1. Statistical Inference About Means and Proportions With Two Populations
Statistics
Statistical Inference About Means and Proportions With Two Populations

2. STATISTICS in PRACTICE
Statistics plays a major role in
pharmaceutical research.
Statistical methods are used to
test and develop new drugs.
In most studies, the statistical method involves hypothesis testing for the difference between the means of the new drug population and the standard drug population.

3. STATISTICS in PRACTICE
In this chapter you will learn how to construct interval estimates and make hypothesis tests about means and proportions with two populations.

4. Contents
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 s2 Unknown
Inferences About the Difference Between Two Population Means: Matched Samples
Inferences About the Difference Between Two Population Proportions

5. Inferences About the Difference Between Two Population Means: s 1 and s 2 Known
Point and Interval Estimation of m 1 – m 2
The Point Estimator of m 1 – m 2 is
Interval Estimation of m 1 – m 2 is

6. Inferences About the Difference Between Two Population Means: s 1 and s 2 Known
Hypothesis Tests About m 1 – m 2
D0: hypothesized difference between m 1 – m 2

7. Estimating the Difference Between Two 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.

8. Estimating the Difference Between Two Population Means
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

9. 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

10. Interval Estimation of 1 - 2: s 1 and s 2 Known
Interval Estimate
where:
1 -  is the confidence coefficient

11. 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.

12. 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 balls
Sample Mean
275 yards 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.

13. 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.

14. Estimating the Difference Between Two Population Means
Rap, Ltd. Golf Balls
m2 = mean driving
distance of Rap
golf balls
Population 1
Par, Inc. Golf Balls
m1 = mean driving
distance of Par
golf balls
μ1– μ2 = difference between
the mean distances
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
= Point Estimate of μ1– μ2

15. 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

16. Interval Estimation of 1 - 2:  1 and  2 Known
or yards to yards
We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls is to yards.

17. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Known
Hypotheses
Left-tailed
Right-tailed
Two-tailed
Test Statistic

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

19. 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
2. Specify the level of significance.
a = .01

20. 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.

21. 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 <
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.

22. 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.

23. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Known
Critical Value Approach
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.

24. Inferences About the Difference Between Two Population Means:
s 1 and s 2 Unknown
Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2
H0: m 1 – m 2 =0
Ha: m 1 – m 2 0

25. 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

26. Interval Estimation of 1 - 2: s 1 and s 2 Unknown
Interval Estimate
where the degrees of freedom for ta/2 are:

27. 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.

28. 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 mpg
Sample Mean
2.56 mpg mpg
Sample Std. Dev.

29. 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.

30. Point Estimate of m 1 - m 2 Point estimate of 1 - 2 = = 29.8 - 27.3
= mpg
where:
1 = mean miles-per-gallon for the
population of M cars
2 = mean miles-per-gallon for the
population of J cars

31. Interval Estimation of m 1 - m 2: s 1 and s 2 Unknown
The degrees of freedom for ta/2 are:
With a/2 = .05 and df = 24, ta/2 =

32. Interval Estimation of m 1 - m 2: s 1 and s 2 Unknown
or to mpg
We are 90% confident that the difference
between the miles-per-gallon performances of
M cars and J cars is to mpg.

33. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
Hypotheses
Left-tailed
Right-tailed
Two-tailed
Test Statistic

34. 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?

35. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: 1 - 2 < 0 
Ha: 1 - 2 > 0
where:
1 = mean mpg for the population of M cars
2 = mean mpg for the population of J cars

36. Hypothesis Tests About m 1 - m 2: s 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.

37. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
p –Value Approach
4. Compute the p –value.
The degrees of freedom for ta/2 are:
Because t = > t.005 = 2.797, the p–value < .005.

38. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
p –Value Approach
5. Determine whether to reject H0.
Because p–value < 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?.

39. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
Critical Value Approach
Determine the critical value and
rejection rule.
For a = .05 and df = 24, t.05 = 1.711
Reject H0 if t > 1.711

40. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Unknown
Critical Value Approach
5. Determine whether to reject H0.
Because > 1.711, 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?.

41. Inferences About the Difference Between Two Population Means: Matched Samples
With a matched-sample design each sampled item provides a pair of data values.
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.

42. Inferences About the Difference Between Two Population Means: Matched Samples
Example: Express Deliveries
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.

43. Inferences About the Difference Between Two Population Means: Matched Samples
Example: Express Deliveries
In testing the delivery times
of the two services, the firm sent
two reports to a random sample of its district
offices with one report carried by UPX and the
other report carried by INTEX. Do the data on
the next slide indicate a difference in mean
delivery times for the two services? Use a .05 level of significance.

44. Inferences About the Difference Between Two Population Means: Matched Samples
Delivery Time (Hours)
District Office
UPX
INTEX
Difference
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver 32 30 19 16 15 18 14 10 7 25 24 15 13 8 9 11 7 6 4 1 2 3 -1 -2 5

45. Inferences About the Difference Between Two Population Means: Matched Samples
p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: d = 0 
Ha: d 
Let d = the mean of the difference values for the
two delivery services for the population
of district offices

46. Inferences About the Difference Between Two Population Means: Matched Samples
p –Value and Critical Value Approaches
a = .05
2. Specify the level of significance.
3. Compute the value of the test statistic.

47. Inferences About the Difference Between Two Population Means: Matched Samples
p –Value Approach
4. Compute the p –value.
For t = 2.94 and df = 9, the p–value is between.02 and (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)

48. Inferences About the Difference Between Two Population Means: Matched Samples
p –Value Approach
5. Determine whether to reject H0.
Because p–value < a = .05, we reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services?

49. Inferences About the Difference Between Two Population Means: Matched Samples
Critical Value Approach
Determine the critical value and
rejection rule.
For a = .05 and df = 9, t.025 =
Reject H0 if t > 2.262

50. Inferences About the Difference Between Two Population Means: Matched Samples
Critical Value Approach
5. Determine whether to reject H0.
Because t = 2.94 > 2.262, we reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services?

51. Inferences About the Difference Between Two Population Proportions
Interval Estimation of p1 - p2
Hypothesis Tests About p1 - p2

52. Sampling Distribution of
Expected Value
Standard Deviation (Standard Error)
where: n1 = size of sample taken from population 1
n2 = size of sample taken from population 2

53. Sampling Distribution of
If the sample sizes are large, the sampling
distribution of can be approximated
by a normal probability distribution.
The sample sizes are sufficiently large if all
of these conditions are met:
n1p1 > 5
n1(1 - p1) > 5
n2p2 > 5
n2(1 - p2) > 5

54. Sampling Distribution of
p1 – p2

55. Interval Estimation of p1 - p2
Interval Estimate

56. Interval Estimation of p1 - p2
Example: Market Research Associates
Market Research Associates is
conducting research to evaluate the
effectiveness of a client’s new adver-
tising 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.

57. Interval Estimation of p1 - p2
Example: Market Research Associates
The new campaign has been
initiated with TV and
newspaper advertisements
running for three weeks.

58. Interval Estimation of p1 - p2
Example: Market Research Associates
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?

59. Point Estimator of the Difference Between Two Population Proportions
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
of the product before the new campaign

60. Interval Estimation of p1 - p2
For α= .05, z.025 = 1.96
(.0510)
Hence, the 95% confidence interval for the
difference in before and after awareness of the
product is -.02 to +.18.

61. Hypothesis Tests about p1 - p2
Hypotheses
We focus on tests involving no difference between the two population proportions (i.e. p1 = p2)
Left-tailed
Right-tailed
Two-tailed

62. Hypothesis Tests about p1 - p2
Pooled Estimate of Standard Error of
where:

63. Hypothesis Tests about p1 - p2
Test Statistic

64. Hypothesis Tests about p1 - p2
Example: Market Research Associates
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?

65. Hypothesis Tests about p1 - p2
p -Value and Critical Value Approaches
1. Develop the hypotheses.
H0: p1 - p2 < 0
Ha: p1 - p2 > 0
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

66. Hypothesis Tests about p1 - p2
p -Value and Critical Value Approaches
2. Specify the level of significance.
a = .05
3. Compute the value of the test statistic.

67. Hypothesis Tests about p1 - p2
p –Value Approach
4. Compute the p –value.
For z = 1.56, the p–value = .0594
5. Determine whether to reject H0.
Because p–value > a = .05, we cannot reject H0.
We cannot conclude that the proportion
of households aware of the client’s product
increased after the new campaign.

68. Hypothesis Tests about p1 - p2
Critical Value Approach
Determine the critical value and
rejection rule.
For a = .05, z.05 = 1.645
Reject H0 if z > 1.645

69. Hypothesis Tests about p1 - p2
Critical Value Approach
5. Determine whether to reject H0.
Because 1.56 < 1.645, we cannot reject H0.
We cannot conclude that the proportion
of households aware of the client’s product
increased after the new campaign.