• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Central limit theorem wikipedia, lookup

Transcript
```A PowerPoint Presentation Package to Accompany
Economics, 4th edition
David P. Doane and Lori E. Seward
Prepared by Lloyd R. Jaisingh
McGraw-Hill/Irwin
Chapter Contents
Chapter 8
Sampling Distributions and Estimation
8.1 Sampling Variation
8.2 Estimators and Sampling Errors
8.3 Sample Mean and the Central Limit Theorem
8.4 Confidence Interval for a Mean (μ) with Known σ
8.5 Confidence Interval for a Mean (μ) with Unknown σ
8.6 Confidence Interval for a Proportion (π)
8.7 Estimating from Finite Populations
8.8 Sample Size Determination for a Mean
8.9 Sample Size Determination for a Proportion
8.10 Confidence Interval for a Population Variance,  2 (Optional)
8-2
Chapter 8
Sampling Distributions and Estimation
Chapter Learning Objectives (LO’s)
LO8-1:
LO8-2:
LO8-3:
LO8-4:
LO8-5:
Define sampling error, parameter, and estimator.
Explain the desirable properties of estimators.
State the Central Limit Theorem for a mean.
Explain how sample size affects the standard error.
Construct a 90, 95, or 99 percent confidence interval for μ.
8-3
Chapter 8
Sampling Distributions and Estimation
Chapter Learning Objectives (LO’s)
LO8-6: Know when to use Student’s t instead of z to estimate μ.
LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.
LO8-8: Construct confidence intervals for finite populations.
LO8-9: Calculate sample size to estimate a mean or proportion.
LO8-10: Construct a confidence interval for a variance (optional).
8-4
•
•
•
Chapter 8
8.1 Sampling Variation
Sample statistic – a random variable whose value depends on
which population items are included in the random sample.
Depending on the sample size, the sample statistic could either
represent the population well or differ greatly from the population.
This sampling variation can easily be illustrated.
8-5
•
Consider eight random samples of size n = 5 from a large
population of GMAT scores for MBA applicants.
•
The sample means tend to be close to the population mean
(m = 520.78).
Chapter 8
8.1 Sampling Variation
8-6
•
Chapter 8
8.1 Sampling Variation
The dot plots show that the sample means have much less variation
than the individual sample items.
8-7
8.2 Estimators and Sampling Distributions
Chapter 8
LO8-1
LO8-1: Define sampling error, parameter and estimator.
Some Terminology
•
•
•
Estimator – a statistic derived from a sample to infer the value of
a population parameter.
Estimate – the value of the estimator in a particular sample.
Population parameters are usually represented by
Greek letters and the corresponding statistic
by Roman letters.
8-8
8.2 Estimators and Sampling Distributions
Examples of Estimators
Chapter 8
LO8-1
Sampling Distributions
•
•
The sampling distribution of an estimator is the probability distribution of
all possible values the statistic may assume when a random sample of
size n is taken.
Note: An estimator is a random variable since samples vary.
8-9
•
8.2 Estimators and Sampling Distributions
Chapter 8
LO8-1
Sampling error is the difference between an estimate and the
corresponding population parameter. For example, if we use the sample
mean as an estimate for the population mean, then the
Bias
•
Bias is the difference between the expected value of the estimator and
the true parameter. Example for the mean,
•
An estimator is unbiased if its expected value is the parameter being
estimated. The sample mean is an unbiased estimator of the population
mean since
•
On average, an unbiased estimator neither overstates nor understates
the true parameter.
8-10
8.2 Estimators and Sampling Distributions
Chapter 8
LO8-1
8-11
8.2 Estimators and Sampling Distributions
Chapter 8
LO8-2
LO8-2: Explain the desirable properties of estimators.
Note: Also, a desirable property for an estimator is for it to be unbiased.
Efficiency
•
•
Efficiency refers to the variance of the estimator’s sampling
distribution.
Figure 8.6
A more efficient estimator has smaller variance.
8-12
8.2 Estimators and Sampling Distributions
Chapter 8
LO8-2
LO8-2: Explain the desirable properties of estimators.
Consistency
A consistent estimator converges toward the parameter being estimated
as the sample size increases.
Figure 8.6
8-13
8.3 Sample Mean and the Central Limit Theorem
Chapter 8
LO8-3
LO8-3: State the Central Limit Theorem for a mean.
The Central Limit Theorem is a powerful result that allows us to
approximate the shape of the sampling distribution of the sample
mean even when we don’t know what the population looks like.
8-14
•
8.3 Sample Mean and the Central Limit Theorem
If the population is exactly
normal, then the sample mean
follows a normal distribution.
•
Chapter 8
LO8-3
As the sample size n increases, the
distribution of sample means narrows
in on the population mean µ.
8-15
•
8.3 Sample Mean and the Central Limit Theorem
If the sample is large enough, the sample means will have
approximately a normal distribution even if your population is not
normal.
Chapter 8
LO8-3
8-16
8.3 Sample Mean and the Central Limit Theorem
Illustrations of Central Limit Theorem
Chapter 8
LO8-3
Using the uniform
and a right skewed
distribution.
Note:
8-17
8.3 Sample Mean and the Central Limit Theorem
Applying The Central Limit Theorem
Chapter 8
LO8-3
The Central Limit Theorem permits us to define an interval within which
the sample means are expected to fall. As long as the sample size n is
large enough, we can use the normal distribution regardless of the
population shape (or any n if the population is normal to begin with).
8-18
8.3 Sample Mean and the Central Limit Theorem
Chapter 8
LO8-4
LO8-4: Explain how sample size affects the standard error.
Sample Size and Standard Error
Even if the population standard deviation σ is large, the sample means
will fall within a narrow interval as long as n is large. The key is the
standard error of the mean:.. The standard error decreases as n
increases.
For example, when n = 4 the standard error is halved. To halve it again
requires n = 16, and to halve it again requires n = 64. To halve the
standard error, you must quadruple the sample size (the law of
diminishing returns).
8-19
Illustration: All Possible Samples from a Uniform Population
•
Consider a discrete uniform population consisting of the integers
{0, 1, 2, 3}.
•
The population parameters are: m = 1.5,  = 1.118.
Chapter 8
8.3 Sample Mean and the Central Limit Theorem
8-20
Illustration: All Possible Samples from a Uniform Population
•
Chapter 8
8.3 Sample Mean and the Central Limit Theorem
The population is uniform, yet the distribution of all possible
sample means of size 2 has a peaked triangular shape.
8-21
Chapter 8
LO8-5
8.4 Confidence Interval for a Mean (m) with
known ()
LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.
What is a Confidence Interval?
8-22
What is a Confidence Interval?
•
Chapter 8
LO8-5
8.4 Confidence Interval for a Mean (m) with
known ()
The confidence interval for m with known  is:
8-23
Choosing a Confidence Level
•
A higher confidence level leads to a wider confidence interval.
•
Greater confidence
implies loss of precision
(i.e. greater margin of
error).
95% confidence is
most often used.
•
Chapter 8
LO8-5
8.4 Confidence Interval for a Mean (m) with
known ()
Confidence Intervals for Example 8.2
8-24
Interpretation
•
•
•
Chapter 8
LO8-5
8.4 Confidence Interval for a Mean (m) with
known ()
A confidence interval either does or does not contain m.
The confidence level quantifies the risk.
Out of 100 confidence intervals, approximately 95% may contain m,
while approximately 5% might not contain m when constructing 95%
confidence intervals.
When Can We Assume Normality?
If  is known and the population is normal, then we can safely use the
formula to compute the confidence interval.
• If  is known and we do not know whether the population is normal, a common
rule of thumb is that n  30 is sufficient to use the formula as long as the
distribution
Is approximately symmetric with no outliers.
• Larger n may be needed to assume normality if you are sampling from a strongly
skewed population or one with outliers.
•
8-25
LO8-6: Know when to use Student’s t instead of z to estimate m.
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Student’s t Distribution
•
Use the Student’s t distribution instead of the normal distribution
when the population is normal but the standard deviation  is
unknown and the sample size is small.
8-26
LO8-6: Know when to use Student’s t instead of z to estimate m.
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Student’s t Distribution
8-27
Student’s t Distribution
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
t distributions are symmetric and shaped like the standard normal
distribution.
The t distribution is dependent on the size of the sample.
Comparison of Normal and Student’s t
Figure 8.11
8-28
Degrees of Freedom
•
•
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Degrees of Freedom (d.f.) is a parameter based on the sample
size that is used to determine the value of the t statistic.
Degrees of freedom tell how many observations are used to
calculate , less the number of intermediate estimates used in
the calculation. The d.f for the t distribution in this case, is given
by d.f. = n -1.
As n increases, the t distribution approaches the shape of the
normal distribution.
For a given confidence level, t is always larger than z, so a
confidence interval based on t is always wider than if z were used.
8-29
Comparison of z and t
•
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
For very small samples, t-values differ substantially from the
normal.
As degrees of freedom increase, the t-values approach the
normal z-values.
For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 =
30.
So for a 90 percent confidence interval, we would use
t = 1.697, which is only slightly larger than z = 1.645.
8-30
Example GMAT Scores Again
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Figure 8.13
8-31
Example GMAT Scores Again
•
Construct a 90% confidence interval for the mean GMAT score of
all MBA applicants.
x = 510
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
s = 73.77
Since  is unknown, use the Student’s t for the confidence interval
with d.f. = 20 – 1 = 19.
First find t/2 = t.05 = 1.729 from Appendix D.
8-32
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
For a 90% confidence
interval, use Appendix
D to find t0.05 = 1.729
with d.f. = 19.
Note: One can use Excel,
Minitab, etc. to
obtain these values
as well as to
construct confidence
Intervals.
We are 90 percent confident
that the true mean GMAT
score might be within the
interval [481.48, 538.52]
8-33
Confidence Interval Width
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Confidence interval width reflects
- the sample size,
- the confidence level and
- the standard deviation.
To obtain a narrower interval and more precision
- increase the sample size or
- lower the confidence level (e.g., from 90% to 80% confidence).
8-34
Using Appendix D
•
•
•
•
Chapter 8
LO8-6
8.5 Confidence Interval for a Mean (m) with
Unknown ()
Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10.
If the table does not give the exact degrees of freedom, use the
t-value for the next lower degrees of freedom.
This is a conservative procedure since it causes the interval to be
slightly wider.
A conservative statistician may use the t distribution for
confidence intervals when σ is unknown because
using z would underestimate the margin of error.
8-35
8.6 Confidence Interval for a Proportion ()
Chapter 8
LO8-7
LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.
•
A proportion is a mean of data whose only values are 0 or 1.
8-36
8.6 Confidence Interval for a Proportion ()
Applying the CLT
•
Chapter 8
LO8-7
The distribution of a sample proportion p = x/n is symmetric if  = .50
and regardless of , approaches symmetry as n increases.
8-37
8.6 Confidence Interval for a Proportion ()
Chapter 8
LO8-7
When is it Safe to Assume Normality of p?
•
Rule of Thumb: The sample proportion p = x/n may be assumed to
be normal if both n 10 and n(1- ) 10.
Sample size to assume
normality:
Table 8.9
8-38
8.6 Confidence Interval for a Proportion ()
Confidence Interval for 
•
Chapter 8
LO8-7
Since  is unknown, the confidence interval for p = x/n
(assuming a large sample) is
8-39
8.6 Confidence Interval for a Proportion ()
Chapter 8
LO8-7
Example Auditing
8-40
8.7 Estimating from Finite Population
Chapter 8
LO8-8
LO8-8: Construct Confidence Intervals for Finite Populations.
N = population size; n = sample size
8-41
8.8 Sample Size determination for a Mean
Chapter 8
LO8-9
LO8-9: Calculate sample size to estimate a mean or proportion.
Sample Size to Estimate m
•
To estimate a population mean with a precision of + E (allowable
error), you would need a sample of size. Now,
8-42
8.8 Sample Size determination for a Mean
How to Estimate ?
Chapter 8
LO8-9
•
Method 1: Take a Preliminary Sample
Take a small preliminary sample and use the sample s in place of
 in the sample size formula.
•
Method 2: Assume Uniform Population
Estimate rough upper and lower limits a and b and set
 = [(b-a)/12]½.
•
Method 3: Assume Normal Population
Estimate rough upper and lower limits a and b and set  = (b-a)/4.
This assumes normality with most of the data with m ± 2 so the
range is 4.
•
Method 4: Poisson Arrivals
In the special case when m is a Poisson arrival rate, then  = m .
8-43
8.9 Sample Size determination for a Proportion
•
To estimate a population proportion with a precision of ± E
(allowable error), you would need a sample of size
•
Since  is a number between 0 and 1, the allowable error E is
also between 0 and 1.
Chapter 8
LO8-9
8-44
8.9 Sample Size determination for a Proportion
How to Estimate ?
•
•
•
Chapter 8
LO8-9
Method 1: Assume that  = .50
This conservative method ensures the desired precision. However,
the sample may end up being larger than necessary.
Method 2: Take a Preliminary Sample
Take a small preliminary sample and use the sample p in place of 
in the sample size formula.
Method 3: Use a Prior Sample or Historical Data
How often are such samples available? Unfortunately,  might be
different enough to make it a questionable assumption.
8-45
LO8-10 8.10 Confidence Interval for a Population Variance (2)
LO8-10: Construct a confidence interval for a variance (optional).
Chi-Square Distribution
•
•
If the population is normal, then the sample variance s2
follows the chi-square distribution (c2) with degrees of
freedom d.f. = n – 1.
Lower (c2L) and upper (c2U) tail percentiles for the chisquare distribution can be found using Appendix E.
8-46
LO8-10 8.10 Confidence Interval for a Population Variance (2)
LO8-10: Construct a confidence interval for a variance (optional).
Confidence Interval
•
Using the sample variance s2, the confidence interval is
•
To obtain a confidence interval for the standard deviation
, just take the square root of the interval bounds.
8-47
LO8-10 8.10 Confidence Interval for a Population Variance (2)
You can use Appendix E to find critical chi-square values.
8-48
LO8-10 8.10 Confidence Interval for a Population Variance (2)
Caution: Assumption of Normality
•
•
The methods described for confidence interval estimation of the
variance and standard deviation depend on the population having a
normal distribution.
If the population does not have a normal distribution, then the
confidence interval should not be considered accurate.
8-49
```
Related documents