Chapter09 - Hypothesis Testing One-Sample Tests

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 9

Learning Objectives

In this chapter, you will learn:



_{The basic principles of hypothesis testing}



_{How to use hypothesis testing to test a mean or}

proportion



_{The assumptions of each hypothesis-testing}

procedure, how to evaluate them, and the

consequences if they are seriously violated



_{How to avoid the pitfalls involved in hypothesis}

testing

The Hypothesis



_{A hypothesis is a claim (assumption) about a}

population parameter:



_{population mean}



_{population proportion}

Example: The mean monthly cell phone bill of this

city is μ = $52

The Null Hypothesis, H

₀

3

μ

:

H

₀

_



_{States the assumption (numerical) to be tested}

Example:

The mean number of TV sets in U.S.

Homes is equal to three.



_{Is always about a population parameter, not about}

The Null Hypothesis, H

₀



_{Begin with the assumption that the null}

hypothesis is true



_{Similar to the notion of innocent until}

proven guilty



_{It refers to the status quo}

The Alternative Hypothesis, H

₁



_{Is the opposite of the null hypothesis}



_{e.g., The mean number of TV sets in U.S.}

homes is not equal to 3 ( H

1

: μ ≠ 3 )



_{Challenges the status quo}



_{Never contains the “=” , “≤” or “}

_

_{” sign}



_{May or may not be proven}



_{Is generally the hypothesis that the}

The Hypothesis Testing

Process



_{Claim: The population mean age is 50.}



_H

₀

_{: μ = 50,}

_H

₁

_{: μ ≠ 50}



_{Sample the population and find sample mean.}

Population

The Hypothesis Testing

Process



_{Suppose the sample mean age was X = 20.}



_{This is significantly lower than the claimed mean}

population age of 50.



_{If the null hypothesis were true, the probability of}

getting such a different sample mean would be very

small, so you reject the null hypothesis

.



_{In other words, getting a sample mean of 20 is so}

unlikely if the population mean was 50, you

The Hypothesis Testing

Process

Sampling

Distribution of X

μ = 50

If H

₀

is true

If it is unlikely that you

would get a sample

mean of this value ...

... then you reject

the null hypothesis

that μ = 50.

20

... if in fact this were

the population mean…

The Test Statistic and

Critical Values



_{If the sample mean is close to the assumed}

population mean, the null hypothesis is not

rejected.



_{If the sample mean is far from the assumed}

population mean, the null hypothesis is

rejected.



_{How far is “far enough” to reject H}

₀

_?



_{The critical value of a test statistic creates a}

The Test Statistic and

Critical Values

Critical Values

Distribution of the test statistic

Region of

Rejection

Errors in Decision Making



_{Type I Error}



_{Reject a true null hypothesis}



_{Considered a serious type of error}



_{The probability of a Type I Error is}

_



_{Called level of significance of the test}



_{Set by researcher in advance}



_{Type II Error}

Errors in Decision Making

Possible Hypothesis Test Outcomes

Actual Situation

Decision

H

₀

True

H

₀

False

Do Not

Reject H

₀

No Error

_{Probability 1 - α}

Type II Error

_Probability

_β

Reject H

₀

Type I Error

Probability α

No Error

Level of Significance, α

Two-tail test

_Rejection

region is

shaded

0

H

₀

: μ = 50

H

₁

: μ ≠ 50

Claim: The population

mean age is 50.



Hypothesis Testing: σ Known

For two tail test for the mean, σ known:



_{Convert sample statistic ( X ) to test statistic}



_{Determine the critical Z values for a specified}

level of significance

_

from a table or by using Excel



_{Decision Rule: If the test statistic falls in the rejection}

region, reject H

0

; otherwise do not reject H

0

n

σ

μ

X

Hypothesis Testing: σ Known

Do not reject H

₀

Reject H

₀

Reject H

₀



_{There are two}

cutoff values

(critical values),

Hypothesis Testing: σ Known

Example: Test the claim that the true mean weight of

chocolate bars manufactured in a factory is 3 ounces.



_{State the appropriate null and alternative hypotheses}



_{H0: μ = 3 H1: μ ≠ 3 (This is a two tailed test)}



_{Specify the desired level of significance}



_{Suppose that}

_

_{= .05 is chosen for this test}



_{Choose a sample size}

Hypothesis Testing: σ Known



_{Determine the appropriate technique}



_{σ is known so this is a Z test}



_{Set up the critical values}



_For

_

_{= .05 the critical Z values are ±1.96}



_{Collect the data and compute the test statistic}



_{Suppose the sample results are}

n = 100, X = 2.84

Hypothesis Testing: σ Known

Reject H

₀

Do not reject H

₀



_{Is the test statistic in the rejection region?}



= .05/2

-Z= -1.96

0

Reject H

₀

if

Z < -1.96 or

Z > 1.96;

otherwise do

not reject H

₀



= .05/2

Reject H

₀

+Z= +1.96

Hypothesis Testing: σ Known



_{Reach a decision and interpret the result}



_{Since Z = -2.0 < -1.96, you reject the null}

hypothesis and conclude that there is sufficient

evidence that the mean weight of chocolate

Hypothesis Testing: σ Known

6 Steps of Hypothesis Testing:

1. State the null hypothesis, H

0

and state the

alternative hypotheses, H

1

2. Choose the level of significance, α, and the sample

size n.

3. Determine the appropriate statistical technique and

the test statistic to use

Hypothesis Testing: σ Known

5.

Collect data and compute the test statistic from the

sample result

6. Compare the test statistic to the critical value to

determine whether the test statistic falls in the

region of rejection. Make the statistical decision:

Reject H

0

if the test statistic falls in the rejection

Hypothesis Testing: σ Known

p-Value Approach



_The

_p-value

_{is the probability of obtaining a}

test statistic equal to or more extreme ( < or

> ) than the observed sample value given H

0

is true



_{Also called observed level of significance}



_{Smallest value of}

_

_{for which H}

₀

_{can be}

Hypothesis Testing: σ Known

p-Value Approach



_{Convert Sample Statistic (ex. X) to Test}

Statistic (ex. Z statistic )



_{Obtain the p-value from a table or by using}

Excel



_{Compare the p-value with}

_



_{If p-value <}

_

_{, reject H}

₀

Hypothesis Testing: σ Known

p-Value Approach



_{Example: How likely is it to see a sample mean of}

Hypothesis Testing: σ Known

reject the null

Hypothesis Testing: σ Known

Confidence Interval Connections

100

confidence interval is:

2.6832 ≤ μ ≤ 2.9968

Hypothesis Testing: σ Known

One Tail Tests



_{In many cases, the alternative hypothesis}

focuses on a particular direction

H

₀

: μ ≥ 3

H

₁

: μ < 3

H

₀

: μ ≤ 3

H

₁

: μ > 3

This is a lower-tail test since the

alternative hypothesis is focused on the

lower tail below the mean of 3

This is an upper-tail test since the

Hypothesis Testing: σ Known

Lower Tail Tests



_{There is only one critical value, since the}

rejection area is in only one tail.

Reject H

₀

Do not reject

H

₀

α

-Z

μ

Z

X

Hypothesis Testing: σ Known

Upper Tail Tests



_{There is only one critical value, since the}

rejection area is in only one tail.

Reject

H

₀

Do not reject

H

₀

α

Z

μ

Critical value

Hypothesis Testing: σ Known

Upper Tail Test Example

A phone industry manager thinks that customer

monthly cell phone bills have increased, and now

average more than $52 per month. The company

wishes to test this claim. Past company records

indicate that the standard deviation is about $10.

H

₀

: μ ≤ 52 the mean is less than or equal to than $52 per month

H

₁

: μ > 52 the mean

is

greater than $52 per month

(i.e., sufficient evidence exists to support the

manager’s claim)

Hypothesis Testing: σ Known

Upper Tail Test Example



_{Suppose that}

_

_{= .10 is chosen for this test}



_{Find the rejection region:}

Reject H

₀

Do not reject H

₀



= .10

Z

0

Reject H

₀

Hypothesis Testing: σ Known

Upper Tail Test Example

What is Z given

a

= 0.10?

Z

.07

.09

1.1 .8790 .8810 .8830

1.2

.8980

.9015

1.3 .9147 .9162 .9177

z

0

1.28

.

08

a

= .10

Critical Value

= 1.28

.90

.8997

.10

Hypothesis Testing: σ Known

Upper Tail Test Example



_{Obtain sample and compute the test statistic.}



_{Suppose a sample is taken with the following}

results: n = 64, X = 53.1 (

_

=10 was

assumed known from past company records)



_{Then the test statistic is:}

Hypothesis Testing: σ Known

Upper Tail Test Example



_{Reach a decision and interpret the result:}



= .10

1.28

0

Reject H

₀

1-

_

= .90

Z = .88

Do not reject H

₀

since Z = 0.88 ≤ 1.28

Hypothesis Testing: σ Known

Upper Tail Test Example



_{Calculate the p-value and compare to}

_

Reject H

₀

p-value = .1894

Hypothesis Testing:

σ Unknown



_{If the population standard deviation is}

unknown, you instead use the sample

standard deviation S.



_{Because of this change, you use the t}

distribution instead of the Z distribution to

test the null hypothesis about the mean.



_{All other steps, concepts, and conclusions are}

Hypothesis Testing:

σ Unknown



_{Recall that the t test statistic with n-1}

degrees of freedom is:

n

S

μ

X

Hypothesis Testing:

σ Unknown Example

The mean cost of a hotel room in New York is said

to be $168 per night. A random sample of 25 hotels

resulted in X = $172.50 and S = 15.40. Test at the



= 0.05 level.

(A stem-and-leaf display and a normal probability plot

indicate the data are approximately normally distributed )

Hypothesis Testing:

σ Unknown Example

H

₀

: μ = 168

H

₁

: μ ≠ 168



_α

₌

_0.05



_n

_{= 25}



_

_{is unknown, so}

use a t statistic



_{Critical Value:}

t

24

= ± 2.0639

Hypothesis Testing:

σ Unknown Example

a/2=.025

Do not reject H

₀

:

not sufficient evidence

that true mean cost is different from $168

Hypothesis Testing:

Connection to Confidence Intervals



_{For X = 172.5, S = 15.40 and n = 25, the 95%}

confidence interval is:

166.14 ≤ μ ≤ 178.86



_{Since this interval contains the hypothesized}

Hypothesis Testing:

σ Unknown



_{Recall that you assume that the sample}

statistic comes from a random sample from a

normal distribution.



_{If the sample size is small (< 30), you should}

use a box-and-whisker plot or a normal

probability plot to assess whether the

assumption of normality is valid.



_{If the sample size is large, the central limit}

theorem applies and the sampling

Hypothesis Testing

Proportions



_{Involves categorical variables}



_{Two possible outcomes}



_{“Success” (possesses a certain characteristic)}



_{“Failure” (does not possesses that}

characteristic)



_{Fraction or proportion of the population in}

Hypothesis Testing



_{Sample proportion in the success category is denoted by p}



_{When both nπ and n(1-π) are at least 5, p}

_{can be}

Hypothesis Testing

Proportions



_{The sampling distribution of p is}

approximately normal, so the test statistic is

a Z value:

n

p

Z

)

1

(

_





Hypothesis Testing

Proportions 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 30 responses. Test at the

_

= .05 significance level.

First, check:

n π = (500)(.08) = 40

Hypothesis Testing

Proportions Example

H

₀

: π = .08 H

₁

: π ≠ .08

α

= .05

n = 500, p = .06

Critical Values: ± 1.96

z

0

Reject

Reject

.025

.025

1.96

-1.96

Hypothesis Testing

Proportions Example

Do not reject H

₀

at



= .05

Test Statistic:

Decision:

Conclusion:

Potential Pitfalls and

Ethical Considerations



_{Use randomly collected data to reduce selection}

biases



_{Do not use human subjects without informed}

consent



_{Choose the level of significance,}

_α,

_{before data}

collection



_{Do not employ “data snooping” to choose between}

one-tail and two-tail test, or to determine the level of

significance



_{Do not practice “data cleansing” to hide}

observations that do not support a stated hypothesis

Chapter Summary

In this chapter, we have



_{Addressed hypothesis testing methodology}



_{Performed Z Test for the mean (σ known)}



_{Discussed critical value and p–value approaches to}

hypothesis testing



_{Performed one-tail and two-tail tests}



_{Performed t test for the mean (σ unknown)}



_{Performed Z test for the proportion}