hypothesis testing

Transform to the power of digital

Hypothesis TestingDecember 2014

Theory vs Hypothesis

A claim(assumption) about a population parameter

What is a hypothesis?

– population mean

– population proportion

Example: The mean monthly cell phone bill of this city is = $42

Example: The proportion of adults in this city with cell phones is p = .68

States the assumption (numerical) to be tested

Example: The average number of TV sets in U.S. Homes is at least 3

Is always about a population parameter, not about a sample statistic

Null Hypothesis(H0)

3μ:H0

3μ:H0 3x:H0

States the assumption (numerical) to be tested

Example: The average number of TV sets in U.S. Homes is at least 3

Is always about a population parameter, not about a sample statistic

The opposite of the null hypothesis

Example: The average number of TV sets in U.S. homes is less than 3 ( HA: < 3 )

Null Hypothesis(H0)

3μ:H0

3μ:H0 3x:H0

Alternative Hypothesis(HA)

CLAIM: The population mean age is 50H0: = 50

Hypothesis Testing Process

Population Sample

Select a random sample

Suppose the sample mean age is 20x = 20

Population Sample

Suppose the sample mean age is 20x = 20

Population Sample

Is x=20 likely if = 50?

Reject the null hypothesis

Do not reject the null hypothesis

Reject the null hypothesis

Reason for rejecting H0

Sampling Distribution of x

If it is unlikely that we would get a sample mean of this value ...

= 50If H0 is true

... if in fact this were the population mean…

= 50If H0 is true

... then we reject the null hypothesis that

... if in fact this were the population mean…

Level of significance,

Defines unlikely values of sample statistic if null hypothesis is true

Defines rejection region of the sampling distribution

Rejection region is shaded

H0: μ ≥ 3

HA: μ < 3 0

H0: μ ≤ 3

HA: μ > 3

Represents critical value

Lower tail test

0Upper tail test

Rejection region is shaded

H0: μ ≥ 3

HA: μ < 3 0

H0: μ ≤ 3

HA: μ > 3

H0: μ = 3

HA: μ ≠ 3

Represents critical value

Lower tail test

Upper tail test

Two tailed test

Critical Value Approach to Testing

Convert sample statistic (e.g.: x ) to test statistic ( Z or t statistic )

Known Unknown

s : population standard deviations : sample standard deviation

n : sample sizeμ : assumed population mean

x : sample mean

Determine the critical value(s) for a specified level of significance from a table or computer

If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0

Example

Test the claim that the true mean # of

TV sets in US homes is at least 3

(Assume σ = 0.8)

Example

(Assume σ = 0.8)

Specify the population value of interest1

The mean number of TVs in US homes

Example

(Assume σ = 0.8)

Formulate the appropriate null and alternative hypotheses2

H0: μ 3

HA: μ < 3 (This is a lower tail test)

Example

(Assume σ = 0.8)

Specify the desired level of significance3Suppose that = .05 is chosen for this test

Example

(Assume σ = 0.8)

Determine the rejection region4

Specify the desired level of significance3

Example

(Assume σ = 0.8)

Reject H0 Do not reject H0

-zα= -1.645

This is a one-tailed test with =.05.

Since σ is known, the cutoff value is a z value

Reject H0 if z < z = -1.645 ;

otherwise do not reject H0

Example

(Assume σ = 0.8)

Obtain sample evidence and compute the test statistic5

Suppose a sample is taken with the following results:

n = 100, x = 2.84

( = 0.8 is assumed known)

2.0.08

0.832.84

Example

(Assume σ = 0.8)

Reach a decision and interpret the result6

-1.645 0

Example

(Assume σ = 0.8)

-1.645 0

Since z = -2.0 < -1.645, we reject the null hypothesis that the mean number of TVs in US homes is at least 3

Example

(Assume σ = 0.8)

-1.645 0

Since z = -2.0 < -1.645, we reject the null hypothesis that the mean number of TVs in US homes is at least 3

Steps to Follow

p – Value Approach to Testing

p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H0 is true

Same as before

Obtain the p-value from a table or computer

Compare the p-value with

If p-value < , reject H0

If p-value , do not reject H0

Example

(Assume σ = 0.8)

Example

(Assume σ = 0.8)

Suppose a sample is taken with the following results:

n = 100, x = 2.84

( = 0.8 is assumed known)

2.8684100

0.81.6453

σzμx αα zα= 1.645

Example

(Assume σ = 0.8)

p-value =.0228

2.8684 3

.02282.0)P(z

1000.8

3.02.84zP

3.0)μ|2.84xP(

Here: p-value = .0228 = .05

Since .0228 < .05, we reject the null hypothesis

Example

(Assume σ = 0.8)

Reach a decision and interpret the result6Compare the p-value with

If p-value < , reject H0

If p-value , do not reject H0

p-value =.0228

2.8684 3

Two-Tailed Test

Reject H0Reject H0

a/2=.025

Do not reject H0

a/2=.025

Hypothesis Tests for Proportions

Involves categorical values

Two possible outcomes

“Success” (possesses a certain characteristic)

“Failure” (does not possesses that characteristic)

Fraction or proportion of population in the “success” category is denoted by p

sizesample

sampleinsuccessesofnumber

sizesample

p can be approximated by a normal distribution with mean

and standard deviation pμP

p)p(1σ

sizesample

p can be approximated by a normal distribution with mean

and standard deviation pμP

p)p(1σ

Example

A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses.

Test at the = .05 significance level.

Example

500.08).08(1

.08.05

np)p(1

Example

500.08).08(1

.08.05

np)p(1

Critical Values: ± 1.96

Reject Reject

.025.025

1.96-2.47

Example

500.08).08(1

.08.05

np)p(1

Critical Values: ± 1.96

Reject Reject

.025.025

1.96-2.47

Reject H0 at = .05

There is sufficient evidence to reject the company’s claim of 8% response rate.

Errors in Making Decisions

Reject a true null hypothesisType I error

Fail to reject a false null hypothesisType II error

The probability of Type I Error is Called level of significance of the test

The probability of Type I Error is β1- β is called the power of the test, i.e. Prob(rejecting the false null hypothesis)

Errors in Making Decisions

Reject a true null hypothesisType I error

Fail to reject a false null hypothesisType II error

The probability of Type I Error is Called level of significance of the test

The probability of Type I Error is β1- β is called the power of the test, i.e. Prob(rejecting the false null hypothesis)

H0 True H0 False

Reject No error (1 - α)

Type II Error ( β )

Do not reject Type I Error(α)

No Error ( 1 - β )

Key:Outcome

(Probability)

State of Nature

Hypothesis Song

Transform to the power of digital

Thank you

hypothesis testing

capgemini consulting

null hypothesish0 copyright

random sample copyright

notion of innocent

sample mean age

proven guiltyrefers

status quoalways

population parameterwhat

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