Transcript Ch 8 - California State University San Marcos

Chapter 8
Hypothesis Tests

What are Hypothesis Tests?
A set of methods and procedure to study the reliability of
claims about population parameters.
Examples of Hypotheses:
The mean monthly cell phone bill of this city
is $42.
The mean dividend return of Oracle stock is
higher than $3 per share.
The mean price of a Cannon Powershot G6
camera on Internet is less than $430.
Why do we do hypothesis tests?
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Constructing
a null hypothesis H0
 A null hypothesis is the basis for testing.
 Null Hypothesis H0
 Mathematical statement of the assumption to be
tested
 Example: The average number of TV sets in U.S.
Homes is at least three ( H0:  ≥ 3 )
 The null hypothesis is always about the
population parameter, not about a sample statistic
H0 : μ  3
H0 : x  3
 Conventionally, it always contains an equal sign.
e.g.  ≥ 4,  ≤ 6, or  = 10
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Alternative Hypothesis
 The opposite of null hypothesis
 Written as HA.
 Example:
1. The mean price of a beach house in
Carlsbad is at least $1million dollars
H0: μ ≥ $1million
HA: μ < $1million
2. The mean gas price in CA is no higher than
$3 per gallon
H0: μ ≤ $3 per gallon
HA: μ > $3 per gallon
3. The mean weight of a football quarterback is
$200lbs.
H0: μ = 200lbs
HA: μ  200lbs
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Exercise
 Problem 8.1 (Page323)
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Hypothesis Testing
Process
 We want to test whether the null
hypothesis is true.
 In statistics, we can never say a hypothesis
is wrong for sure.
 We can only evaluate the probability that the
hypothesis is true
 If the probability is too small, we say we
reject the null hypothesis
 Otherwise, we say we fail to reject the null
hypothesis.
x  5.5
sample
The mean height of male
students at Cal State San
Marcos is 6 feet
H0 :μ  6
H A :μ  6
Not likely.
Reject the
hypothesis
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Types of errors
 Type I error
 Rejecting the null hypothesis when it is, in fact,
true.
 It may happen when you decide to reject the
hypothesis.
-- you decide to reject the hypothesis when your result
suggests that the hypothesis is not likely to be true.
However, there is a chance that it is true but you get a
bad sample.
 Type II error
 Failing to reject the null hypothesis when it is, in
fact, false.
 It may happen when you decide not to reject.
 Whatever your decision is, there is always a
possibility that you make at least one mistake.
 The issue is which type error is more serious
and should not be made.
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Exercise
 Problem 8.7 (Page 323)
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Two kinds of tests
 One-tailed test:
 Upper tail test (e.g.  ≤ $1000)
Reject when the sample
mean is too high
 Lower tail test (e.g.  ≥$800)
Reject when the sample
mean is too low
Two-tailed test:
  =$1000
Reject when the sample
mean is either too high
or too low
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Information needed in
hypothesis tests
 When  is known
 The claimed range of mean  (i.e. H0 and HA)
 When to reject: level of significance 
• i.e. if the probability is too small (even smaller
than ), I reject the hypothesis.
 Sample size n
 Sample mean
x
 When  is unknown
 The claimed range of mean  (i.e. H0 and HA)
 When to reject: level of significance 
• i.e. if the probability is too small (even smaller
than ), I reject the hypothesis.
 Sample size n
 Sample mean x
 Sample variance (or standard deviation):
s2 or s
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Upper tail test
H0: μ ≤ 3
H A: μ > 3
Reject when the sample
mean is too high
z
 Level of Significance: 
 Generally given in the task
 The maximum allowed probability of type I error
 In other words, the size of the blue area
 The cutoff z-score. z
 The corresponding z-score which makes
P(z> z)= 
 In other words, P(0<z< z) = 0.5 - 
 Decision rule
 If zx > z, reject H0
 If zx ≤ z, do not reject H0
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Example
 Problem 8.3 (P323)
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An alternative way to test:
use p-value
 p-value:
 The probability of getting the sample mean or
higher.
H 0: μ ≤ 3
HA: μ > 3
The p-value of
the sample mean
3
x
 Reject if the p-value is too small
• i.e. even smaller than 
• It is too insignificant.
 Exercise:
 Use the p-value method to test the hypotheses in
Problem 8.3
 Think: what is the probability of making type 1
and type 2 errors
 if you reject the hypothesis
 If you fail to reject the hypothesis
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More Exercise
 Problem 8.4
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Lower tail test
H0: μ ≥ 3
H A: μ < 3
Reject when the
sample mean is too low
 The cutoff z score is negative
 z <0
 Decision rule:
 If zx < z, reject H0
 If zx ≥ z, do not reject H0
 The hypothesis is rejected only when you get a
sample mean too low to support it.
 Exercise: Problem 8.5 (Page 323)
assuming that =210
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Two-tailed tests
H 0: μ = 3
HA: μ  3
/2
/2
 The null hypothesis is rejected when the
sample mean is too high or too low
 Given a required level of significance 
 There are two cutoffs. (symmetric)
 The sum of the two blue areas is .
 So each blue area has the size /2.
 The z-scores:
z and -z
2
2
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Decision Rule for twotailed tests
H 0: μ = 3
HA: μ  3
/2
/2
 Decision rule for two-tailed tests
 If zx > z/2, reject H0
 Or, if zx < -z/2, reject H0
 Otherwise, do not reject H0
Exercise 8.8
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When  is unknown
 Now we use the sample standard deviation (i.e.
s) to estimate the population standard deviation
 The distribution is a t-distribution,
Not Normal !
You should check the t-table P597
Pay attention to the degree of freedom: n-1
 The rest of the calculations are the same.
Exercise 8.5 – lower tail test
Exercise 8.14 – upper tail test
Exercise 8.16 – two-tailed test
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Summary of Hypothesis
testing Steps

Step 1: Construct the hypotheses pair H0 and HA.

Step 2: Whether  is given?
 Given: use z-score (page 595)
 Unknown: use t-score (page 597)
• Need to have s (sample standard deviation)
• Degree of freedom: n-1

Step 3: Determine the decision rule
 One-tailed? Upper or lower?
 Two-tailed?
 Write down the decision rule based on the type of tests.

Step 5: Find out the cutoff z-score or t-score
(z

or t for one tailed. z  or t  for two-tailed.)
Drawing always help!
2
2

Step 6: Find out the z-score or t-score for sample

mean ( z x or t x )
Step 7: compare and make the right decision.
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