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Chapter 8 Hypothesis Tests. What are Hypothesis Tests ? A set of methods and procedure to s tudy the reliability of claim s about population parameter s. Examples of Hypotheses :. The mean monthly cell phone bill of this city is \$42.

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Chapter 8 Hypothesis Tests

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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

### Exercise

• Problem 8.1 (Page323)

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

### Exercise

• Problem 8.7 (Page 323)

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<z< z) = 0.5 - 

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

### Example

• Problem 8.3 (P323)

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

### More Exercise

• Problem 8.4

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