welcome to mm207 – statistics hypothesis testing with one sample chapter 7 7.1 – 7.4 anthony j....

Welcome to MM207 – Statistics

Hypothesis Testing with One SampleChapter 7 7.1 – 7.4

Anthony J. Feduccia

Chapter Outline

• 7.1 Introduction to Hypothesis Testing

• 7.2 Hypothesis Testing for the Mean (Large Samples)

• 7.3 Hypothesis Testing for the Mean (Small Samples)

• 7.4 Hypothesis Testing for Proportions

• 0mit Section 7.5

2Larson/Farber 4th ed.

Guidelines -- Hypothesis Testing Steps:

•State H0 and Ha.

•Specify the level of significance alpha .

•Determine the test statistic, either z or t. Find the test statistic using the given data.

•Find the P-value or the critical value(s) z0 or t0. Use the method specified in the problem statement.

•Define the rejection region using either the P-value method or critical values from the Normal distribution.

•Make a decision to reject or fail to reject the null hypothesis.

•Interpret the decision in the context of the original claim.

Section 7.1 Objectives

• State a null hypothesis and an alternative hypothesis

• Identify type I and type I errors and interpret the level of significance

• Determine whether to use a one-tailed or two-tailed statistical test and find a p-value

• Make and interpret a decision based on the results of a statistical test


Hypothesis TestsHypothesis test

• A process that uses sample statistics to test a claim about the value of a population parameter.

• For example: An automobile manufacturer advertises that its new hybrid car has a mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the sample mean differs enough from the advertised mean, you can decide the advertisement is wrong.


Hypothesis TestsStatistical hypothesis

• A statement, or claim, about a population parameter.

• Need a pair of hypotheses• one that represents the claim

• the other, its complement

• When one of these hypotheses is false, the other must be true.


Example: Stating the Null and Alternative Hypotheses

Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.1.A university publicizes that the proportion of its students who graduate in 4 years is 82%.

Equality condition

Complement of H0

H0:

Ha:

(Claim)p = 0.82

p ≠ 0.82

Solution:


μ ≥ 2.5 gallons per minute


Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.2.A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute.

Inequality condition

Complement of HaH0:

Ha:(Claim)μ < 2.5 gallons per minute

Solution:


μ ≤ 20 ounces


Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.3.A cereal company advertises that the mean weight of the contents of its 20-ounce size cereal boxes is more than 20 ounces.

Inequality condition

Complement of HaH0:

Ha:(Claim)μ > 20 ounces

Solution:


1.a.A candidate for mayor of a city claims to be favored by at least half the voters. Test this candidates claim. (4 points)

claim: H0 : Ha :

Test :

b.The mean age of bus drivers in Chicago is 50.9 years. Test the claim that the mean age differs from this. (4 points)

claim: H0 : Ha :

Test :

Types of Errors• No matter which hypothesis represents the claim, always

begin the hypothesis test assuming that the equality condition in the null hypothesis is true.

• At the end of the test, one of two decisions will be made:

– reject the null hypothesis

– fail to reject the null hypothesis

• Because your decision is based on a sample, there is the possibility of making the wrong decision.


Types of Errors

• A type I error occurs if the null hypothesis is rejected when it is true.

• A type II error occurs if the null hypothesis is not rejected when it is false.

Actual Truth of H0

Decision H0 is true H0 is false

Do not reject H0 Correct Decision Type II Error

Reject H0 Type I Error Correct Decision


P-values(see page 370)

P-value (or probability value)

• Find test statistic from the data. Either a z or a t.

• For a left-tail test: p value is area to the left of the test statistic.

• For a right-tail test: p value is area to the right of the test statistic (recall tables give area to the left)

• For a double-tail test: p value is twice the area to the left of a negative test statistic or twice the area to the right of a positive test statistic.

13

Left-tailed Test• The alternative hypothesis Ha contains the

less-than inequality symbol (<).

z0 1 2 3-3 -2 -1

Test statistic

H0: μ k

Ha: μ < k

P is the area to the left of the test statistic.

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• The alternative hypothesis Ha contains the greater-than inequality symbol (>).

z0 1 2 3-3 -2 -1

Right-tailed Test

H0: μ ≤ k

Ha: μ > k

Test statistic

P is the area to the right of the test statistic.


Two-tailed Test

• The alternative hypothesis Ha contains the not equal inequality symbol (≠). Each tail has an area of ½P.

z0 1 2 3-3 -2 -1

Test statistic

Test statistic

H0: μ = k

Ha: μ kP is twice the area to the left of the negative test statistic.

P is twice the area to the right of the positive test statistic.


In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random selection of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Is there enough evidence to support the claim at = 0.01? Use a P-value.


Example Using P-Value

• H0:

• Ha:

• = • Test Statistic:

μ ≥ 30 min

μ < 30 min (claim)

0.01

28.5 30

3.5 36

2.57

xz

n

• Decision:

At the 1% level of significance, you have sufficient evidence to conclude the mean delivery time is less than 30 minutes.

z0-2.57

0.0051

• P-value

0.0051 < 0.01Reject H0

18

In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random selection of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes

Employees in a large accounting firm claim that the mean salary of the firm’s accountants is less than that of its competitor’s, which is $45,000. A random sample of 30 of the firm’s accountants has a mean salary of $43,500 with a standard deviation of $5200. At α = 0.05, test the employees’ claim.


An alternative approach to Hypothesis testingUsing Rejection regions (page 386)

• H0:

• Ha:

• = • Rejection Region:

μ ≥ $45,000

μ < $45,000

0.05

43,500 45,000

5200 30

1.58

xz

n

• Decision:At the 5% level of significance, there is not sufficient evidence to support the employees’ claim that the mean salary is less than $45,000.

• Test Statistic

z0-1.645

0.05

-1.58

-1.645

Fail to reject H0


The critical z based on alpha, Critical value of z has area of (0.0500) to it’s left

Example: Testing μ with a Small Sample

A used car dealer says that the mean price of a 2005 Honda Pilot LX is at least $23,900. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $23,000 and a standard deviation of $1113. Is there enough evidence to reject the dealer’s claim at α = 0.05? Assume the population is normally distributed. (Adapted from Kelley Blue Book)


Solution: Testing μ with a Small Sample

• H0:

• Ha:

• α =

• df = • Rejection Region:

• Test Statistic: use t

• Decision:

μ ≥ $23,900 (claim)μ < $23,900

0.0514 – 1 = 13

23,000 23,9003.026

1113 14

xt

s n

t0-1.771

0.05

-3.026

-1.771

At the 0.05 level of significance, there is enough evidence to reject the claim that the mean price of a 2005 Honda Pilot LX is at least $23,900

Reject H0


A used car dealer says that the mean price of a 2005 Honda Pilot LX is at least $23,900. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $23,000 and a standard deviation of $1113. Is there enough evidence to reject the dealer’s claim at α = 0.05

•A fast food outlet claims that the mean waiting time in line is less than 4.9 minutes. A random sample of 60 customers yield a sample mean of 4.8 minutes. From past studies it is know that the standard deviation is 0.6 minutes. At alpha = 0.05, test the fast food outlet’s claim a. Use the critical value z0 method from the normal distribution.

Known: u = 4.9, x bar = 4.8, sigma = 0.6; alpha = 0.05; n =60

Ho:Ha:

Test statistic:

Critical zo

Rejection Region:

Decision:

60,05.0,6.0,8.4,9.4 nx

9.49.4

29.160/6.0

9.48.4

/

n

xz

Use the P-value method.

•H0 :

Ha :

•: Test statistic

claim

z-Test for a Population Proportion

z-Test for a Population Proportion• A statistical test for a population proportion. • Can be used when a binomial distribution is given such

that np 5 and nq 5.• The test statistic is the sample proportion . • The standardized test statistic is z.

ˆ

ˆ

ˆ ˆp

p

p p pzpq n

p̂


Example: Hypothesis Test for Proportions

Zogby International claims that 45% of people in the United States support making cigarettes illegal within the next 5 to 10 years. You decide to test this claim and ask a random sample of 200 people in the United States whether they support making cigarettes illegal within the next 5 to 10 years. Of the 200 people, 49% support this law. At α = 0.05 is there enough evidence to reject the claim?

Solution:•Verify that np ≥ 5 and nq ≥ 5.

np = 200(0.45) = 90 and nq = 200(0.55) = 11026Larson/Farber 4th ed.

• H0:

• Ha:

• = • Rejection Region:

p = 0.45

p ≠ 0.45

0.05

ˆ 0.49 0.45

(0.45)(0.55) 200

1.14

p pz

pq n

• Decision:At the 5% level of significance, there is not enough evidence to reject the claim that 45% of people in the U.S. support making cigarettes illegal within the next 5 to 10 years.

• Test Statistic

z0-1.96

0.025

1.96

0.025

-1.96 1.96

1.14

Fail to reject H0


Use Normal z for proportions

welcome to mm207 – statistics hypothesis testing with one sample chapter 7 7.1 – 7.4 anthony j....

Documents

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original claim

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alternative hypotheseswrite

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