Confidence Intervals for

1

• Confidence Intervals for
• µ1 - µ2 and p1 - p2

2
Inference about Two Populations

• We are interested in
• Confidence intervals for the difference between
  two means.
• Confidence intervals for the difference between
  two
• proportions.

3
Confidence Intervals for the Difference between
Two Population Means µ1 - µ2 Independent Samples

• Two random samples are drawn from the two
  populations of interest.
• Because we compare two population means, we use
  the statistic .

4

• Population 1 Population 2
• Parameters µ1 and ?12 Parameters µ2 and
  ?22
• (values are unknown) (values are
  unknown)
• Sample size n1 Sample size n2
• Statistics x1 and s12 Statistics x2 and s22
• Estimate µ1? µ2 with x1? x2

5
Confidence Interval for m1 m2
Note when the values of ?12 and ?22 are unknown,
the sample variances s12 and s22 computed from
the data can be used.
6
Example confidence interval for m1 m2

• Do people who eat high-fiber cereal for breakfast
  consume, on average, fewer calories for lunch
  than people who do not eat high-fiber cereal for
  breakfast?
• A sample of 150 people was randomly drawn. Each
  person was identified as a consumer or a
  non-consumer of high-fiber cereal.
• For each person the number of calories consumed
  at lunch was recorded.

7
Example confidence interval for m1 m2

• Solution
• The parameter to be tested is
• the difference between two means.
• The claim to be tested is
• The mean caloric intake of consumers (m1)
• is less than that of non-consumers (m2).
• Use s12 4,103 for ?12 and s22 10,670
• for ?22

8
Example confidence interval for m1 m2

• The confidence interval estimator for the
  difference between two means is

9
Interpretation

• The 95 CI is (-56.59, -1.83).
• We are 95 confident that the interval
• (-56.59, -1.83) contains the true but unknown
  difference m1 m2
• Since the interval is entirely negative (that is,
  does not contain 0), there is evidence from the
  data that µ1 is less than µ2. We estimate that
  non-consumers of high-fiber breakfast consume on
  average between 1.83 and 56.59 more calories for
  lunch.

10
Does smoking damage the lungs of children exposed
to parental smoking? Forced vital capacity (FVC)
is the volume (in milliliters) of air that an
individual can exhale in 6 seconds. FVC was
obtained for a sample of children not exposed to
parental smoking and a group of children exposed
to parental smoking.
Parental smoking FVC s n
Yes 75.5 9.3 30
No 88.2 15.1 30
We want to know whether parental smoking
decreases childrens lung capacity as measured by
the FVC test. Is the mean FVC lower in the
population of children exposed to parental
smoking?
11
Parental smoking FVC s n
Yes 75.5 9.3 30
No 88.2 15.1 30
95 confidence interval for (µ1 - µ2)

• 1 mean FVC of children with a smoking parent
• 2 mean FVC of children without a smoking parent

We are 95 confident that lung capacity in
children of smoking parents is between 19.05 and
6.35 milliliters LESS than in children without a
smoking parent.
12
Bunny Rabbits and Pirates on the Box

• The data below show the sugar content (as a
  percentage of weight) of 10 brands of cereal
  randomly selected from a supermarket shelf that
  is at a childs eye level and 8 brands selected
  from the top shelf.

Eye level 40.3 55 45.7 43.3 50.3 45.9 53.5 43 44.2 44
Top 20 2.2 7.5 4.4 22.2 16.6 14.5 10
Create and interpret a 95 confidence interval
for the difference ?1 ?2 in mean sugar content,
where ?1 is the mean sugar content of cereal at a
childs eye level and ?2 is the mean sugar
content of cereal on the top shelf.
13
Eye level 40.3 55 45.7 43.3 50.3 45.9 53.5 43 44.2 44
Top 20 2.2 7.5 4.4 22.2 16.6 14.5 10
14
Interpretation

• We are 95 confident that the interval
• (28.46, 40.22) contains the true but unknown
  value of ?1 ?2.
• Note that the interval is entirely positive (does
  not contain 0) therefore, it appears that the
  mean amount of sugar ?1 in cereal on the shelf at
  a childs eye level is larger than the mean
  amount ?2 on the top shelf.

15

• Do left-handed people have a shorter
  life-expectancy than right-handed people?
• Some psychologists believe that the stress of
  being left-handed in a right-handed world leads
  to earlier deaths among left-handers.
• Several studies have compared the life
  expectancies of left-handers and right-handers.
• One such study resulted in the data shown in the
  table.

left-handed presidents
Handedness Mean age at death s n
Left 66.8 25.3 99
Right 75.2 15.1 888
star left-handed quarterback Steve Young
We will use the data to construct a confidence
interval for the difference in mean life
expectancies for left-handers and
right-handers. Is the mean life expectancy of
left-handers less than the mean life expectancy
of right-handers?
16
Handedness Mean age at death s n
Left 66.8 25.3 99
Right 75.2 15.1 888
95 confidence interval for (µ1 - µ2)
The Bambino,left-handed hitter Babe Ruth,
baseballs all-time best hitter

• 1 mean life expectancy of left-handers
• 2 mean life expectancy of right-handers

We are 95 confident that the mean life
expectancy for left-handers is between 3.32 and
13.48 years LESS than the mean life expectancy
for right-handers.
17
Example confidence interval for m1 m2

• Example
• An ergonomic chair can be assembled using two
  different sets of operations (Method A and Method
  B)
• The operations manager would like to know whether
  the assembly time under the two methods differ.

18
Example confidence interval for m1 m2

• Example
• Two samples are randomly and independently
  selected
• A sample of 25 workers assembled the chair using
  method A.
• A sample of 25 workers assembled the chair using
  method B.
• The assembly times were recorded
• Do the assembly times of the two methods differs?

19
Example confidence interval for m1 m2
Assembly times in Minutes

• Solution
• The parameter of interest is the difference
• between two population means.
• The claim to be tested is whether a difference
• between the two methods exists.
• Use s12 .848 for ?12 and s22 1.303
• for ?22

20
Example confidence interval for m1 m2
A 95 confidence interval for m1 - m2 is
calculated as follows
We are 95 confident that the interval (-0.3029 ,
0.8469) contains the true but unknown m1 -
m2 Notice Zero is included in the confidence
interval
21
Confidence Intervals for the difference p1 p2
between two population proportions

• In this section we deal with two populations
  whose data are qualitative.
• For qualitative data we compare the population
  proportions of the occurrence of a certain event.
• Examples
• Comparing the effectiveness of new drug versus
  older one
• Comparing market share before and after
  advertising campaign
• Comparing defective rates between two machines

22
Parameter and Statistic

• Parameter
• When the data are qualitative, we can only count
  the occurrences of a certain event in the two
  populations, and calculate proportions.
• The parameter we want to estimate is p1 p2.
• Statistic
• An estimator of p1 p2 is (the
  difference between the sample proportions).

23
Point Estimator

• Two random samples are drawn from two
  populations.
• The number of successes in each sample is
  recorded.
• The sample proportions are computed.

Sample 1 Sample size n1 Number of successes
x1 Sample proportion
Sample 2 Sample size n2 Number of successes
x2 Sample proportion
24
Confidence Interval for p1 ? p2
25
Example confidence interval for p1 p2

• Estimating the cost of life saved
• Two drugs are used to treat heart attack victims
• Streptokinase (available since 1959, costs 460)
• t-PA (genetically engineered, costs 2900).
• The maker of t-PA claims that its drug
  outperforms Streptokinase.
• An experiment was conducted in 15 countries.
• 20,500 patients were given t-PA
• 20,500 patients were given Streptokinase
• The number of deaths by heart attacks was
  recorded.

26
Example confidence interval for p1
p2(cont.)

• Solution
• The problem objective Compare the outcomes of
  two treatments.
• The data are qualitative (a patient lived or
  died)
• The parameter to be estimated is p1 p2.
• p1 death rate with Streptokinase
• p2 death rate with t-PA

27
Example confidence interval for p1
p2(cont.)

• Experiment results
• A total of 1497 patients treated with
  Streptokinase died.
• A total of 1292 patients treated with t-PA died.
• Estimate the difference in the death rates when
  using Streptokinase and when using t-PA.

28
Example confidence interval for p1
p2(cont.)

• Compute Manually
• Sample proportions
• The 95 confidence interval estimate is

29
Example confidence interval for p1
p2(cont.)

• Interpretation
• The interval (.0051, .0149) for p1 p2 does not
  contain 0 it is entirely positive, which
  indicates that p1, the death rate for
  streptokinase, is greater than p2, the death rate
  for t-PA.
• We estimate that the death rate for streptokinase
  is between .51 and 1.49 higher than the death
  rate for t-PA.

30
Example 95 confidence interval for p1 p2
The age at which a woman gives birth to her first
child may be an important factor in the risk of
later developing breast cancer. An international
study conducted by WHO selected women with at
least one birth and recorded if they had breast
cancer or not and whether they had their first
child before their 30th birthday or after.
Cancer Sample Size
Age at First Birth gt 30 683 3220 21.2
Age at First Birth lt 30 1498 10,245 14.6
The parameter to be estimated is p1 p2. p1
cancer rate when age at 1st birth gt30 p2 cancer
rate when age at 1st birth lt30
We estimate that the cancer rate when age at
first birth gt 30 is between .05 and .082 higher
than when age lt 30.