MA135Chapter81-Tagged.pdf

Chapter 8HYPOTHESIS TEST

In statistics, a hypothesis is a claim or statement about a property of a population.

A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population.

Examples of Hypotheses that can be Tested

• Genetics: The Genetics & IVF Institute claims that its XSORT method allows couples to increase the probability of having a baby girl.

• Business: A newspaper headline makes the claim that most workers get their jobs through networking.

• Medicine: Medical researchers claim that when people with colds are treated with echinacea, the treatment has no effect.

Examples of Hypotheses that can be Tested

• Aircraft Safety: The Federal Aviation Administration claims that the mean weight of an airline passenger (including carry-on baggage) is greater than 185 lb, which it was 20 years ago.

• Quality Control: When new equipment is used to manufacture aircraft altimeters, the new altimeters are better because the variation in the errors is reduced so that the readings are more consistent. (In many industries, the quality of goods and services can often be improved by reducing variation.)

Components of aFormal Hypothesis

Test

Null Hypothesis:

• The null hypothesis (denoted by ) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.

0H

0H

Alternative Hypothesis:

1H

1 a AH or H or H

Null and Alternative Hypotheses; Hypothesis Test

Null hypothesis: A hypothesis to be tested. We use the symbol H0 to represent the null hypothesis.

Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol Ha to represent the alternative hypothesis.

Hypothesis test: The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.

Example:

Consider the claim that the mean weight of airline passengers (including carry-on baggage) is differ 195 lb (the current value used by the Federal Aviation Administration). Follow the two-steps procedure outlined in to identify the null hypothesis and the alternative hypothesis.

Example:

Step 2:

The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

Test Statistic

Test Statistic – Formulas

Test statistic for mean

or x x

z ts

n n

 



  

Significance Level

The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices for are 0.05, 0.01, and 0.10.





Critical Value

A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis. The critical values depend on the nature of the null hypothesis, the sampling distribution that applies, and the significance level . See the previous figure where the critical value of z = 1.645 corresponds to a significance level of .



0.05 

Two-tailed Test

Means less than or greater than

 is divided equally between the two tails of the critical

region

0 :H 

1 :H 

Conclusions in Hypothesis Testing

We always test the null hypothesis. The initial conclusion will always be one of the following:

a. Reject the null hypothesis.

b. Fail to reject the null hypothesis.

Traditional method:

If the test statistic falls within the critical region, reject .

If the test statistic does not fall within the critical region, fail to reject .

Decision Criterion

0H

0H

Decision Criterion

Confidence Intervals:

A confidence interval estimate of a population parameter contains the likely values of that parameter. If a confidence interval does not include a claimed value of a population parameter, reject that claim.