C H A P T E R 8 : H Y P O T H E S I S T E S T I N G

8.1 INTRODUCTION TO STATISTICAL TESTS

INFERENTIAL STATISTICS

• Inferential statistics involves methods of using information from a sample to draw conclusions regarding the population.• Most statistical inference centers around the parameters

of a population (usually the mean or probability of success in a binomial trial)

• In Chapter 7, we focused on estimating the value of the parameter

• In this chapter, we will focus on decisions concerning the value of a parameter and hypothesis testing• To introduce statistical tests, we will discuss the mean,

however the methods apply to other parameters as well

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STATING HYPOTHESES

• There are two types of hypotheses that we must identify• Null hypothesis - • This is the statement that is under investigation or being

tested. Usually it represents a statement of “no effect”, “no difference”, or “things haven’t changed”. The value specified is usually a historical value, a claim, or a production specification.

• Alternate hypothesis - • Any hypothesis that differs from the null hypothesis is called

an alternate hypothesis. It is constructed in such a way that it is the hypothesis to be accepted when the null hypothesis must be rejected.

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Note: The alternate hypothesis is sometimes represented by

• In statistical testing, the null hypothesis always contains the equals symbol.

• The alternate hypothesis will contain symbols of “inequality”

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EXAMPLE 1: NULL AND ALTERNATE HYPOTHESES

A car manufacturer advertises that its new subcompact models get 47 miles per gallon (mpg). Let be the mean of the mileage distribution for these cars. You assume that the manufacturer will not underrate the car, but you suspect that the mileage might be overrated. State the null and alternate hypotheses.

SOLUTIONWe want to see if the manufacturer’s claim that mpg can be rejected. Therefore, our null hypothesis is From experience with this manufacturer, we have every reason to believe that the advertised mileage is too high. If is not 47 mpg, we are sure it is less than 47 mpg. Therefore, the alternate hypothesis is

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TYPES OF TESTS

• We categorize a statistical test according to the alternate hypothesis

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Table 8-1

The Null and Alternate Hypotheses for Tests of the Mean

IMPORTANT VOCABULARY FOR HYPOTHESIS TESTS

• A population parameter is a numerical descriptive measurement on the entire population (such as: )• Parameters are fixed values• The null hypothesis makes a statement about a population

parameter

• A statistic is a numerical descriptive measurement of a sample (such as: )• Statistics usually vary from one sample to the next

• The probability distribution of the statistic is called the sampling distribution.

• A sample test statistic is a numerical summary of a set of data that can be used to perform a hypothesis test. The formula for the sample test statistic depends on the type of hypothesis test.

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BASIC PROCESS OF HYPOTHESIS TESTS

1. State the null and alternate hypotheses2. Take a simple random sample and compute a

sample test statistic3. Use the sampling distribution of the test statistic

and the type of test to compute the P-value4. Conclude the test5. Interpret the results

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P – VALUE

• Assuming is true, the P-value of the test is the probability that the test statistic will take on values as extreme as or more extreme than the observed (computed)test statistic. • The P-value can be thought of as the probability that the

results of a statistical experiment are dues only to chance.

• The smaller the P-value computed from sample data, the stronger the evidence against .• The P-value takes on different values depending on the

alternate hypothesis and the type of test.

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P – VALUE AND TYPES OF TESTS

• Let represent the standardized sample test statistic for testing a mean using the standard normal distribution.

Left-Tailed Test

P-value

Right-Tailed Test

P-value

Two-Tailed Test

P-value

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TYPES OF ERRORS

• If we reject when it is true, we have made a type I error.• If we accept when it is false, we have made a

type II error.

Table 8-2

Type I and Type II Errors

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TYPES OF ERRORS

• For tests of hypotheses to be well constructed, they must be designed to minimize possible errors of decision. (Usually, we do not know if an error has been made, and therefore, we can talk only about the probability of making an error.)

TYPES OF ERRORS

• Good statistical practice requires that we announce in advance how much evidence against H0 will be required to reject H0.

• The probability with which we are willing to risk a type I error is called the level of significance of a test. • The level of significance is denoted by the Greek letter

(“alpha”).

• The probability of making a type II error is (Greek letter: “beta”)• The probability of a type II error is treated in advanced

statistics courses, we will focus on the probability of a type I error.

TYPES OF ERRORS

• Summary of the probabilities or errors associated with a statistical test

Table 8-3

Probabilities Associated with a Statistical Test

CONCLUDING A STATISTICAL TEST

• is specified in advance before any samples are drawn

• Compare with the P-value• If P-value , we reject and say the data are statistically

significant at the level • If P-value > , we fail to reject

Note: “significant” has a special meaning – at the level of risk, the evidence (sample data) against is sufficient to discredit , so we adopt

We DO NOT claim that we proved or disproved .We can say that the probability of a type I error

(rejecting , when it is true) is .

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• When we “accept” or “fail to reject” , we need to understand that we are not proving . We are only saying that the sample evidence (data) is not strong enough to justify rejection of the null hypothesis.

• When cannot be rejected, a confidence interval us often used to estimate the parameter in question because it gives the statistician a range of possible values for the parameter.

CONCLUDING A STATISTICAL TEST

Table 8-4

Meaning of the Terms Fail to Reject H0 and Reject H0

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EXAMPLE 2: STATISTICAL TESTING PREVIEW

Rosie is an aging sheep dog in Montana who gets regular checkups from her owner, the local veterinarian. Let be a random variable that represents Rosie’s resting heart rate (in beats per minute). From past experience, the vet knows that has a normal distribution with . The vet checked the Merck Veterinary Manual and found for dogs of this breed beats per minute.

Over the past six weeks, Rosie’s heart rate (beats/min) measured: 93, 109, 110, 89, 112, 117. The sample mean is . The vet is concerned that Rosie’s heart rate may be slowing. Do the data indicate that this is the case? Use

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SOLUTION1. State the null and alternate hypotheses

2. Take a simple random sample and compute a sample test statistic

Test Statistic 3. Use the sampling distribution of the test

statistic and the type of test to compute the P-value

P-value = P-value = Normalcdf

4. Conclude the test (compare the P-value to )

.0207 the sample evidence is sufficient at the

0.05 level to justify rejecting .5. Interpret the results

It seems that Rosie’s heart rate is slowing.