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統計學 Spring 2004

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統計學 Spring 2004

授課教師：統計系余清祥

日期：2004年3月2日

第三週：比較期望值

1 = 2 ?

ANOVA

- Estimation of the Difference Between the Means of
Two Populations: Independent Samples

- Hypothesis Tests about the Difference between the
Means of Two Populations: Independent Samples

- Inferences about the Difference between the Means
of Two Populations: Matched Samples

- Inferences about the Difference between the
Proportions of Two Populations:

- Point Estimator of the Difference between the Means of Two Populations
- Sampling Distribution
- Interval Estimate of Large-Sample Case
- Interval Estimate of Small-Sample Case

- 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 .

- 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

- Interval Estimate with 1 and 2 Known
where:

1 - is the confidence coefficient

- Interval Estimate with 1 and 2 Unknown
where:

- Interval Estimate of 1 - 2: Large-Sample 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.

- Interval Estimate of 1 - 2: Large-Sample Case
- Sample Statistics
Sample #1 Sample #2

Par, Inc. Rap, Ltd.

Sample Sizen1 = 120 balls n2 = 80 balls

Mean = 235 yards = 218 yards

Standard Dev. s1 = 15 yards s2 = 20 yards

- Sample Statistics

- 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.

Simple random sample

of n1 Par golf balls

x1 = sample mean distance

for sample of Par golf ball

Simple random sample

of n2 Rap golf balls

x2 = sample mean distance

for sample of Rap golf ball

x1 - x2 = Point Estimate of m1 –m2

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

- 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown
Substituting the sample standard deviations for the population standard deviation:

= 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.

- Interval Estimate with 2 Known
where:

- Interval Estimate with 2 Unknown
where:

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:

Sample #1 Sample #2

M CarsJ 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

- 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.

- 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample 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 2.

- The miles per gallon rating must be normally

- 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case
= 2.5 + 2.2 or .3 to 4.7 miles per gallon.

We are 95% confident that the difference between the

mean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car having the higher mpg).

- Hypotheses
H0: 1 - 2< 0 H0: 1 - 2> 0 H0: 1 - 2 = 0

Ha: 1 - 2 > 0 Ha: 1 - 2 < 0 Ha: 1 - 2 0

- Test Statistic
Large-Sample Small-Sample

- Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample 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.

- Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case
- Sample Statistics
Sample #1 Sample #2

Par, Inc. Rap, Ltd.

Sample Sizen1 = 120 balls n2 = 80 balls

Mean = 235 yards = 218 yards

Standard Dev. s1 = 15 yards s2 = 20 yards

- Sample Statistics

- 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

- HypothesesH0: 1 - 2< 0
Ha: 1 - 2 > 0

- HypothesesH0: 1 - 2< 0

- Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case
- Rejection RuleReject H0 if z > 2.33
- 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.

- Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case
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?

1 = mean mpg for the population of M cars

2 = mean mpg for the population of J cars

- HypothesesH0: 1 - 2< 0
Ha: 1 - 2 > 0

- HypothesesH0: 1 - 2< 0

- Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case
- Rejection Rule
Reject H0 if t > 1.734

(a = .05, d.f. = 18)

- Test Statistic
where:

- Rejection Rule

- 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.

- 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?

Delivery Time (Hours)

District OfficeUPXINTEXDifference

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

- 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

- HypothesesH0: 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 < -2.262 or if t > 2.262

- Inference About the Difference Between the Means of Two Populations: Matched Samples
- ConclusionReject H0.
There is a significant difference between the mean delivery times for the two services.

- ConclusionReject H0.

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

- Expected Value
- Standard Deviation
- Distribution Form
If the sample sizes are large (n1p1, n1(1 - p1), n2p2,

and n2(1 - p2) are all greater than or equal to 5), the

sampling distribution of can be approximated

by a normal probability distribution.

- Interval Estimate
- Point Estimator of

MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s 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 client’s 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 client’s product.

Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

- 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

- 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 client’s product before and after the new advertising campaign is -.02 to +.18.

- Conclusion

- Hypotheses
H0: p1 - p2< 0

Ha: p1 - p2 > 0

- Test statistic
- Point Estimator of where p1 = p2
where:

- Hypothesis Tests about p1 - p2
Can we conclude, using a .05 level of significance, that the proportion of households aware of the client’s 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

- HypothesesH0: p1 - p2< 0
Ha: p1 - p2 > 0

- HypothesesH0: p1 - p2< 0

- Hypothesis Tests about p1 - p2
- Rejection RuleReject H0 if z > 1.645
- Test Statistic
- ConclusionDo not reject H0.