Confidence Intervals

1
Confidence Intervals
2
Inference

• In statistical inference we are concerned with
  methods for using random samples to gain
  information about population parameters, such as
  a population mean or a population proportion.

3

• Researchers use the methods of inferential
  statistics for two related purposes
• 1. Estimate of the value of a population
  parameter. Example
• Estimate the proportion of college students who
  smoke cigarettes.
• Estimate the mean pulse rate of adult women.
• 2. Test hypotheses about population parameters.
  Example
• Test whether the success rate of a new asthma
  treatment is better than the success rate of a
  standard treatment.
• Test whether the mean weight-loss resulting from
  a diet program is greater than 10 pounds.

4
Concepts

• A confidence interval is a range of values that
  is likely to contain the true population value.
• Every confidence interval has an associated
  confidence level, the probability that the
  interval actually captures the true population
  value.

5

• The margin of error indicates the possible size
  of the sampling error. The latter is the
  difference between the estimate and the
  population value that may occur due to the act of
  sampling.
• We cannot know the actual sampling error for a
  particular sample. So the margin of error
  indicates how severe a sampling error might be
  for samples of a particular size.

6

• In general, most confidence intervals are
  calculated by adding and subtracting the margin
  of error to the sample estimate. That is, most
  confidence intervals are of the form
• sample estimate margin of error
• Therefore, we only need to know the sample
  estimate and the margin of error in order to
  calculate the confidence interval. It is standard
  practice for polling organizations and news media
  to report a margin of error for the 95
  confidence level.

7
Example
In 1996, an article in Victoria, British
Columbia's primary newspaper, the Victoria Times
Colonist, reported the results of a poll about
Canadians' view of alien life. The article
reported
"Of 1,501 Canadians surveyed for an Angus Reid
poll last month, one in four said they believed
there is definitely intelligent life elsewhere in
the universe. The survey was conducted Aug. 27 to
Aug. 30 and is considered 95 percent accurate
within 2.5 percentage points."
8
Conclusions

• From this report, we know
• the sample estimate ("one in four" or 25)?
• the margin of error ("within 2.5 percentage
  points")
• the confidence level ("95 percent accurate")
• Therefore, for a 95 confidence interval from
  this data set
• the lower limit is 25 - 2.5, or 22.5
• the upper limit is 25 2.5, or 27.5
• In other words, we are 95 confident that between
  22.5 and 27.5 percent of all Canadians believe
  there is definitely intelligent life elsewhere in
  the universe.

9
What Is "95 Confident"?
10
Using Confidence Intervals

• Confidence intervals can be used to answer
  research questions that involve
• Estimating a population parameter
• Considering whether a population parameter equals
  a certain value
• Comparing the parameters of two different
  populations.

11
Confidence Interval for the Mean

• A confidence interval for a mean is a range of
  values that is likely to capture the population
  mean.

12

• When we use the mean of a large sample to
  estimate a population mean
• the sample estimate is , the sample mean
• the margin of error for a 95 confidence level is
  (approximately)
• So, an approximate 95 confidence interval for a
  population mean is

13
95 Confidence Interval
If after observing X1 x1,, Xn xn, we compute
the observed sample mean , then a 95
confidence interval for can be expressed as
14
Other Levels of Confidence
0
15
Other Levels of Confidence
A confidence interval for
the mean of a normal population when the
value of is known is given by
16
Sample Size
The general formula for the sample size n
necessary to ensure an interval width w is
17
Large-Sample Confidence Interval
If n is sufficiently large (ngt30), the
standardized variable
has approximately a standard normal distribution.
This implies that
is a large-sample confidence interval for
with level
18
Population Proportions

• P is the population proportion
• p is the sample proportion
• px/n where x is the number of successes and n
  is the sample size.
• q1-p

19
Sampling Proportion of the Mean

• Suppose a large sample of size n is taken from a
  population with parameter p, then the following
  is true
• The random variable p is approximately normally
  distributed
• µpp and spsqrt(pq/n)

20
Confidence Interval for a Population Proportion p
with level
Lower() and upper() limits
21
Confidence Interval

• Simplifies, when n is large, to

22
Problem

• In a survey of 1250 developers, 650 favored a
  certification for software engineers.   Find the
  99 confidence interval to estimate p for the
  population.

23
Error
24
Problem

• A pollster found with a small sample that 59 out
  of a 100 people would like the special council
  to continue his investigation.  Find how large
  the sample would need to be for a 1 percent
  margin of error, E, for a 95 confidence interval

25
Large-Sample Confidence Bounds for
Upper Confidence Bound
Lower Confidence Bound
26
Normal Distribution
The population of interest is normal, so that
X1,, Xn constitutes a random sample from a
normal distribution with both
27
t Distribution
When is the mean of a random sample of size n
from a normal distribution with mean the rv
has a probability distribution called a t
distribution with n 1 degrees of freedom (df).
28
Properties of t Distributions
Let tv denote the density function curve for v df.

• Each tv curve is bell-shaped and centered at 0.
• Each tv curve is spread out more than the
  standard normal (z) curve.

29
Properties of t Distributions

• 3. As v increases, the spread of the
  corresponding tv curve decreases.
• 4. As , the sequence of tv curves
  approaches the standard normal curve (the z curve
  is called a t curve with df

30
t Critical Value
Let the number on the measurement axis
for which the area under the t curve with v df to
the right of
is called a t critical value.
31
Pictorial Definition of
0
32
Confidence Interval
Let and s be the sample mean and standard
deviation computed from the results of a random
sample from a normal population with mean The

33
Normal Population
Let X1,, Xn be a random sample from a normal
distribution with parameters
Then the rv
has a chi-squared probability
distribution with n 1 df.
34
Chi-squared Critical Value
Let , called a chi-squared critical value,
denote the number of the measurement axis such
that of the area under the chi-squared curve
with v df lies to the right of
35
Notation Illustrated
36
Confidence Interval
confidence interval for
for the variance of a normal population has
lower limit
upper limit
For a confidence interval for , take the
square root of each limit above.