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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
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 h 0
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
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
Exercise
  • Problem 8.1 (Page323)

BUS304 – Chapter 8 Hypothesis for Mean

hypothesis testing process
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
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

exercise1
Exercise
  • Problem 8.7 (Page 323)

BUS304 – Chapter 8 Hypothesis for Mean

two kinds of tests
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
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
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

example
Example
  • Problem 8.3 (P323)

BUS304 – Chapter 8 Hypothesis for Mean

an alternative way to test use p value
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
More Exercise
  • Problem 8.4

BUS304 – Chapter 8 Hypothesis for Mean

lower tail test
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

two tailed tests
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

decision rule for two tailed tests
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
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
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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