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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

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

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

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

Not likely. Reject the hypothesis

The mean height of male students at Cal State San Marcos is 6 feet

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

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

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

Upper tail test

- The cutoff z-score. z
- The corresponding z-score which makes

P(z> z)=

- In other words, P(0

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

- Decision rule
- If zx > z, reject H0
- If zx≤z, do not reject H0

BUS304 – Chapter 8 Hypothesis for Mean

An alternative way to test: use p-value

- 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

H0: μ≥ 3

HA: μ < 3

Reject when the

sample mean is too low

Lower tail test- 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

H0: μ= 3

HA: μ 3

/2

/2

Two-tailed tests- 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

H0: μ= 3

HA: μ 3

/2

/2

Decision Rule for two-tailed tests- 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

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

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

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