1. 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?
BUS304 – Chapter 8 Hypothesis for Mean

2. 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
Conventionally, it always contains an equal sign.
e.g.  ≥ 4,  ≤ 6, or  = 10
BUS304 – Chapter 8 Hypothesis for Mean

3. Alternative Hypothesis
The opposite of null hypothesis
Written as HA.
Example:
The mean price of a beach house in Carlsbad is at least $1million dollars
The mean gas price in CA is no higher than $3 per gallon
The mean weight of a football quarterback is $200lbs.
H0: μ ≥ $1million
HA: μ < $1million
H0: μ ≤ $3 per gallon
HA: μ > $3 per gallon
H0: μ = 200lbs
HA: μ  200lbs
BUS304 – Chapter 8 Hypothesis for Mean

4. BUS304 – Chapter 8 Hypothesis for Mean
Exercise
Problem 8.1 (Page323)
BUS304 – Chapter 8 Hypothesis for Mean

5. 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.
sample
The mean height of male students at Cal State San Marcos is 6 feet
Not likely. Reject the hypothesis
BUS304 – Chapter 8 Hypothesis for Mean

6. BUS304 – Chapter 8 Hypothesis for Mean
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.
BUS304 – Chapter 8 Hypothesis for Mean

7. BUS304 – Chapter 8 Hypothesis for Mean
Exercise
Problem 8.7 (Page 323)
BUS304 – Chapter 8 Hypothesis for Mean

8. BUS304 – Chapter 8 Hypothesis for Mean
Two kinds of tests
One-tailed test:
Upper tail test (e.g.  ≤ $1000)
Lower tail test (e.g.  ≥$800)
Reject when the sample mean is too high
Reject when the sample mean is too low
Two-tailed test:
 =$1000
Reject when the sample mean is either too high or too low
BUS304 – Chapter 8 Hypothesis for Mean

9. 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
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
Sample variance (or standard deviation):
s2 or s
BUS304 – Chapter 8 Hypothesis for Mean

10. BUS304 – Chapter 8 Hypothesis for Mean
Upper tail test
H0: μ ≤ 3
HA: 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) = 
Decision rule
If zx > z, reject H0
If zx ≤ z, do not reject H0
BUS304 – Chapter 8 Hypothesis for Mean

11. BUS304 – Chapter 8 Hypothesis for Mean
Example
Problem 8.3 (P323)
BUS304 – Chapter 8 Hypothesis for Mean

12. An alternative way to test: use p-value
The probability of getting the sample mean or higher.
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
H0: μ ≤ 3
HA: μ > 3
The p-value of
the sample mean
BUS304 – Chapter 8 Hypothesis for Mean

13. BUS304 – Chapter 8 Hypothesis for Mean
More Exercise
Problem 8.4
BUS304 – Chapter 8 Hypothesis for Mean

14. BUS304 – Chapter 8 Hypothesis for Mean
Lower tail test
Reject when the
sample mean is too low
H0: μ ≥ 3
HA: μ < 3
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
BUS304 – Chapter 8 Hypothesis for Mean

15. BUS304 – Chapter 8 Hypothesis for Mean
Two-tailed tests
H0: μ = 3
HA: μ  3
/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:
BUS304 – Chapter 8 Hypothesis for Mean

16. Decision Rule for two-tailed tests
/2
H0: μ = 3
HA: μ  3
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
BUS304 – Chapter 8 Hypothesis for Mean

17. BUS304 – Chapter 8 Hypothesis for Mean
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
BUS304 – Chapter 8 Hypothesis for Mean

18. 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
( )
Drawing always help!
Step 6: Find out the z-score or t-score for sample mean ( )
Step 7: compare and make the right decision.
BUS304 – Chapter 8 Hypothesis for Mean