Unit 6 Chapter: Confidence Intervals
Chapter 6 begins the study of inferential statistis. Inferential Statistics
The branch of statistics concerned with drawing conclusions about a
population from a sample. This is generally done through random sampling,
followed by inferences made about central tendency, or any of a number of
other aspects of a distribution.
You will leran how to make a more meaning estamimate by specifying an interval of numbers on
x
a number line, together with a stement which staes how confident you are thar the interval
contains the population parameter.
μ = ΣX/N
The standard error of the mean is the standard deviation of the sampling
distribution of the mean. The formula for the standard error of the mean in a
population is:
where σ is the standard deviation and N is the sample size. When computed
in a sample, the estimate of the standard error of the mean is:
Point estimate - A point estimate of a population parameter is a single value used to estimate the
population parameter. For example, the sample mean x is a point estimate of the population mean μ.
The most unbiased point estimate of the population mean μ is the sample mean x.
Point Estimate for Population μ
Point Estimate
•
A single value estimate for a population parameter
•
Most unbiased point estimate of the population mean μ is the sample mean x
Estimate Population Parameter…
Mean:
with Sample Statistic
μ
x
Interval estimate- an interval, range of values, used to estimate a population parameter. It is defined
by two numbers, between which a population parameter is said to lie. Interval estimate -An interval,
or range of values, used to estimate a population parameter.
For example, a < μ < b is an interval estimate for the population mean μ. It indicates that the
population mean is greater than a but less than b.
Level of confidence - Is the probability (likelihood) that the interval estimate contains the population
parameter. The confidence level is usually expressed as a percentage. Thus, a 95% confidence level
implies that the probability that the true population parameter lies within the confidence interval is
0.95. Here, the confidence level (95%) implies a probability equal to 0.95.
c is the area under the standard normal
curve between the critical values
c
½(1 – c)
½(1 – c)
Critical values
Use the Standard Normal Table to find the
corresponding z-scores.
The level of confidence c is the area under the standard normal curve between critical
Sampling error - The difference between the point estimate and actual parameter value. It is the error
that results from taking one sample rather than taking a census of the entire population.
Sampling error
•
•
The difference between the point estimate and the actual population parameter value.
For μ:
 the sampling error is the difference – μ
 μ is generally unknown

varies from sample to sample
Margin of error - The margin of error sometimes also called the maximum of estimate or error
tolerance) E greatest possible difference between the point value and the values of the parameter it is
estimating.
Margin of error
•
The greatest possible distance between the point estimate and the value of the parameter it is
estimating for a given level of confidence, c.
•
Denoted by E.
E  zcσ x  zc
σ
When n  30, the sample standard deviation, s, can be
used for .
n
C-confidence interval - Used to express the degree of uncertainty associated with a sample statistic.
A confidence interval is an intervalestimate combined with a probability statement. The probability
that the confidence interval contains μ is c.
A c-confidence interval for the population mean μ
xE  xE
where E  zc

n
Constructing Confidence Intervals for μ
Finding a Confidence Interval for a Population Mean
(n  30 or σ known with a normally distributed population)
In Words
1. Find the sample statistics n and
2. Specify , if known. Otherwise, if n  30, find the sample standard deviation s and use it as an
estimate for .
3. Find the critical value zc that corresponds to the given level of confidence.
3. Find the margin of error E.
4. Find the left and right endpoints and form the confidence interval.
In Symbols
x
x
n
(x  x )2
s
n 1
Use the Standard Normal Table.
E  zc

n
Left endpoint: x  E
Right endpoint: Interval: x  E
xE  xE
Finding a confidence Interval for a Population Mean (p 313)
Sample size (P 316)
Find a minimum sample size to estimate μ
Sample Size
•
Given a c-confidence level and a margin of error E, the minimum sample size n needed to
estimate the population mean  is
z
n c 
 E 
•
2
If  is unknown, you can estimate it using s provided you have a preliminary sample with at
least 30 members.
The t distribution (aka, Student t distribution) is the sampling distribution of the t statistic. The values
of the t statistic are given by:
t = [ x - μ ] / [ s / sqrt( n ) ]
When the population standard deviation is unknown, the sample size is less than 30, and the random
variable x is approximately normally distributed, it follows a t-distribution.
t
x -
s
n
Critical values of t are denoted by tc.
Properties of the t-Distribution
The t-distribution is bell shaped and symmetric about the mean.
The t-distribution is a family of curves, each determined by a parameter called the degrees of
freedom. The degrees of freedom are the number of free choices left after a sample statistic such as
is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom
are equal to one less than the sample size.
d.f. = n – 1
Degrees of freedom
The total area under a t-curve is 1 or 100%.
The mean, median, and mode of the t-distribution are equal to zero.
As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30
d.f., the t-distribution is very close to the standard normal z-distribution.
Confidence Intervals for the Population Mean
A c-confidence interval for the population mean μ
xE  xE
where E  tc
s
n
In Words
The probability that the confidence interval contains μ is c.
1. Identify the sample statistics n, , and s.
2. Identify the degrees of freedom, the level of confidence c, and the critical value tc.
3. Find the margin of error E
4. Find the left and right endpoints and form the confidence interval.
In Symbols
1.
x
x
n
(x  x )2
s
n 1
2. d.f. = n – 1
3.
E  tc
s
n
4. Left endpoint: x  E
Right endpoint: x  E
5. Interval: x  E    x  E
Constructing a Confidnce Interval for the Mean: t-Distribution (pg 327)
Proportion- A proportion refers to the fraction of the total that possesses a certain attribute.
For example, suppose we have a sample of four pets - a bird, a fish, a dog, and a cat. We might ask
what proportion has four legs. Only two pets (the dog and the cat) have four legs. Therefore, the
proportion of pets with four legs is 2/4 or 0.50. Another example, we might ask what proportion of
women in our sample weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the
proportion would be 3/5 or 0.60.
Point Estimate for Population p
Population Proportion
•
•
The probability of success in a single trial of a binomial experiment.
Denoted by p
Point Estimate for p
•
•
The proportion of successes in a sample.
Denoted by read as “p hat”
pˆ 
x number of successes in sample

n
number in sample
Estimate Population with Sample
Parameter…
Statistic
p̂
Proportion: p
Point Estimate for q, the proportion of failures
•
Denoted by
qˆ  1  pˆ
Read as “q hat”
A c-confidence interval for the population proportion p
pˆ  E  p  pˆ  E
pq
where E  zc ˆ ˆ
n
The probability that the confidence interval contains p is c.
In Words
1.
2.
3.
4.
5.
6.
Identify the sample statistics n and x.
Find the point estimate
Verify that the sampling distribution of
can be approximated by the normal distribution.
Find the critical value zc that corresponds to the given level of confidence c.
Find the margin of error E.
Find the left and right endpoints and form the confidence interval.
p̂
p̂
In Symbols
1. Identify the sample statistics n and x.
3.
x
n
npˆ  5, nqˆ  5
4.
Use the Standard Normal Table
2.
pˆ 
pq
E  zc ˆ ˆ
n
6. Left endpoint: p̂  E
5.
p̂  E
pˆ  E  p  pˆ  E
Right endpoint:
Interval:
Sample Size
•
Given a c-confidence level and a margin of error E, the minimum sample size n needed to
estimate p is
2
•
z 
ˆ ˆ c 
n  pq
E
This formula assumes you have an estimate for
p̂
and
qˆ.
If not, use pˆ  0.5 and qˆ  0.5.
The Chi-Square Distribution
•
The point estimate for 2 is s2
•
The point estimate for  is s
•
s2 is the most unbiased estimate for 2
Estimate Population Parameter…
Variance:
with Sample Statistic
σ2
s2
Standard deviation: σ
•
s
You can use the chi-square distribution to construct a confidence interval for the variance
and standard deviation.
•
If the random variable x has a normal distribution, then the distribution of
2 
(n  1)s 2
σ2
forms a chi-square distribution for samples of any size n > 1.
Properties of The Chi-Square Distribution
1. All chi-square values χ2 are greater than or equal to zero.
2. The chi-square distribution is a family of curves, each determined by the degrees of freedom.
To form a confidence interval for 2, use the χ2-distribution with degrees of freedom equal to
one less than the sample size.
•
d.f. = n – 1
Degrees of freedom
3. The area under each curve of the chi-square distribution equals one.
4. Chi-square distributions are positively skewed.
chi-square distributions
Critical Values for χ2
•
There are two critical values for each level of confidence.
•
The value χ2 represents the right-tail critical value
R
•
The value χ2 represents the left-tail critical value
L
Confidence Intervals for 2 and 
Confidence Interval for 2:
(n  1)s 2
 R2
σ2 
(n  1)s 2
 L2
Confidence Interval for :
(n  1)s 2
 R2
σ 
(n  1)s 2
 L2
The probability that the confidence intervals contain σ2 or σ is c.
Confidence Intervals for 2 and 
In Words
1. Verify that the population has a normal distribution.
2. Identify the sample statistic n and the degrees of freedom.
3. Find the point estimate s2.
4. Find the critical value χ2R and χ2L that correspond to the given level of confidence c.
5. Find the left and right endpoints and form the confidence interval for the population variance.
6. Find the confidence interval for the population standard deviation by taking the square root of
each endpoint.
In Symbols
1. Verify that the population has a normal distribution.
2. d.f. = n – 1
3.
( x  x )2
s 
n 1
2
4. Use Table 6 in Appendix B
5.
(n  1)s 2
 R2
6.
σ 
(n  1)s 2
 R2
2
σ 
(n  1)s 2
 L2
(n  1)s 2
 L2