chapter 8 hypothesis testing. section 8-1: steps in hypothesis testing – traditional method...

Chapter 8

Hypothesis Testing

Section 8-1: Steps in Hypothesis Testing – Traditional Method

• Learning targets– IWBAT understand the definitions used in

hypothesis testing.– IWBAT state the null and alternative hypotheses. – IWBAT find critical values for the z-test

Vocabulary

• Statistical hypothesis – a conjecture about a population parameter. This conjecture may or may not be true.

• Null hypothesis – symbolized as H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

• Alternative hypothesis – symbolized as H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters.

Practice Problems

Solutions

• A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.

• The numerical value obtained from a statistical test is called the test value.

Errors

In hypothesis testing there are 2 types of errors:- Type I Error – you reject the null hypothesis

when it is true- Type II Error – you do not reject the null

hypothesis when it is false

Example: jury trial outcomes

Level of Significance

• represented by alpha (α) • the value used to determine the critical value

that helps determine whether or not to reject the null hypothesis

• Also referred to as the P-value which is the area under the curve

Critical Value and RegionCritical value – the z-value that separates the critical

region from the noncritical region (symbol C.V.)Critical/rejection region – range of values of the test

value that indicates that there is a significant difference and that the null hypothesis should be rejected

Noncritical/nonrejection region – range of values of the test value that indicates that the difference was probably due to chance and the null hypothesis should not be rejected

One-Tailed Test

Two-Tailed Test

% Left-tailed Right-tailed Two-tailed

.20 80 -.84 .84 1.28

.15 85 -1.03 1.03 1.44

.10 90 -1.28 1.28 1.645

.05 95 -1.645 1.645 1.96

.02 98 -2.05 2.05 2.33

.01 99 -2.33 2.33 2.575

This chart contains the z-scores for the most used α

The z-scores are found the same way they were in Section 6-1.

Practice Problems

Section 8-2

Z Test for a Mean

Steps

1. State null and alternative hypotheses.2. Find the critical values3. Compute the test value4. Make decision to reject or accept5. Summarize the results

Critical Value and RegionCritical value – the z-value that separates the critical

region from the noncritical region (symbol C.V.)Critical/rejection region – range of values of the test

value that indicates that there is a significant difference and that the null hypothesis should be rejected

Noncritical/nonrejection region – range of values of the test value that indicates that the difference was probably due to chance and the null hypothesis should not be rejected

% Left-tailed Right-tailed Two-tailed

.20 80 -.84 .84 1.28

.15 85 -1.03 1.03 1.44

.10 90 -1.28 1.28 1.645

.05 95 -1.645 1.645 1.96

.02 98 -2.05 2.05 2.33

.01 99 -2.33 2.33 2.575

This chart contains the z-scores for the most used α

The z-scores are found the same way they were in Section 6-1.

Formula

X = value = mean = Standard deviationn = sample size

One-Tailed Test

Two-Tailed Test

Summarize Results

• To summarize the results you need to state whether there is or is not sufficient evidence to support the claim (the alternative hypothesis)

- If we reject the null there is sufficient evidence to support the claim- If we fail to reject the null there is not sufficient evidence to support the claim. Example: There is sufficient evidence to support the claim that students will have an average score of 19 on the ACT.

Test StatisticTwo-tailed with α=.05, therefore critical region starts at 1.96.

Since the situation is two tailed, we have a tail to the right and a tail to the left. If we compare the two z-scores, we notice that the test statistic is greater than the critical value.

Therefore, our decision is to reject the null hypothesis.

Thus, there is sufficient evidence to support the claim that the valve does not perform to specifications.

Test Statistic

Since the claim is “less than” the situation is one-tailed.

The z-score critical value for α=.01 is -2.33

When we compare the two z-scores, we notice that the test statistic is less than the critical value and falls in the rejection region. Therefore, we will reject the null hypothesis.

Since we have rejected the null, we can conclude there is sufficient evidence to support the claim that the state employees earn on average less than the federal employees.