Chapter 8 Hypothesis Testing

1
Chapter 8- Hypothesis Testing
In chapter 7, we estimated the values of
population parameters in this chapter we will
learn how to test claims (hypotheses) made about
parameters.
2

• In statistics, a hypothesis is a claim or
• statement about a property of a population.
• A hypothesis test (or test of significance) is
• a standard procedure for testing a claim
• about a property of a population.

3
Rare Event Rule for Inferential Statistics

• If, under a given assumption, the probability
• of a particular observed event is
• exceptionally small (but the event DOES
• occur), we conclude that the assumption is
• probably not correct.

4
Section 8-2 Components of a Formal Hypothesis
Test

• The null hypothesis (HO) is a statement
• that the value of a population parameter
• is equal to some claimed value (k)
• HO ? k
• HO ? k
• HO ? k
• ? We will either reject or fail to reject
• the null hypothesis.

5

• The alternative hypothesis (H1 or Ha) is
• the statement that the parameter has a
• value that somehow differs from the null
• hypothesis
• H1 ? ? k
• H1 ? lt k
• H1 ? gt k
• Note H1 must be true if H0 is false

6
Note about Forming Your Own Claims

• If you are conducting a study and want to use
  a hypothesis test to support your claim, the
  claim must be worded so that it becomes the
  alternative hypothesis.

Note about Testing the Validity of Someone Elses
Claim
Someone elses claim may become the null
hypothesis (because it contains equality), and it
sometimes becomes the alternative hypothesis
(because it does not contain equality).
7

• The significance level, ?, is the
• probability that we will reject the null
• hypothesis is actually true.
• Common choices for ? are 0.05, 0.01, and 0.10.

8

• Conclusions in Hypothesis Testing
• We always test the null hypothesis and
• reach one of the following initial
• conclusions
• 1. Reject HO
• 2. Fail to reject HO

9
Decision Criteria

• The P-value (or p-value or probability value) is
  the probability of getting a value of the test
  statistic that is at least as extreme as the one
  representing the sample data, assuming that the
  null hypothesis is true.
• Reject HO if P-value ? ? (where ? is the
  significance level, such as 0.05).
• Fail to reject HO if P-value gt ?.

10

• In writing conclusions for the hypotheses
• tests, we will either reject the null
• hypothesis or fail to reject the null
• hypothesis this does NOT mean that we
• are proving the null hypothesis is true!
• See pg. 380 for the appropriate wording for the
  written conclusions.

11
Example 1

• Examine the given statement, then express the
  null hypothesis HO and alternative hypothesis H1
  in symbolic form.
• The percentage of men who watch golf on TV is
  not 70, as is claimed by the Madison Advertising
  Company
• The mean IQ of statistics students is at least
  110.
• Remember, the null hypothesis must contain
  equality .

12
Example 2

• Use the given information to find the P-value
• a. left-tailed, test statistic is z -1.72
• ?
• b. 2 tailed, test statistic is z 1.63
• ?
• c. H1 p gt 0.34, test statistic is z 2.35
• ?

13
Example 3

• State the final conclusion in simple
  non-technical terms. Be sure to address the
  original claim.
• Original Claim The proportion of college
  graduates who smoke is less
• than 0.27.
• Initial Conclusion Reject the null hypothesis.

14

• Original Claim The proportion of MMs that
  are blue is equal to 0.10.
• Initial Conclusion Reject the null hypothesis.

15
Types of Errors
16

• Well constructed hypothesis tests must be
• designed to minimize possible decision
• errors. The following considerations may
• be relevant
• 1. A larger sample size will lessen the
• chance that you make the error of not
• rejecting the null hypothesis when its
• actually false (type II error).

17

• 2. For any fixed sample size n, a
• decrease in ? will cause an increase in
• ?, and vice versa.
• 3. To decrease both ? and ?, increase the
• sample size.
• NOTE Since ? is the probability of failing
• to reject a false null hypothesis, 1 - ? is the
• probability of rejecting a false null
• hypothesis and is known as the power of
• a test.

18
Types of Errors