1
SNOTES 4
Chapter 8 Sampling methods and Distributions
Objectives:
Note: Sampling distributions should be used whenever information
about a sample is used to make inferences about the population.
Sampling error - The difference between a sample statistic and its
corresponding population parameter. Page 269
Sampling error comes from:
1. Sample size
2. Sample Bias
1. Characteristics of sample are not the same as the
population
2. Cognition issues-see page 8-9 of snotes 3
How do you solve for sampling bias:
1. Random samples
2. Systematic sampling
3. Cluster sampling
4. Stratified sampling
Sampling distribution of sample means - A probability distribution
of all possible sample means of a given sample size. A sampling
distribution includes every possible sample statistic of a certain
sample size that can be drawn from a population. Pg 259
Number in a sampling distribution - combination
Population
Samples
Sampling Distribution
Distribution
nCx
------------------------------------------------_
_
MEAN

X
E(X) = 
STANDARD DEVIATION
σ
Sx
Standard Deviation of the
Mean
Excel - Tools/sampling
STANDARD DEVIATION OF THE MEAN 8-1
𝜎𝑥̅ =
𝜎𝑥
√𝑛
2
Pg 280
Rule for Distribution of Sample Means for Normal Population
If the population for X is normally distributed with mean  and
_
standard deviation σ, the sample mean X is normally distributed with
mean  and standard deviation shown above.
8-2 Population standard deviation is known page 282
𝑧=
𝑥− 𝜇
𝜎
⁄ 𝑛
√
Central Limit Theorem: If all samples of a particular size are
selected from any population, the sampling distribution of the sample
means is approximately a normal distribution. This approximation
improves with larger samples. Pg 274
Chapter 9 Estimation and Confidence Intervals
1. The student will be able to explain and interpret a
confidence interval and confidence level. Pg 294296
2. Student will be able to compute and interpret
confidence intervals for a unknown population
standard
deviation. Pg 302
2. The student will be able to estimate a confidence
interval for proportions. Pg 309
3. The student will be able to estimate the proper
sample size necessary for a desired confidence
level, width and standard deviation of the
population. Pg 315-317
Confidence Interval A range of values constructed from sample data
so the population parameter occurs within that range at a specified
probability. The specified probability is called the level of
confidence.
3
Formula 9-1 Confidence Interval for : σ Known Pg 298
𝑥̅ ± 𝑧
𝜎
√𝑛
The value 1 - α is termed the confidence coefficient.
the value 100(1 - ά)% is known as the confidence level. Type one error
is called α
Population Standard Deviation σ Unknown page 302 9-2
𝑥̅ ± 𝑡
𝑠
√𝑛
Excel - fx paste function/statistics/confidence
USING A SAMPLING DISTRIBUTION Z VALUES
Average Height of males in U.S. U = 70"
N = 70,000,000
P =X/n
9-3
n = 10,000
page 310
Confidence Interval For a Population Proportion
𝑝 ± 𝑧√
9-4
𝑝(1−𝑃)
𝑛
Sample Size for estimating the population mean Pg 316
𝑧𝜎 2
𝑛 = (𝐸)
Estimating sample size for proportions p Pg 317
𝑧 2
𝑛 = 𝑝(1 − 𝑃) (𝐸)
4
PROPORTIONS
N = 100,000,000
n = 1000 200 smokers
Chapter 10 HYPOTHESIS TESTING
1. The student should know the five steps of
hypothesis testing. Pg 332
2. The student will be able to define type I and type
errors and p-values. Pg 334-335,356-357
II
3. The student will be able to perform and interpret a
one or two sided test using a large population Pg
337-343
4. The student will be able to perform and interpret
the results of a one and two sided hypothesis test
for
a population mean or proportion. Pg 353
The goal in estimation is to estimate the value of some
population parameter(u).
The goal in significance testing is to decide if a claim about
a population parameter is true.
Hypothesis testing involves using sample data to test
statements, claims, or assumptions about population parameters.
Steps of Hypothesis Testing Pg 332
1. Formulate the null hypothesis Ho.
Formulate the alternative hypothesis Ha in statistical terms.
2. Set the level of significance α and the sample size n.
3. Select the appropriate test statistic and rejection rule.
4. Collect the data and calculate the test statistic.
5. If the calculated value of the test statistic falls in the
rejection region, then reject Ho. If the calculated value of the
test statistic does not fall in the rejection region, then do not
reject Ho.
Testing a mean σ known
10-1 page 335
5
𝑧=
𝑥̅ − 𝜇
𝜎
⁄ 𝑛
√
t Distribution testing a mean σ unknown
𝑡=
10-2 page 345
𝑥̅ − 𝜇
𝑠
⁄ 𝑛
√
NULL HYPOTHESIS - OVERHEAD OF ERRORS
The null hypothesis usually states that the difference between
the sample statistic and its claimed population parameter is due to
chance variation in sampling.
Ho: B=0
H1: B0
CRITICAL VALUES - OVERHEAD FOR HYPOTHESIS TESTING
Type I error occurs if we reject Ho when Ho is true. Pg 468
Type II error occurs if we do not reject Ho when Ho is false.
In our legal system the null hypothesis is the individual is presumed
innocent. The prosecution gathers a sample of evidence and presents
this to a judge or jury. Form this sample of evidence the legal system
attempts to disprove the null hypothesis. If the null hypothesis is
rejected, the court accepts the alternative hypothesis the
individual is not innocent.
Ho:
Ha:
α = P(Rejecting Ho when Ho is true) = P(type I error)
β = P(Not rejecting Ho when Ho is false) = P(type II error)
[What is the cost of a type one error to society?]
[What is the cost of a type II error to society?]
TTEST mu=K C
ONESIDED TESTS
ENGINES - PISTON CLEARANCE
Ho: U  .001
6
H1: U D .001
TYPE I ERROR COST IS TO TEAR DOWN AND RE-ENGINEER ALL ENGINES
TYPE II WARRANTY WORK
Ho: U  .001
H1: U < .001
IF YOU ACCEPT THE NULL HYPOTHESIS THE FIRM IS NOT SURE
WHETHER THE ENGINE IS ADEQUATE OR NOT.
PERFUME IF SOLUTION IS GREATER THAN 1% CAUSES SKIN IRRITATION
Ho: U  .01
H1: U > .01
TYPE I THROW OUT AND START OVER
TYPE II LAWSUITS
[Test of the painkiller Vioxx for heart problems showed 45 of Vioxx
users had a heart attack out of 1287 test group. The placebo group
had 25 heart attacks out of 1299. Is there a statistically significant
difference? WSJ 2/7/05]
Test of hypothesis, one proportion 10-3
𝑧=
𝑝− 𝜋
√
4/2/11
page 354
𝜋(1−𝜋)
𝑛
Making a Stat Less Significant WSJ A3
Reference:
7/28/93 AStatisticians Occupy Front Lines in Battle Over Passive
Smoking@ WSJ Confidence intervals
2/25/00 ANielson ratings Spark a Battle Over Just Who Speaks Spanish@
WSJ
10/24/06 ACounting War Dead Is Difficult-Therefore, Let=s Not
Exaggerate@ WSJ A19