hypothesis testing betsy farber ppg_2

7/29/2019 Hypothesis Testing Betsy Farber PPG_2

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Chapter

7

Elementary Statistics

Larson Farber

Hypothesis Testing


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A Statistical Hypothesis

Alternative

hypothesis Hacontains a statement

ofinequality, such as.

Null hypothesis H0Statistical hypothesis

hat contains a

tatement ofequality,uch as , = or.

If I am false,you are true

If I am false,

you are true

H0Ha

Complementary Statements

A claim about a population.


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Write the claim about the population. Then,write its complement. Either hypothesis, the

ull or the alternative, can represent the claim

A hospital claims its ambulance responsetime is less than 10 minutes.

H0 : 10 min

Ha : 10 min claim

Ha : 60.0

p

H0 : 60.0p claim

Writing Hypotheses

A consumer magazine claims theproportion of cell phone calls made during

evenings and weekends is at most 60%.


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A type I error: Null hypothesis is actually

rue but the decision is to reject it.

Level of significance, aMaximum probability of committing

a type I error.

Decisi

on

Actual Truth of H0

Errors and Level ofSignificance

H0 True H0 False

Do not

reject H0

Reject H0

Correct

Decision

Correct

Decision

Type II

Error

Type I

Error


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Sampling distribution for x

The rejection region is the range of

values for which the null hypothesis is not

probable. It is always in the direction of

the alternative hypothesis. Its area is equalto a.

Rejection Region

0z z0

A critical value separates the rejection

region from the non-rejection region

Critical Value z0

Rejection Regions


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z00

-z0

0 z0

Right-tail test Ha: >valu

Reject H0 ifz > z0otherwise fail to reject H0.

Two-tail testHa:value

Reject H0 ifz>z0 or z


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Claim is H0

There is not

enoughevidence to

reject the claim

There is

enoughevidence to

reject the claim

Interpreting the Decision


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Claim is Ha

There is not

enoughevidence to

support the

claim

There is

enoughevidence to

support the

claim

Interpreting the Decision


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1. Write the null and alternative hypothesi

2. State the level of significance

3. Identify the sampling distribution

Write H0 and Ha as mathematical statements.

Remember H0 always contains the = symbol.

This is the maximum probability of rejecting the

null hypothesis when it is actually true. (Makinga type I error.)

The sampling distribution is the distribution for

the test statistic assuming that H0 is true and

that the experiment is repeated an infinite

number of times.

8 Steps in a Hypothesis Test


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7. Make your decision

6. Find the test statistic

5. Find therejection region

4. Find thecritical value

8. Interpret your decision

The critical value separates the rejection region

of the sampling distribution from the non-

rejection region.

Perform the calculations to standardize your

sample statistic.

If the test statistic falls in the critical region,reject H0. Otherwise, fail to reject H0.


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The critical value z0 separates the rejection region

rom the non-rejection region. The area of the

ejection region is equal to a.

z0 0

ejection

egion

z00

Rejection

region

-z0 0 z0

Rejection

region

Rejection

region

ind z0 for a left-tail

est with a =.01Find z0 for a right-tail

test with a =.05

Find - z0

and z0for a two-tail test with a =.01

z0=-2.33

-z0=-2.575 and z0 =2.575

z0=1.645

Critical Values


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A cereal company claims the meansodium content in one serving of its cereal is

no more than 230 mg. You workfora

national health service and are asked to test

this claim. You find that a random sample of52 servings has a mean sodium content of

232 milligrams and a standard deviation of

10 mg. At a= 0.05, do you have enough

evidence to reject the companys claim?1. Write the null and alternative hypothesis

H0: 230 mg.(claim) Ha: > 230 mg.

2. State the level of significance

a= 0.05

3. Determine the sampling distribution

Since the sample size is at least 30, the sampling

distribution is normal.

The z-test for a Mean


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5. Find the rejection

region

Rejection

region

Since Ha contains the > symbol, this is a right tail tes

n=52

x = 232s=10

44.1387.1

2

52

10

230232

z

= 1.44 does not fall in the rejection region, soail to reject H0

There is not enough evidence to reject the

ompanys claim that there is at most 230mg ofodium in one serving of its cereal.

z00

1.645

4. Find the critical

value


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Find a c20 critical value for a left-tail test when

=17 and a = 0.05.

2 is the test statistic for the population variance. Its

ampling distribution is a c2 distribution with n-1 d.f.

Find critical values c20 for a two-tailed test when= 12 and a = 0.01.

The standardized test statistic is2

22 )1(

csn

c20 =7.962

c2l =2.603 and c2

R=26.757

0 1 0 2 0 3 0 4 0

Critical Values for 2


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A state school administrator says that the standard

deviation of test scores for 8th grade students who

took a life-science assessment test is less than 30.

You work for the administrator and are asked to test

this claim. You find that a random sample of 10 testshas a standard deviation of 28.8. At a = 0.01, do you

have enough evidence to support the administrators

claim? Assume test scores are normally distributed.

. Write the null and alternative hypothesis

H0

: 30 Ha: < 30 (claim)

. State the level of significance a= 0.01

3. Determine the sampling distribution

The sampling distribution is c2 with 10 - 1 = 9 d.f.

Test for


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n=10s = 28.8

c2 = 8.2944 does not fall in the rejection region,

so fail to reject H0

There is not enough evidence to support the

dministrators claim that the standard deviation is

2944.8

30

8.28)110()1(2

2

2

22

csn

0 1 0 2 0 3 0 4 0

5. Find the rejection

region

4. Find the critical

value

2.088

hypothesis testing betsy farber ppg_2

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