Section 8 - Breazeal



Chapter 10 Requirements from the Generic Syllabus

The student will be able to:

• Understand the concepts behind hypothesis testing.

• Use hypothesis testing to test claims made about population proportions, and means

• Use the P-value approach that is commonly used in research.

• State real world conclusions to hypothesis tests using appropriate terminology.

• Distinguish between statistical significance and practical significance.

• Use a graphing calculator when testing hypotheses.

Section 10.1 – The Language of Hypothesis Testing

Objectives

1. Determine the null and alternative hypotheses

2. Explain Type I and Type II errors

3. State conclusions to hypothesis tests

Objective 1 – Determine the null and alternative hypothesis

A hypothesis is a statement regarding a characteristic of one or more populations.

Examples

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

We test the types of statements above using sample data because it is usually impossible or impractical to gain access to the entire population. If population data are available, there is no need for inferential statistics.

Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

Because we are using sample data to test the claim, we cannot say with 100% certainty that the statement is true; we can only determine if the sample data support the claim or not.

Steps in Hypothesis Testing

1. Make a statement regarding the nature of the population.

2. Collect evidence (sample data) to test the statement.

3. Analyze the data to assess the plausibility of the statement.

[pic]

In other words, the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. The null hypothesis is assumed to be true until we have evidence to the contrary. We seek evidence that supports the statement in the alternative hypothesis.

Some textbooks present the alternative hypothesis as HA instead of H1.

[pic]

Examples

For each of the following claims, determine the null and alternative hypotheses. State whether the test is two-tailed, left-tailed or right-tailed.

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

Objective 2 – Explain Type I and Type II Errors

[pic]

Examples

For each of the following claims, explain what it would mean to make a Type I error. What would it mean to make a Type II error?

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

[pic]

Objective 3 - State Conclusions to Hypothesis Tests

We never “accept” the null hypothesis. Rather, we say that we do not reject the null hypothesis.

Generally the conclusions are stated as one of

• “This study provides evidence at the ___ significance level that “ followed by the verbal statement of H1.

• “This study does not provide evidence at the ___ significance level that" followed by the verbal statement of H1.

Example

According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

a) Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion.

b) Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.

Hypothesis Testing Procedures

1. State the null hypothesis H0 and H1 mathematically (in terms of parameters).

H0 should include the “=” statement

H1 should include either “” or “[pic]”

2. State the significance level ( to be used and the corresponding critical value of the test statistic. Define the rejection region(s). The critical value is any value that separates the critical region from the values of the test statistic that do not lead to a rejection of Ho.

3. Draw the appropriate graph showing the critical (rejection) region(s).

4. Calculate the test statistic. This is calculated from the data given in the problem. Label the test statistic on the graph.

|Proportion |Mean |Standard Deviation |

|[pic] |[pic] |[pic] |

5. Calculate the P-value (comes from a calculator)

6. Compare the test statistic to the rejection region, or, compare the P-value to (.

7. Make a decision about the null hypothesis, either Reject H0 or Do Not Reject H0:

a. If the test statistic is in the rejection region, or the P-value is smaller than (, state "reject H0".

Classical: If the test statistic falls in the critical region, Reject H0

If the test statistic does not fall in the critical region, Do Not Reject H0

P-Value: If P–value < α, Reject H0

If P–value > α, Do Not Reject H0

b. If not, state "do not reject H0".

8. Form a scientific conclusion based on that decision.

a. If H0 is rejected, then start with "This study provides evidence" followed by "at the ___ significance level that" followed by the verbal statement of H1.

b. If H0 is not rejected, then start with "This study does not provide evidence" followed by "at the ___ significance level that" followed by the verbal statement of H1.

The critical region (rejection region) is the set of all values of the test statistic that will cause us to reject Ho.

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Four Outcomes from Hypothesis Testing

1. Reject the null hypothesis when the alternative hypothesis is true. Correct decision

2. Do not reject the null hypothesis when the null hypothesis is true. Correct decision

3. Reject the null hypothesis when the null hypothesis is true. Incorrect decision, Type I error

4. Do not reject the null hypothesis when the alternative hypothesis is true. Incorrect decision, Type II error

Definitions – null and alternative hypothesis

The null hypothesis, denoted H0 (H-naught), is a statement to be tested. The null hypothesis is a statement of no change, no effect or no difference and is assumed true until evidence indicates otherwise.

The alternative hypothesis, denoted H1 (H-one), is a statement that we are trying to find evidence to support.

In this chapter, there are three ways to set up the null and alternative hypotheses:

Equal versus not equal hypothesis (two-tailed test)

H0: parameter = some value

H1: parameter ≠ some value

Equal versus less than (left-tailed test)

H0: parameter = some value

H1: parameter < some value

Equal versus greater than (right-tailed test)

H0: parameter = some value

H1: parameter > some value

The level of significance, α, is the probability of making a Type I error and is chosen by the researcher before the sample data is collected.

α = P(Type I Error) = P(rejecting H0 when H0 is true)

β = P(Type II Error) = P(not rejecting H0 when H1 is true)

β is the Greek letter ‘beta’

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