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IS 310 Business Statistics CSU Long Beach. Inferences on Two Populations. In the past, we dealt with one population mean and one population proportion. However, there are situations where two populations are involved dealing with two means. Examples are the following:

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IS 310 Business Statistics CSU Long Beach

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IS 310

CSU

Long Beach

### Inferences on Two Populations

In the past, we dealt with one population mean and one population proportion. However, there are situations where two populations are involved dealing with two means.

Examples are the following:

O We want to compare the mean salaries of male and female graduates (two populations and two means).

O We want to compare the mean miles per gallon(MPG) of two comparable automobile makes (two populations and two means)

### Statistical Inferences About Meansand Proportions with Two Populations

• Inferences About the Difference Between

Two Population Means: s1 and s2 Known

• Inferences About the Difference Between

Two Population Means: s1 and s2 Unknown

### Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Known

• Interval Estimation of m1 – m2

• Hypothesis Tests About m1 – m2

• 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 .

Estimating the Difference BetweenTwo Population Means

• 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.

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

Interval Estimation of 1 - 2:s 1 and s 2 Known

• Interval Estimate

where:

1 -  is the confidence coefficient

Interval Estimation of 1 - 2:s 1 and s 2 Known

• 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. The sample statistics appear on the

next slide.

Interval Estimation of 1 - 2:s 1 and s 2 Known

• Example: Par, Inc.

Sample #1

Par, Inc.

Sample #2

Rap, Ltd.

Sample Size

120 balls 80 balls

Sample Mean

275 yards 258 yards

Based on data from previous driving distance

tests, the two population standard deviations are

known with s 1 = 15 yards and s 2 = 20 yards.

Interval Estimation of 1 - 2:s 1 and s 2 Known

• Example: Par, Inc.

Let us develop a 95% confidence interval estimate

of the difference between the mean driving distances of

the two brands of golf ball.

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

Simple random sample

of n1 Par golf balls

x1 = sample mean distance

for the Par golf balls

Simple random sample

of n2 Rap golf balls

x2 = sample mean distance

for the Rap golf balls

x1 - x2 = Point Estimate of m1 –m2

Estimating the Difference BetweenTwo Population Means

m1 –m2= difference between

the mean distances

Point Estimate of 1 - 2

Point estimate of 1-2 =

= 275 - 258

= 17 yards

where:

1 = mean distance for the population

of Par, Inc. golf balls

2 = mean distance for the population

of Rap, Ltd. golf balls

Interval Estimation of 1 - 2: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 is 11.86 to 22.14 yards.

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

• Hypotheses

Left-tailed

Right-tailed

Two-tailed

• Test Statistic

Hypothesis Tests About m 1-m 2:s 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?

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

• p –Value and Critical Value Approaches

1. Develop the hypotheses.

H0: 1 - 2< 0

Ha: 1 - 2 > 0

where:

1 = mean distance for the population

of Par, Inc. golf balls

2 = mean distance for the population

of Rap, Ltd. golf balls

a = .01

2. Specify the level of significance.

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

• p –Value and Critical Value Approaches

3. Compute the value of the test statistic.

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

• p –Value Approach

4. Compute the p–value.

For z = 6.49, the p –value < .0001.

5. Determine whether to reject H0.

Because p–value <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.

Hypothesis Tests About m 1-m 2:s 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> 2.33

5. Determine whether to reject H0.

Because z = 6.49 > 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.

### Sample Problem

Problem # 7 (10-Page 401; 11-Page 414)

a. H : µ = µ H : µ > µ

0 1 2 a 1 2

• Point reduction in the mean duration of games during 2003 = 172 – 166

= 6 minutes

_ _ 2 2

c. Test-statistic, z = [( x - x ) – 0] /√ [ (σ / n ) + (σ / n )]

1 2 1 1 2 2

=(172 – 166)/√[ (144/60 + 144/50)]

= 6/2.3 = 2.61

Critical z at  = 1.645 Reject H

0.05 0

Statistical test supports that the mean duration of games in 2003 is less than that in 2002.

p-value = 1 – 0.9955 = 0.0045

Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Unknown

• Interval Estimation of m1 – m2

• Hypothesis Tests About m1 – m2

Interval Estimation of 1 - 2:s 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.

### Interval Estimation of µ - µ 1 2

• (Unknown  and  )

• 1 2

• Interval estimate

• _ _ 2 2

• (x - x ) ± t √ (s /n + s /n )

• 1 2 /2 1 1 2 2

• Degree of freedom = n + n - 2

• 1 2

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.

Difference Between Two Population Means:

s 1 and s 2 Unknown

• Example: Specific Motors

Sample #1

M Cars

Sample #2

J Cars

24 cars 28 cars

Sample Size

29.8 mpg 27.3 mpg

Sample Mean

2.56 mpg 1.81 mpg

Sample Std. Dev.

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.

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

### Interval Estimate of µ - µ 1 2

• Interval estimate

• 2 2

• 29.8 – 27.3 ± t √ (2.56) /24 + (1.81) /28)

• 0.1/2

• 2.5 ± 1.676 (0.62)

• 2.5 ± 1.04

• 1.46 and 3.54

• We are 90% confident that the difference between the average miles per gallon between the J cars and M cars is between 1.46 and 3.54.

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

• Hypotheses

Left-tailed

Right-tailed

Two-tailed

• Test Statistic

Hypothesis Tests About m 1-m 2:s 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?

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

• p –Value and Critical Value Approaches

a = .05

2. Specify the level of significance.

3. Compute the value of the test statistic.

### Hypothesis Tests of µ - µ 1 2

• H : µ = µ H : µ > µ

• 0 1 2 a 1 2

• Where µ average miles per gallon of M cars

• 1

• µ average miles per gallon of J cars

• 2

• At  = 0.05 with 50 degree of freedom, critical t = 1.676

• Since t-statistic (4.003) is larger than critical t (1.676), we reject the null hypothesis. This means that the average MPG of M cars is not equal to that of J cars