Chapters 6

1
(No Transcript)
2
Chapters 6 7 Overview
Created by Erin Hodgess, Houston, Texas
3
Chapter 6Confidence Intervals

• 6-1 Overview
• 6-3 Estimating a Population Mean s Known
• 6-4 Estimating a Population Mean s Not Known
• 6-2 Estimating a Population Proportion

4
Inferential Statistics
Chapters 6 7 mark the start of inferential
statistics (drawing conclusions about population
using sample data). Two major types of
inferential statistics Use sample data to

1. Estimate the value of a population parameter
   (Confidence Intervals)
2. Test some claim about a population parameter
   (Hypothesis Testing).

5
Quantitative vs. Qualitative

• Quantitative Data (means)
• Population Mean, µ
• Sample Mean,
• Qualitative Data (proportions)
• Population Proportion, p
• Sample Proportion,

6
Estimating a Population Parameter (µ or p)

• An estimate of a population parameter can be
  either
• Point Estimate, or
• Interval Estimate
• ex. Estimate µ, the mean weight of an adult
  male.
• ex. Estimate p, the proportion of adults who
  have health insurance.

7
Estimating a Population Parameter, µ or p

• Point Estimate a single value estimate.
• For µ, the best point estimate is x.
• µ x.
• For p, the best point estimate is
• p
• Question How can we improve our estimate?

8
Estimating a Population Parameter

• Interval Estimate (Confidence Interval) a range
  of values that is likely to contain the
  parameter.
• Format µ x E or p p E
• Comes with a level of confidence (1-a) which is
  the probability that the interval actually
  contains the parameter.

9
Common Confidence Levels
Confidence Levels (1-a) 90 95 99
1-a 0.90 0.95 0.99
a 0.10 0.05 0.01
10
Section 6-3 Estimating a Population Mean, µ (?
Known)
Created by Erin Hodgess, Houston, Texas
11
Estimating a Population Mean, µ x E

• Margin of Error, E the maximum likely difference
  observed between sample mean x and population
  mean µ.
• How to find E?
• Back to the Central Limit Theorem
• For n 30 or x N, x N (µ, s/vn)

12
Confidence Levels (1-a) and Critical Values

• Critical Values za/2 z-scores with an area of
  a/2 in the upper tail.

(1-a) a a/2 za/2
90 or 0.90 0.10 0.05 1.645
95 or 0.95 0.05 0.025 1.96
99 or 0.99 0.01 0.005 2.575
13
Critical Values

• 1. We know from Section 5-6 that under certain
  conditions, the sampling distribution of sample
  proportions can be approximated by a normal
  distribution, as in Figure 6-2.
• 2. Sample proportions have a relatively small
  chance (with probability denoted by ?) of falling
  in one of the red tails of Figure 6-2.
• 3. Denoting the area of each shaded tail by ?/2,
  we see that there is a total probability of ?
  that a sample proportion will fall in either of
  the two red tails.

14
Critical Values

• 4. By the rule of complements (from Chapter 3),
  there is a probability of 1? that a sample
  proportion will fall within the inner region of
  Figure 6-2.
• 5. The z score separating the right-tail is
  commonly denoted by z? /2, and is referred to as
  a critical value because it is on the borderline
  separating sample proportions that are likely to
  occur from those that are unlikely to occur.

15
The Critical Value
z??2
Figure 6-2
16
Notation for Critical Value
The critical value z?/2 is the positive z value
that is at the vertical boundary separating an
area of ?/2 in the right tail of the standard
normal distribution. (The value of z?/2 is at
the vertical boundary for the area of ?/2 in the
left tail). The subscript ?/2 is simply a
reminder that the z score separates an area of
?/2 in the right tail of the standard normal
distribution.
17
DefinitionCritical Value

• A critical value is the number on the borderline
  separating sample statistics that are likely to
  occur from those that are unlikely to occur. The
  number z?/2 is a critical value that is a z score
  with the property that it separates an area of
  ?/2 in the right tail of the standard normal
  distribution. (See Figure 6-2).

18
Finding z??2 for 95 Degree of Confidence
19
Finding z??2 for 95 Degree of Confidence
? 0.05
Use Table A-2 to find a z score of 1.96
20
Procedure for Constructing a (1-a)100
Confidence Interval for µ when ? is known

• Assumptions a) x comes from a SRS
• b) s is known
• c) n 30 or x N
• 1. µ x E where E za/2.s/vn
• 2. Write as an interval

21
Confidence Interval Notation

• Interval Notation
• Rounding Round statistics and confidence
  intervals to one more decimal place than used in
  original set of data.

µ x E
(x E, x E)
22
Notes on Confidence Intervals

• Interpretation
• Theres a 95 chance that the interval actually
  contains µ or,
• 95 of all such samples would generate an
  interval that contained µ
• Desired Properties
• High Confidence
• Small Margin of Error
• To Decrease Margin of Error
• Lower level of confidence
• Increase sample size

23
Determining Sample Size Needed to Estimate a
Population Mean ?
First Choose 1. Confidence 2. Margin of
Error Then, the minimum sample size needed to
estimate µ is
24
Notes on Sample Size n

• n depends only on confidence level and E.
  Population size is irrelevant.
• 2. Since the formula gives the minimum sample
  size needed to achieve the desired confidence and
  margin of error, always round up to next larger
  whole number.
• 3. Must know s in advance. Either
• a. Use s from a previous study
• b. Do a preliminary sample n 30 and s s
• c. If the range is known, use the range rule to
  estimate s. (s range/4)

25
Section 6-4 Estimating a Population Mean ? Not
Known
Created by Erin Hodgess, Houston, Texas
26
? Not KnownAssumptions

• 1) The sample is a simple random sample.
• 2) Either the sample is from a normally
  distributed population, or n gt 30.
• Use Students t distribution

27
Student t Distribution (with n -1 degrees of
freedom)

• If the distribution of a population is
  essentially normal, then the distribution of

x - µ
t
s
n

• is a Student t Distribution, with n -1 degrees
  of
• freedom

28
Definition

• Degrees of Freedom (df )
• corresponds to the number of sample values that
  can vary after certain restrictions have been
  imposed on all data values

df n 1 in this section.
29
Important Properties of the Student t
Distribution

• 1. The Student t distribution is different for
  different sample sizes (see Figure 6-5 for the
  cases n 3 and n 12).
• 2. The Student t distribution has the same
  general symmetric bell shape as the normal
  distribution but it reflects the greater
  variability (with wider distributions) that is
  expected with small samples.
• 3. The Student t distribution has a mean of t 0
  (just as the standard normal distribution has a
  mean of z 0).
• 4. The standard deviation of the Student t
  distribution varies with the sample size and is
  greater than 1 (unlike the standard normal
  distribution, which has a ??? 1).
• 5. As the sample size n gets larger, the Student
  t distribution gets closer to the normal
  distribution.

30
Student t Distributions for n 3 and n 12
Figure 6-5
31
Confidence Interval for the Estimating µ, s
unknown Based on an Unknown ? and a Simple
Random Sample from a Normally Distributed (or
roughly symmetric) Population

• t?/2 with n-1 degrees of freedom from
• Table A-3.

32
Using the Normal and t Distribution
Figure 6-6
33
Section 6-2 Estimating a Population Proportion, p
Created by Erin Hodgess, Houston, Texas
34
Qualitative Data and Proportions

• Population Proportion, p
• proportion of the population that has the
  attribute
• Sample Proportion
• proportion of the sample that has the attribute
• Other notation
• q 1 p proportion of the population that
  does not have the attribute.
• q 1 p proportion of the sample that does
  not have the attribute.

35
Estimating a Population Proportion, p

• Point Estimate p p
• Confidence Interval p p E
• Notation
• p p E
• p E lt p lt p E
• (p E, p E)

36
Confidence Interval for Population Proportion

p p E
where
37
(1-a) Confidence Interval for Population
Proportion


Notes 1. Use z-distribution provided
) 2. Round proportions and
confidence intervals to significant digits
38
Sample Size for Estimating Proportion p