11. More about Hypothesis Testing

Question 1

Two-Tailed or Nondirectional Test

Answer 1

Rejection regions are located in both tails of the sampling distribution.

Question 2

One-Tailed or Directional Test

Answer 2

Rejection region is located in just one tail of the sampling distribution.

Question 3

The following statement could represent the point of departure for a hypothesis test. Given only the information in the statement, would you use a two-tailed (or nondirectional) test, a one-tailed (or directional) test with the lower tail critical, or a one-tailed (or directional) test with the upper tail critical?
Indicate your decision by specifying the appropriate H0 and H1. Furthermore, whenever you conclude that the test is one-tailed, indicate the precise word (or words) in the statement that justifies the one-tailed test.

An investigator wishes to determine whether, for a sample of drug adicts, the mean score on the depression scale of a personality test differs from a score of 60, which, according to the test documentation, represents the mean score for the general population.

Answer 3

H0: µ = 60

H1: µ ≠ 60

Question 4

The following statement could represent the point of departure for a hypothesis test. Given only the information in the statement, would you use a two-tailed (or nondirectional) test, a one-tailed (or directional) test with the lower tail critical, or a one-tailed (or directional) test with the upper tail critical?
Indicate your decision by specifying the appropriate H0 and H1. Furthermore, whenever you conclude that the test is one-tailed, indicate the precise word (or words) in the statement that justifies the one-tailed test.

To increase rainfall, extensive cloud-seeding experiments are to be concluded, and the results are to be compared with a baseline figure of 0.54 inch of rainfall (for comparable periods when cloud seeding was not done).

Answer 4

H0: µ ≤ 0.54

H1: µ > 0.54

Justification: to increase rainfall

Question 5

The following statement could represent the point of departure for a hypothesis test. Given only the information in the statement, would you use a two-tailed (or nondirectional) test, a one-tailed (or directional) test with the lower tail critical, or a one-tailed (or directional) test with the upper tail critical?
Indicate your decision by specifying the appropriate H0 and H1. Furthermore, whenever you conclude that the test is one-tailed, indicate the precise word (or words) in the statement that justifies the one-tailed test.

Public health statistics indicate, we will assume, that American males gain an average of 23 lbs during the 20-year period after age 40. An ambitious weight-reduction program, spanning 20 years, is being tests with a sample of 40 year-old men.

Answer 5

H0: µ ≥ 23

H1: µ < 23

Justification: weight-reduction program

Question 6

The following statement could represent the point of departure for a hypothesis test. Given only the information in the statement, would you use a two-tailed (or nondirectional) test, a one-tailed (or directional) test with the lower tail critical, or a one-tailed (or directional) test with the upper tail critical?
Indicate your decision by specifying the appropriate H0 and H1. Furthermore, whenever you conclude that the test is one-tailed, indicate the precise word (or words) in the statement that justifies the one-tailed test.

When untreated during their lifetime, cancer-susceptible mice have an average life span of 134 days. To determine the effects of a potentially life-prolonging (and cancer-retarding) drug, the average life span is determined for a group of mice that receives this drug.

Answer 6

H0: µ ≤ 134

H1: µ > 134

Justification: life-prolonging drug

Question 7

Should H0 be retained or rejected?

Given a one-tailed test, lower tail critical with α = .01 and z = –2.34

Answer 7

Reject H0 at the .01 level of significance because z = –2.34 is more negative than –2.34.

Question 8

Should H0 be retained or rejected?

Given a one-tailed test, lower tail critical with α = .01 and z = –5.13

Answer 8

Reject H0 at the .01 level of significance because z = –5.13 is more negative than –2.33.

Question 9

Should H0 be retained or rejected?

Given a one-tailed test, lower tail critical with α = .01 and z = 4.04

Answer 9

Retain H0 at the .01 level of significance because z = 4.04 is less negative than –2.33.

(The value of the observed z is in the direction of no concern.)

Question 10

Should H0 be retained or rejected?

Given a one-tailed test, uppertail critical with α = .05 and z = 2.00

Answer 10

Reject H0 at the .05 level of significance because z = 2.00 is more positive than 1.65.

Question 11

Should H0 be retained or rejected?

Given a one-tailed test, uppertail critical with α = .05 and z = –1.80

Answer 11

Retain H0 at the .05 level of significance because z = –1.80 is less positive than 1.65.

(The value of the observed z is in the direction of no concern.)

Question 12

Should H0 be retained or rejected?

Given a one-tailed test, uppertail critical with α = .05 and z = 1.61

Answer 12

Retain H0 at the .05 level of significance because z = 1.61 is less positive than 1.65.

Question 13

Specify the decision rule for the following situation:

a two-tailed test with α = .05

Answer 13

Reject H0 at the .05 level of significance if z equals or is more positive than 1.96 or if z equals or is more negative than –1.96.

Question 14

Specify the decision rule for the following situation:

a one-tailed test, upper tail critical, with α = .01

Answer 14

Reject H0 at the .01 level of significance if z equals or is more positive than 2.33.

Question 15

Specify the decision rule for the following situation:

a one-tailed test, lower tail critical, with α = .05

Answer 15

Reject H0 at the .05 level of significance if z equals or is more negative than –1.65.

Question 16

Specify the decision rule for the following situation:

a two-tailed test with α = .01

Answer 16

Reject H0 at the .01 level of significance if z equals or is more positive than 2.58 or if z equals or is more negative than –2.58.

Question 17

Type I Error

Answer 17

Rejecting a true null hypothesis.

Question 18

Type II Error

Answer 18

Retaining a false null hypothesis.

Question 19

List the four possible outcomes for any hypothesis.

Answer 19

Correct decision (True H0 is retained)

Type I Error

Correct decision (False H0 is rejected)

Type II Error

Question 20

Under the U.S. Criminal Code, a defendant is presumed innocent until proven guilty. Viewing a criminal trial as a hypothesis test (with H0 specifying that the defendant is innocent), describe each of the four possible outcomes.

Answer 20

Correct Decision: Innocent defendant is released

Type I Error: Innocent defendant is sentenced (False Alarm).

Correct Decision: Guilty defendant is sentenced

Type II Error: Guilty defendant is released (Miss)

Question 21

Alpha (α)

Answer 21

The probability of a type I error, that is, the probability of rejecting a true null hypothesis.

Question 22

In order to eliminate the type I error, someone decides to use the .00 level of significance. What’s wrong with this procedure?

Answer 22

A false H0 will never be rejected.

Question 23

Effect

Answer 23

Any difference between a true and a hypothesized population mean.

Question 24

Hypothesized Sampling Distribution

Answer 24

Centered about the hypothesized population mean, this distribution is used to generate the decision rule.

Question 25

True Sampling Distribution

Answer 25

Centered about the true population mean, this distribution produces the one observed mean (or z).

Question 26

Beta (β)

Answer 26

The probability of a type II error, that is, the probability of retaining a false null hypothesis.

Question 27

Comment critically on the following experimental report:

Using a group of 4 subjects, an investigator announces that H0 was retained at the .05 level of significance.

Answer 27

Because of the small sample size, only very large effects will be detected.

Question 28

Comment critically on the following experimental report:

Using a group of 600 subjects, an investigator reports that H0 was rejected at the .05 level of significance.

Answer 28

Because of the large sample size, even small, unimportant effects will be detected.

Question 29

Power (1 – β)

Answer 29

The probability of detecting a particular effect.

Question 30

Power

Answer 30

Shows how the likelihood of detecting any possible effect varies for a fixed sample size.

Question 31

An investigator consults a chart to determine the sample size required to detect an eight-point effect with a probability of .80. What happens to this detection rate of .80–will it actually be smaller, the same, or larger–if, unknown to the investigator, the true effect actually equals…

twelve points?

Answer 31

The power for the 12-point effect is larger than .80 because the true sampling distribution is shifted further into the rejection region for the false H0.

Question 32

An investigator consults a chart to determine the sample size required to detect an eight-point effect with a probability of .80. What happens to this detection rate of .80–will it actually be smaller, the same, or larger–if, unknown to the investigator, the true effect actually equals…

five points?

Answer 32

The power for the 5-point effect is smaller than .80 because the true sampling distribution is shifted further into the retention region for the false H0.