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 ≠.
Hypothesis-Testing Common Phrases
In hypothesis testing, the idea is to give the benefit of the doubt to the null hypothesis. H0 will be rejected only if the sample data suggest beyond a reasonable doubt that Ho is false.
Example 1: For the claims in part (a) – (f), (i) find H0 and H1, (ii) give the type of critical region (right-tailed, left-tailed, or two tailed).
(a) The mean temperature in a coastal town is less than 78°F.
(b) The mean grade point average of graduating seniors at a university
is at least 2.3.
(c) The mean income for L.A. Mission faculty is $4500 per month.
(d) The proportion of voters in a state who favor the death penalty
under prescribed conditions is .53.
(e) The proportion of full-time college students younger than 26 years
of age is at most .90.
(f) The proportion of of high school graduates in 1990 seeking full-time
employment was more than .30.
Design of the Study
• After stating the hypotheses, the researcher’s next step is to design the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study.
Statistical Test
• A statistical test uses the data obtained from a sample to make a decision about whether or not the null hypothesis should be rejected.
• The numerical value obtained from a statistical test is called the test value.
II. Possible Outcomes of a Hypothesis Test
Summary of Possible Outcomes
• A type I error occurs if one rejects the null hypothesis when it is true.
• A type II error occurs if one does not reject the null hypothesis when it is false.
Error Probabilities
• The level of significance is the maximum probability of committing a type I error. This probability is symbolized by a; that is ,
• The probability of a type II error is symbolized by b. That is,
a and b Probabilities
• In most hypothesis testing situations, b cannot easily be computed; however, a and b are related in that decreasing one increases the other.
Hypothesis Testing
• In a hypothesis testing situation, the researcher generally agree on using three arbitrary significance levels: the 0.10, 0.05, and 0.01 levels. That is, if the null hypothesis is rejected, the probability of a type I error will be 10, 5, or 1%, depending on which level of significance is used.
Ex) If a = 0.10, there is a 10% chance of rejecting a true null hypothesis.
If a = 0.05, there is a 5% chance of rejecting a true null hypothesis.
If a = 0.01, there is a 1% chance of rejecting a true null hypothesis.
Critical Values
• After a significance level is chosen, a critical value is selected from a table for the appropriate test.
• The critical value(s) separates the critical region from the noncritical region. The symbol for critical value is C.V.
• The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected.
• The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.
One-Tailed Test
• A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean.
• A one-tailed test is either right-tailed or left-tailed, depending on the direction of the inequality of the alternative hypothesis.
Left-Tailed Test
Right-Tailed Test
Two-Tailed Test
• In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions.
Hypothesis-Testing (Traditional Method)
• Step 1 State the hypothesis, and identify the claim.
• Step 2 Find the critical value from the appropriate table.
• Step 3 Compute the test value.
• Step 4 Make the decision to reject or not reject the null hypothesis.
• Step 5 Summarize the results.
Section 8.2 z Test for a Mean
I. Test Statistic
• The z test is a statistical test for the mean of a population. It can be used when , or when the population is normally distributed and s is known.
• Many hypotheses are tested using a statistical test based on the general formula
The z Test
The formula for the z test is:
The z Test When s is Unknown
The central limit theorem states that when the population standard deviation s is unknown, the sample standard deviation s can be used in the formula as long as the sample size is 30 or more.
Test statistic for Testing a Claim about a Mean
If s is known ,
If s is unknown but n ³ 30,
If s is unknown and n < 30,
II. The Classical Approach to Hypothesis Testing
Step 1: Identify H0 and H1
Ho will contain =
H1 will contain > , < , or ¹
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 a = 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 ¹
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 £ a , reject H0.
If the P-value > a , do not reject H0.
Interpret your decision in terms of the claim.
Decision Rule When Using a P-Value
If the P-value £ a , reject the null hypothesis.
If the P-value > a , do not reject the null hypothesis.
Example 1: Find the P-value and decide whether to reject or not to reject the H0.
Assume a = 0.05.
(a) H0: m = 80; Ha: m > 80 Observed value: z = 1.75
(b) H0: m = 15; Ha: m < 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 a = 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, s 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 with d.f. = 16 for a right-tailed t test.
Example 2: Find the critical t value for 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.
Example 4: Find the P-value and decide whether to reject or not to reject Ho.
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 a = 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 , 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: = 0.60; Ha: ¹ 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.