Download The Standard Normal Distribution

6/19/2009
Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
The Standard Normal Distribution
• ‘Bell Shaped’ & Symmetric
• The mean, μ, is always 0
• The standard deviation, σ , is always 1
To find P(z<a)
=NORMSDIST(a)
p(Z)
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To standardize
=STANDARDIZE(a,
Introduction to Hypothesis
Testing
Z
0
)
One-Sample Hypothesis Testing
To find a: P(z<a) = p
=NORMSINV(p)
…a is a value from the sample (for what we’re doing, a refers to your sample mean)

Hypothesis testing uses data to test a model
GENERAL PROCEDURE FOR HYPOTHESIS TESTING
1. Define the hypotheses
2. Calculate the test statistic
3. Make a decision

Hypothesis testing uses data to test a model
GENERAL PROCEDURE FOR HYPOTHESIS TESTING
1. Define the hypotheses
A hypothesis is a claim or assumption
about a population parameter


The first step in hypothesis testing is to translate your conjectures into
hypothesis statements.
Every statistical test tests the null hypothesis against an alternate hypothesis.
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The Null Hypothesis

NULL
Begin with the assumption that the null hypothesis is true
The Alternative Hypothesis

Similar to the notion of innocent until proven guilty



It refers to the status quo
Always contains “=” , “≤” or “ ” sign
May or may not be rejected
The type of test depends on what you want to prove
ONE-TAILED
 Rejection region in one of the tails
 The null is FALSE in one direction of your original claim
Is the opposite of the null hypothesis
Covers the area of the population not covered in the NULL



One-Tailed vs. Two-Tailed
Hypothesis Statements
ALT
Challenges the status quo
Never contains the “=” , “≤” or “ ” sign
Is generally the hypothesis that the researcher is trying to
prove
One-Tailed vs. Two-Tailed
Hypothesis Statements
The type of test depends on what you want to prove
ONE-TAILED
TWO-TAILED
 Rejection region in both of the tails
 The null can be FALSE in both direction of your original claim
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The Test Statistic and Critical Values


If the sample mean is close to the assumed population
mean, DO NOT REJECT the null hypothesis.
If the sample mean is far from the assumed population
mean, REJECT the null hypothesis.
The Test Statistic and Critical Values





If the sample mean is close to the assumed population
mean, DO NOT REJECT the null hypothesis.
If the sample mean is far from the assumed population
mean, REJECT the null hypothesis.
How far is “far enough” to reject H0?
The critical value of a test statistic creates a “line in the
sand” for decision making.
Hypothesis testing uses data to test a model
OBSERVED VALUES
GENERAL PROCEDURE FOR HYPOTHESIS TESTING
 Define the hypotheses
2. Calculate the test statistics

The benefit of a normally distributed sampling distribution is the
ability to standardize data and calculate associated probabilities…
We need to be able to answer the following:

How likely(or unlikely) is this set of data?

How likely (or unlikely) do we need it to be?
Z-score
P-value
Observed values
Critical values
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ONE-SAMPLE HYPOTHESIS TESTING
For the population mean, n > 30
CRITICAL VALUES
Distribution of the test statistic
Define the NULL & ALT hypotheses
1.
Implies
some
direction
Region of
Rejection
Region of
Rejection
…uses
comparisons like
‘bigger’, ‘less than’,
etc….
One-
Z-critical
or two-tailed?
Implies
equality
…uses
comparisons like
‘the same as’…
Critical Values
-value
ONE-SAMPLE HYPOTHESIS TESTING
For the population mean, n > 30
1.
Define the NULL & ALT hypotheses
ONE-SAMPLE HYPOTHESIS TESTING
For the population mean, n > 30
1.
Define the NULL & ALT hypotheses
2.
Calculate the test statistics
One- or two-tailed?
2.
Calculate the test statistics
zOBS = STANDARDIZE(x, , / n)
One- or two-tailed?
zOBS, zCRIT, p-value
zOBS = STANDARDIZE(x, , / n)
zCRIT = NORMSINV(1- /# of tails)
zOBS, zCRIT, p-value
zCRIT = NORMSINV(1- /# of tails)
p-value =(# of tails) * (1-NORMSDIST(|zOBS|))
p-value =(# of tails) * (1-NORMSDIST(|zOBS|))
Make a decision
3.


|zOBS| > zCRIT?
p-value < α-level?
…then REJECT the NULL
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ONE-SAMPLE HYPOTHESIS TESTING
For the population mean, n < 30
Define the NULL & ALT hypotheses
1.
ONE-SAMPLE HYPOTHESIS TESTING
For the population proportion
Define the NULL & ALT hypotheses
1.
One- or two-tailed?
One- or two-tailed?
NULL:
= 0.a
ALT:
≠ 0.a
----------------------------------------------------------------NULL:
≥ 0.a
NULL:
≤ 0.a
ALT:
< 0.a
ALT:
> 0.a
Calculate the test statistics
2.
tOBS, tCRIT, p-value
Use t-test!
(watch the video lecture & video example)
Make a decision
3.


|tOBS| > tCRIT?
p-value < α-level?
…then REJECT the NULL
ONE-SAMPLE HYPOTHESIS TESTING
ONE-SAMPLE HYPOTHESIS TESTING
For the population proportion
1.
Define the NULL & ALT hypotheses
For the population proportion
1.
Define the NULL & ALT hypotheses
2.
Calculate the test statistics
One- or two-tailed?
2.
Calculate the test statistics
(1
zOBS = STANDARDIZE(p, ,
One- or two-tailed?
zOBS, zCRIT, p-value
)
(1
n )
zOBS = STANDARDIZE(p, ,
zCRIT = NORMSINV(1- /# of tails)
zOBS, zCRIT, p-value
)
n )
zCRIT = NORMSINV(1- /# of tails)
p-value =(# of tails) * (1-NORMSDIST(|zOBS|))
p-value =(# of tails) * (1-NORMSDIST(|zOBS|))
Make a decision
3.


|zOBS| > zCRIT?
p-value < α-level?
…then REJECT the NULL
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