Section 9.1 The Nature of Hypothesis Testing We are now going to be comparing a sample to the population Sample was from the population

Population

Sample

Sample was taken from a population one year or four years from now

Population

Sample

Is the sample similar to the population or is it different that is what we will be focusing on in this chapter. Hypothesis: is a statement that something is true. Null and Alternative Hypotheses: Hypothesis Test Null hypothesis: A hypothesis to be tested. We use the symbol H O to represent the null hypothesis. Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol H a 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.

Symbolically Null Hypothesis: The null hypothesis is expressed as HO :   0 In our course, we will have a single value for our parameter and number.

0 is some

Alternative Hypothesis:  In this type of alternative hypothesis, we are testing if the population mean



is different from a specified value

0

H a :   0

An alternative hypothesis of this form is called a two-tailed test.  In this type of alternative hypothesis, we are testing if the population mean



is less than a specified value

0

H a :   0

An alternative hypothesis of this form is called a left-tailed test.  In this type of alternative hypothesis, we are testing if the population mean



is greater than a specified value

0

H a :   0

An alternative hypothesis of this form is called a right-tailed test. One tailed test: A hypothesis test is called this if it is left tailed or right tailed.

Example: Hypothesis test are proposed. For the hypothesis test, a. Determine the null hypothesis. b. Determine the alternative hypothesis. c. Classify the hypothesis test as two tailed, left tailed, or right tailed. 1. Dementia is the loss of the intellectual and social abilities severe enough to interfere with judgment, behavior, and daily functioning. Alzheimer’s disease is the most common type of dementia. In the article “Living with Early Onset Dementia: Exploring the Experience and Development Evidence-Based Guidelines for Practice” the researchers explored the experience and struggles of people diagnosed with dementia and their families. A hypothesis test is to be performed to decide whether the mean age at diagnosis of all people with early-onset dementia is less than 55 years old. 2. A study on “Heat Stress Evaluation and Worker Fatigue in a Steel Plant” assessed fatigue in steel-plant workers due to heat stress. Among other things, the researchers monitored the heart rates of a random sample of 29 casting workers. A hypothesis test is to be conducted to decide whether the mean post-work heart rate of casting workers exceeds the normal resting heart rate of 72 beats per minute (bpm). Basic Logic of Hypothesis Testing Take a random sample from the population. If the sample data are consistent with the null hypothesis, do not reject the null hypothesis; if the sample data are not inconsistent with the null hypothesis and supportive of the alternative hypothesis, reject the null hypothesis in favor of the alternative hypothesis. Beginning in Section 9.2 we will learn about the test statistic which will help us to decide if we should reject the null hypothesis or not.

Type I and Type II Errors Type I error: Rejecting the null hypothesis when it is in fact true. Type II error: Not rejecting the null hypothesis when it is in fact false. HO is

Decision:

Do not reject HO

True Correct decision

False Type II error

Reject HO

Type I error

Correct decision

Example 1. Refer to the Early-Onset Dementia problem Explain what each of the following would mean. a. Type I error b. Type II error c. Correct decision Now suppose that the results of carrying out the hypothesis test lead to nonrejection of the null hypothesis. Classify that conclusion by error type or as a correct decision if in fact the mean age at diagnosis of all people with early-onset dementia d. is 55 years old. e. is less than 55 years old.

Significance level: is the probability of making a Type I error, that is, of rejecting a true null hypothesis. This probability is 

 :is the probability of making a Type II error, that is, accepting (not rejecting) a false null hypothesis. Relation between Type I and Type II Error Probabilities: For a fixed sample size, the smaller we specify the significance level

 the larger the probability of  of not rejecting a false null hypothesis will be.

Possible Conclusions for a Hypothesis Test Suppose that a hypothesis test is conducted at a small significance level.  If the null hypothesis is rejected, we conclude that the data provide sufficient evidence to support the alternative hypothesis  If the null hypothesis is not rejected, we conclude that the data does not provide sufficient evidence to support the alternative hypothesis.