Chapter 8 hypothesis tests
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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

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<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

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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