Chapter 7 Part II Hypothesis Testing



CHAPTER 8

Hypothesis Testing

Objectives

Understand the definitions used in hypothesis testing.

State the null and alternative hypotheses.

Find critical values for the z test.

State the five steps used in hypothesis testing.

Test means for large samples using the z test.

Test means for small samples using the t test.

Test proportions using the z test.

Test variances or standard deviations using the chi square test.

Test hypotheses using confidence intervals.

Explain the relationship between type I and type II errors and the power of a test.

Introduction

Statistical hypothesis testing is a decision-making process for evaluating claims about a population.

In hypothesis testing, the researcher must define the population under study, state the particular hypotheses that will be investigated, give the significance level, select a sample from the population, collect the data, perform the calculations required for the statistical test, and reach a conclusion.

Hypothesis Testing

Hypotheses concerning parameters such as means and proportions can be investigated.

The z test and the t test are used for hypothesis testing concerning means.

Methods to Test Hypotheses

The three methods used to test hypotheses are:

1. The traditional method.

2. The P-value method.

3. The confidence interval method.

Section 8.1 Steps in Hypothesis Testing-Traditional Method

I. Setting up a hypothesis testing.

A statistical hypothesis is a conjecture about a population parameter which may or may not be true.

There are two types of statistical hypotheses for each situation: the null hypothesis and the alternative hypothesis.

Hypotheses

The null hypothesis, symbolized by 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. The null hypothesis contains an equal sign.

The alternative hypothesis, symbolized by 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. The alternative hypothesis usually contains the symbol >, , < , or (

Step 2: Select the test statistic and determine its value from the sample data.

See the formulas given for II.

Step 3: Use a given level of significance and H1 to determine the critical

region(s).

Step 4: Make your decision. If the test statistic falls in the critical region,

reject Ho. If the test statistic does not fall in the critical region,

we do not reject Ho. Interpret your decision in terms of the claim.

Example 1: Full-time Ph.D. students receive an average salary of $12,837 according to the U.S. Department of Education. The dean of graduate studies at a large state university feels that Ph.D. students earn more than this. He surveys 44 randomly selected students and finds their average salary is $14,445 with a standard deviation of $1,500. With ( = 0.05, is the dean correct?

Example 2: The average U.S. wedding includes 125 guests. A random sample of 35 weddings during the past year in a particular county had a mean of 110 guests and a standard deviation of 30. Is there sufficient evidence at the 0.01 level of significance that the average number of guests differs from the national average?

III. The P-value

The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.

The P-value is the actual area under the standard normal distribution curve (or other curve depending on what statistical test is being used) representing the probability of a particular sample statistic or a more extreme sample statistic occurring if the null hypothesis is true.

The P-value Approach to Hypothesis Testing

Step 1: Identify H0 and H1

Ho will contain =

Ha will contain > , < , or (

Step 2: Select the test statistic and determine its value from the sample data.

See the formulas given for I.

Step 3: Omit the step of finding the critical region. Find the P-value

associated with the value of the test statistic.

Step 4: Make your decision. If the P-value ( ( , reject H0.

If the P-value > ( , do not reject H0.

Interpret your decision in terms of the claim.

Decision Rule When Using a P-Value

If the P-value ( ( , reject the null hypothesis.

If the P-value > ( , do not reject the null hypothesis.

Example 1: Find the P-value and decide whether to reject or not to reject the H0.

Assume ( = 0.05.

(a) H0: ( = 80; Ha: ( > 80 Observed value: z = 1.75

(b) H0: ( = 15; Ha: ( < 15 Observed value: z = -.83

Example 2: A college professor claims that the average cost of a paperback

textbook is greater than $27.50. A sample of 50 books has an average cost of $29.30. The standard deviation of the sample is $5.00. Find the P-value for the test. On the basis of the P-value, should the null hypothesis be rejected at ( = 0.05? Is there sufficient evidence to conclude that the

professor’s claim is correct?

8.3 The t Test

The t test is a statistical test of the mean of a population and is used when the population is normally or approximately normally distributed, ( is unknown, and the sample size is less than 30.

The formula for the t test is:

The degrees of freedom are d.f. = n–1.

Example 1: Find the critical t value for [pic] with d.f. = 16 for a right-tailed t test.

Example 2: Find the critical t value for [pic] with d.f. = 18 for a right-tailed t test.

Example 3: Find the P-value and decide whether to reject or not to reject Ho.

[pic]

Example 4: Find the P-value and decide whether to reject or not to reject Ho.

[pic]

Example 5: Data were obtained from one section of a mathematics course taught

by a good instructor. For past classes, the mean final exam score for this

course is 70. Prior to the final exam, the instructor was interested in

whether the mean final exam score for his section containing 14 students

would be statistically significantly higher than 70.

A stem-and-leaf plot is given.

5 0 0

6 1 1 2 3

7 0 3 4 5

8 0 2

9 2 9

(a) Is the t procedure appropriate in this case? If so, proceed to part (b).

(b) Find the mean and standard deviation for this class.

(c) Using a 5% level of significance, can the teacher conclude that the mean

performance of the class is larger than 70?

Example 6: The average amount of rainfall during the summer months for the

summer months for the northeast part of the United States is 11.52 inches. A researcher selects a random sample of 10 cities in the northeast and finds that the average amount of rainfall for 1995 was 7.42 inches. The standard deviation of the sample is 1.3 inches. At ( = 0.05, can it be concluded that for 1995 the mean rainfall was below 11.52 inches?

8.4 z Test for a Proportion

A hypothesis test involving a population proportion can be considered as a binomial experiment when there are only two outcomes and the probability of a success does not change from trial to trial.

Formula for the z Test for Proportions

Example 1: Dropout Rates for High School Seniors. An educator estimates that the dropout rate for seniors at high schools in Ohio is 15%. Last year, 38 seniors from a random sample of 200 Ohio seniors withdrew. At [pic], is there enough evidence to reject the educator’s claim?

Example 2: A retailer has received a large shipment of VCRs. He decided to

accept the shipment provided there is no evidence to suggest that the

shipment contains more than 5% non-acceptable items. The retailer found 14 non-acceptable items in a random selection of 160 items. At a 5% level of significance, what is the retailer’s decision?

Example 3: In a recent presidential election, 611 voters were surveyed, and

308 of them said that they voted for the candidate who won.

Use a 0.04 significance level to test the claim that among all voters, 43%

say that they voted for the winning candidate.

Example 4: Find the P-value and decide whether to reject or not to reject Ho

H0: [pic]= 0.60; Ha: [pic]( 0.60 Observed value: z = -1.22

Summary

A statistical hypothesis is a conjecture about a population.

There are two types of statistical hypotheses: the null hypothesis states that there is no difference, and the alternative hypothesis specifies a difference.

The z test is used when the population standard deviation is known and the variable is normally distributed or when ( is not known and the sample size is greater than or equal to 30.

When the population standard deviation is not known and the variable is normally distributed, the sample standard deviation is used, but a t test should be conducted if the sample size is less than 30.

Researchers compute a test value from the sample data in order to decide whether the null hypothesis should or should not be rejected.

Statistical tests can be one-tailed or two-tailed, depending on the hypotheses.

The null hypothesis is rejected when the difference between the population parameter and the sample statistic is said to be significant.

The difference is significant when the test value falls in the critical region of the distribution.

The critical region is determined by (, the level of significance of the test.

The significance level of a test is the probability of committing a type I error.

A type I error occurs when the null hypothesis is rejected when it is true.

The type II error can occur when the null hypothesis is not rejected when it is false.

One can test a single variance by using a chi-square test.

Conclusions

Researchers are interested in answering many types of questions. For example:

“Will a new drug lower blood pressure?”

“Will seat belts reduce the severity of injuries caused by accidents?”

These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population.

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