ 2 ? Sample 1 Sample 2 x1 , s1 , n1 x2 , s2 , n2 Sampling Distribution of ¯1  ¯2 xx Slide 2 Expectation and Variance of X 1 − X 2
s Expected Value: Variance: E ( X 1 − X 2 ) = µ1 − µ2 s Var( X 1 − X 2 ) = Var( X 1 ) + Var( X 2 )
2 σ 12 σ 2 = + n1 n2 s Standard Deviation: σ X1 − X 2 σ 12 σ 22 = + n1 n2 Slide 3 Summary of Interval Estimation Procedures About the Difference in Popln. Means Slide 4 σ 1 and σ 2 KNOWN: Interval Estimate About µ 1 µ 2
LargeSample Case (n1 > 30 and n2 > 30)
s Invoke Central Limit Theorem (CLT). Interval Estimate: s x1 − x2 ± zα / 2 σ x1 − x2
where: σ x1 − x2 2 σ1 σ 2 = +2 n1 n2 NOTE: Use z distribution Slide 5 σ 1 and σ 2 KNOWN: Interval Estimate About µ 1 µ 2
SmallSample Case (n1 < 30 and/or n2 < 30)
s s Assume Both Populations have Normal Distributions • That is they assume bellshape! Interval Estimate: x1 − x2 ± zα / 2 σ x1 − x2
where: σ x1 − x2 2 σ1 σ 2 = +2 n1 n2 NOTE: Use z distribution Slide 6 σ 1 and σ 2 UNKNOWN: Interval Estimate About µ 1 µ 2
LargeSample Case (n1 > 30 and n2 > 30)
s Interval Estimate (Assume Normal Distribution) x1 − x2 ± zα / 2 sx1 − x2 Where: OR
2 2 s1 s2 = + n1 n2 x1 − x2 ± tα / 2 sx1 − x2 sx1 − x2 1 α = Confidence Coefficient NOTE: Use Z or t distribution Slide 7 σ 1 and σ 2 UNKNOWN: Interval Estimate About µ 1 µ 2
SmallSample Case (n1 < 30 and/or n2 < 30)
s If σ and σ Unknown, make following assumptions: s σ1 = σ 2 = σ • Both populations have normal distributions • Variances are equal: + 1 ) 1 σ x1 − x2 = σ 2 ( Then n1 n2
2 2 2 1 2 s Interval Estimate with σ and σ unknown and small x1 − x2 ± tα / 2 sx1 − x2
1 2 sample case:
sx1 − x2 11 = s( + ) n1 n2
2 2 2 ( n1 − 1) s1 + ( n2 − 1) s2 s= n1 + n2 − 2 2 NOTE: Use t distribution Slide 8 Interval Estimate of µ 1 µ 2
Example 1: Par, Inc. Par, Inc. is a manufacturer of golf equipment. Par 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 data is given below. Sample #1
Par, Inc. Par, Sample Size Mean Standard Deviation Standard n1 = 120 balls = 235 yards x1 = 15 yards s1 Sample #2 Sample Rap, Ltd. Rap, n2 = 80 balls = 218 yards x22= 20 yards s Slide 9 Example 1: Par, Inc.
s Point Estimate of the Difference Between Two Population Means µ 1 = mean distance for the population of Par, Inc. golf balls µ 2 = mean distance for the population of Rap, Ltd. golf balls (a) What is the point estimate of µ 1 µ 2? Slide 10 Example 1: Par, Inc.
(b) Compute the 95% Confidence Interval Estimate of the Difference Between Two Population Means.
s 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 Slide 11 Example 2 : 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 #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 Sample Size Mean Standard Deviation s2 = 1.81 mpg Slide 12 Example 2: Specific Motors
s Point Estimate of the Difference Between Two Population Means M cars µ 2 = mean milespergallon for the population of J cars (a) What is the point estimate of µ 1 µ 2? µ 1 = mean milespergallon for the population of Slide 13 Example 2: 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 14 Example 2: Specific Motors
(b) Construct the 95% Confidence Interval Estimate of the Difference Between Two Population Means.
s We are 95% confident that the difference between the mean mpg ratings of the two car types is from Slide 15 Summary of Interval Estimation Procedures About the Difference in Popln. Means Slide 16 Independent Samples (LargeSample and SmallSample Cases)
s σ1, σ2 KNOWN: Hypothesis Tests About µ 1 µ 2: Hypotheses H0: µ 1 µ 2 < 0 H0: µ 1 µ 2 > 0 H0: µ 1 µ 2 = 0 Ha: µ 1 µ 2 > 0 Ha: µ 1 µ 2 < 0 Ha: µ 1 µ 2 ≠ 0 Test Statistic n1 > 30 n2 < 30
s Invoke CLT Assume Normal Dist. ( x1 − x2 ) − ( µ1 − µ 2 ) ( x1 − x2 ) − ( µ1 − µ 2 ) z= z= 2 2 2 σ1 n1 + σ 2 n2 σ1 n1 + σ 2 n2 2 NOTE: Use z distribution Slide 17 σ 1, σ 2 KNOWN (Large and Small Samples)
s Rejection Rule z > zα , I. Reject if H 0 : µ1 − µ 2 ≤ 0 Or, pvalue = P (Z>z) < α. H 0 : µ1 − µ 2 ≥ 0 z < − zα , II. Reject if Or, pvalue = P (Z<z) < α . z < − zα / 2 H 0 : µ1 − µ2 = 0 III. Reject if Or z > zα / 2 , α. Or, pvalue = 2P (Z>z) < Slide 18 σ 1, σ 2 UNKNOWN: Hypothesis Tests About µ 1 µ 2: Independent Samples (LargeSample Case: n1 > 30 and n2 > 30)
s Hypotheses H0: µ 1 µ 2 < 0 H0: µ 1 µ 2 > 0 H0: µ 1 µ 2 = 0 Ha: µ 1 µ 2 > 0 Ha: µ 1 µ 2 < 0 Ha: µ 1 µ 2 ≠ 0 Test Statistic s z= distribution( µ − µ ) ( x1 − x2 ) − 1 2
2 σ1 n1 + σ 2 n2 2 Z distribution Z distribution OR
t= t t
2 s12 n1 + s2 n2 ( x1 − x2 ) − ( µ1 − µ 2 ) NOTE: Use Z or t distribution Slide 19 Independent Samples (SmallSample Case: n1 < 30 and/or n2 < 30)
s σ1, σ2 UNKNOWN: Hypothesis Tests About µ 1 µ 2: Test Statistic: Assume Normal Distributions
t= ( x1 − x2 ) − ( µ1 − µ 2 ) s2 (1 n1 + 1 n2 ) where 2 2 ( n1 − 1) s1 + ( n2 − 1) s2 s= n1 + n2 − 2 2 s For σ and σ Unknown (and smallsample case), make the following assumptions: 2 σ 12 = σ 2 = σ 2 • Both populations have normal distributions • Variances are equal: NOTE: Use t distribution
1 2 Slide 20 σ 1, σ 2 UNKNOWN: Hypothesis Tests About µ 1 µ 2: Independent s Rejection Rule (when σ , σ known): Same as Slide 15.
1 2 Samples (SmallSample Case: n1 < 30 and/or n2 < 30) s t > tα , n1 + Rejection Rule (when ≤ 0 , σ unknown)n2 −2 , H 0 : µ1 − µ2 σ
1 2 P(t > t obs ) < α . I. Reject if t < −tα, H 0 : µ1 − µ 2 ≥ 0 Or, pvalue = n1 +n2 −2 , P(t < t obs ) < α . II. Reject if H 0 : µ1 − µ2 = 0 Or, pvalue = t < −tα / 2,n1 +n2 −2 Or t > tα / 2,n1 +n2 −2 , 2P(t > t obs ) < α. III. Reject if Slide 21 Or, pvalue = Hypothesis Tests About the Difference Between the Means of Two Populations
Example 3: Par, Inc. Par, Inc. is a manufacturer of golf equipment. Par 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 data is given below. Sample #1
Par, Inc. Par, Sample Size Mean Standard Deviation Standard n1 = 120 balls = 235 yards x1 = 15 yards s1 Sample #2 Sample Rap, Ltd. Rap, n2 = 80 balls = 218 yards x2= 20 yards s2 Slide 22 Example 3: Par, Inc. 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? µ 1 = mean distance for the population of Par, Inc. golf balls µ 2 = mean distance for the population of Rap, Ltd. golf balls (a) What are the Hypotheses H and H ?
0 a Slide 23 Example 3: Par, Inc.
(b) Compute the Test Statistic. (c) What is the conclusion? 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. Slide 24 Example 4: 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 #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 Sample Size Mean Standard Deviation s2 = 1.81 mpg Slide 25 Example 4: Specific Motors
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? µ 1 = mean mpg for the population of M cars µ 2 = mean mpg for the population of J cars
(a) What are the Hypotheses H and H ? 0 a Slide 26 Example 4: Specific Motors
(b) Find the Test Statistic (c) What is the conclusion? We are at least 95% confident that the mean mpg of M cars is greater than the mean mpg of J cars. Slide 27 Inference About the Difference Between the Means of Two Populations: Matched Samples
s With a matchedsample design each sampled item provides a pair of data values. By considering the difference of each pair of the data values, we can apply the same procedures as the single population case. s Slide 28 Example 5: Express Deliveries
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 5: Express Deliveries
District Office
Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee 7 Denver Delivery Time (Hours) UPX INTEX Difference 32 25 30 24 19 15 16 15 15 13 18 15 14 15 10 8 9 16 11 Slide 30 Example 5: Express Deliveries
s Let µ d = the mean of the difference values for the two delivery services for the population of district offices Hypotheses: H0: µ d = 0, Ha: µ d ≠ 0 s s 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). Rejection Rule: Reject H0 if t < 2.262 or if t > 2.262 s Slide 31 Example 5: Express Deliveries
s Inference About the Difference Between the Means of Two Populations: Matched Samples ∑d d=
n i sd = (di − d ) 2 ∑ n −1 d − µd t= sd n • What is the conclusion?
There is a significant difference between the mean delivery times for the two services. Slide 32 Confidence Interval Approach to a TwoTailed Test about the Difference Between the Means of Two Populations: Matched Samples
s 100(1α)% confidence interval for µ d: d ± tα / 2 sd n s If the confidence interval contains the hypothesized value of µ d, do not reject H0. Otherwise, reject H0. Slide 33 ...
View
Full Document
 Summer '08
 Priya
 Standard Deviation, σ, interval estimate, Specific Motors

Click to edit the document details