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Chapter 7 Hypothesis Testing

Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

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Hypothesis testing Definitions & steps hypothesis testing One-tailed test Two-tailed test p-value Type I & Type II error

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Page 1: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Chapter 7

Hypothesis Testing

Page 2: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

• Define a hypothesis and hypothesis testing.• Describe the five step hypothesis testing procedure.• Distinguish between a one-tailed and a two-tailed test of

hypothesis.• Conduct a test of hypothesis about a population mean. • Conduct a test of hypothesis about a population

proportion.• Define Type I and Type II errors.

GOALS:

Page 3: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Hypothesis testing

Definitions & steps

hypothesis testing

One-tailed test

Two-tailed test p-value Type I &

Type II error

Page 4: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

What is a Hypothesis? A statement about the value of a population

parameter Data are then used to check the reasonableness

of the statementExamples

The mean monthly income for systems analysts is Rs15,000.

Twenty percent of all customers at Pizza Hut return for another meal within a month.

Page 5: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

What is Hypothesis Testing?

A procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.

Page 6: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Definitions

Null Hypothesis (H0): A statement about the value of a population parameter

Alternative Hypothesis (H1): A statement that is accepted if the sample data provide evidence that the null hypothesis is false.

Level of Significance: The probability of rejecting the null hypothesis when it is actually true.

Page 7: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Definitions

Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

Page 8: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Five step procedure for testing a hypothesis

Step 1:State null & alternate hypothesis

Step 2:Select a level of significance

Step 3:Identify the test statistic

Step 4:Formulate a decision rule

Step 5:Take a sample to arrive at a decision

Do not reject null

Reject null

Page 9: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

State null & alternate hypothesis

Example 1: A recent article indicated the mean age of US commercial aircraft is 15 years

The null hypothesis represents the current or reported condition. It is written as:

H0: = 15 The alternative hypothesis is that the statement

is not true, that is:H1: 15

Page 10: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

State null & alternate hypothesis

Example 2: A recent article indicated the mean age of US commercial aircraft is no more than15 years

The null hypothesis represents the current or reported condition. It is written as:

H0: 15 The alternative hypothesis is that the statement

is not true, that is:H1: > 15

Page 11: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Select a level of significance

The level of significance (sometimes called the level of risk or critical value) is designated as (Greek letter alpha)

Page 12: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Select the test statistic for mean (for large sample) For the mean () when is known or the sample

size is large (n 30), use Z distribution as test statistic

Z value is based on the sampling distribution of X, which is normally distributed when the sample is reasonably large with a mean (x) equal to and a standard deviation x, which is equal to /n

n

XZ

Page 13: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Select the test statistic for mean (for small sample) When the sample is less than 30 and the

population standard deviation is not known we use the t distribution

nsX

t

Page 14: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

One-Tailed Tests A test is one-tailed when the alternate

hypothesis, H1 , states a direction H1: The mean yearly commissions earned by full-

time Insurance agent is more than Rs35,000. (µ>Rs35,000)

H1: The mean speed of trucks traveling on A1 route is less than 60 km per hour. (µ<60)

H1: Less than 20 percent of the customers pay cash for their gasoline purchase. (µ<.20)

Note; when region of rejection is only in one tail/end

Page 15: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

One-Tailed Tests

Sampling Distribution for the test statistic (Z) for a one-Tailed Test (0.05 Level of Significance)[H0: h & H1: >h ]

1.650

0.95 0.05

Rejection regionDo not reject H0

Critical value

Page 16: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

One-Tailed Tests

Sampling Distribution for the test statistic (Z) for a one-Tailed Test (0.05 Level of Significance)[H0: h & H1: <h ]

1.65 0

0.950.05

Rejection regionDo not reject H0

Critical value

Page 17: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

One-tailed test (Rejection of null hypothesis)

H0: h & H1: >h

when+ critical value < + test statistic

H0: h & H1: <h

when- critical value > - test statistic

0

Critical valueTest statistic

0

Critical valueTest statistic

Page 18: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Two-Tailed Tests

A test is two-tailed when the alternate hypothesis, H1 , does not state a direction H1: The mean yearly commissions earned by

full-time Insurance agent is not equal Rs35,000. (µRs35,000)

H1: The mean speed of trucks traveling on A1 route is not equal 60 km per hour. (µ60)

Page 19: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Two-Tailed Tests

Sampling Distribution for the test statistic (Z) for a one-Tailed Test (0.05 Level of Significance)

1.960

0.95 0.025

Rejection regionDo not reject H0

Critical value

Rejection region

Critical value0.025

-1.96

Page 20: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Two-tailed test (Rejection of null hypothesis) H0: =h & H1: h

when+ critical value < + test statistic

or - critical value > - test statistic

0+ Critical value- Critical value

+ Test statistic- Test statistic

Page 21: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 1 The Glen valley Steel Company manufactures steel bars. If the

production process is working properly, it turns out steel bars with mean length of at least 2.8 feet with a standard deviation of 0.20 feet. Longer steel bars can be used or altered but shorter steel bars must be scrapped. A sample of 50 bars is selected from the production line. The sample indicates a mean of length 2.73 feet. The company wants to determine whether the production equipment needs to be adjusted.

a) State the null and alternative hypotheses.b) If the company wants to test the hypothesis at the 0.05 level of

significance, what decision would be made using the critical value approach to hypothesis testing?

Page 22: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 1

Step 1 - Null hypothesis & alternative hypothesisH0: 2.8H1: < 2.8

One-tailed test because alternative hypothesis shows a directionn = 50 >30 use Z test and = 0.2

Page 23: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 1

Step 2 – Level of significancelevel of significance is 0.05Z value related to 0.05 = -1.645

Step 3 – Identify the test statistic

2.47500.202.82.73

nσμX

Z

0.02σ 2.8,μ 50,n 2.73,X

Page 24: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 1

Step 4 & 5 – Formulate a decision rule & take a decision

-1.645 0

Rejection regionDo not reject H0

-2.47

Critical valueTest statistic

Decision RuleTest statistic < critical valueReject null hypothesis

Page 25: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 2 The Glen valley Steel Company manufactures steel bars. If the

production process is working properly, it turns out steel bars with mean length of at least 2.8 feet. Longer steel bars can be used or altered but shorter steel bars must be scrapped. A sample of 20 bars is selected from the production line. The sample indicates a mean of length 2.73 feet and a standard deviation of 0.20 feet. The company wants to determine whether the production equipment needs to be adjusted.

a) State the null and alternative hypotheses.b) If the company wants to test the hypothesis at the 0.05 level of

significance, what decision would be made using the critical value approach to hypothesis testing?

Page 26: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 2

Step 1 - Null hypothesis & alternative hypothesisH0: 2.8H1: < 2.8

One-tailed test because alternative hypothesis shows a directionn = 20 <30 use t-distribution and = 0.2

Page 27: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 2

Step 2 – Level of significancelevel of significance is 0.05 & degree of freedom = 19t value related to 0.05 = -1.729

Step 3 – Identify the test statistic

57.1200.202.82.73

nsμX

t

0.02σ 2.8,μ 50,n 2.73,X

Page 28: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Example 2

Step 4 & 5 – Formulate a decision rule & take a decision

-1.729 0

Rejection regionDo not reject H0

-1.57

Critical value Test statisticDecision RuleTest statistic > critical valueDo not reject null hypothesis

Page 29: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Exercise 1 The director of manufacturing at a clothing factory needs

to determine whether a new machine is producing a particular type of cloth according to the manufacturers specifications, which indicate that the cloth should have a mean breaking strength of 70 pounds and a standard deviation of 3.5 pounds. A sample of 49 pieces of cloth reveals a sample mean breaking strength of 69.1 pounds

a) State the null and alternative hypotheses.b) Is there evidence that the machine is not meeting the

manufacturers specifications for average breaking strength? (Use a 0.05 level of significance)

Page 30: Chapter 7 Hypothesis Testing. Define a hypothesis and hypothesis testing. Describe the five step hypothesis testing procedure. Distinguish between a one-tailed

Exercise 2 The director of manufacturing at a clothing factory needs

to determine whether a new machine is producing a particular type of cloth according to the manufacturers specifications, which indicate that the cloth should have a mean breaking strength of 60 pounds and a standard deviation of 2.5 pounds. A sample of 19 pieces of cloth reveals a sample mean breaking strength of 59.1 pounds

a) State the null and alternative hypotheses.b) Is there evidence that the machine is not meeting the

manufacturers specifications for average breaking strength? (Use a 0.05 level of significance)