11. More about Hypothesis Testing Flashcards
Two-Tailed or Nondirectional Test
Rejection regions are located in both tails of the sampling distribution.
One-Tailed or Directional Test
Rejection region is located in just one tail of the sampling distribution.
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.
H0: µ = 60
H1: µ ≠ 60
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).
H0: µ ≤ 0.54
H1: µ > 0.54
Justification: to increase rainfall
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.
H0: µ ≥ 23
H1: µ < 23
Justification: weight-reduction program
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.
H0: µ ≤ 134
H1: µ > 134
Justification: life-prolonging drug
Should H0 be retained or rejected?
Given a one-tailed test, lower tail critical with α = .01 and z = –2.34
Reject H0 at the .01 level of significance because z = –2.34 is more negative than –2.34.
Should H0 be retained or rejected?
Given a one-tailed test, lower tail critical with α = .01 and z = –5.13
Reject H0 at the .01 level of significance because z = –5.13 is more negative than –2.33.
Should H0 be retained or rejected?
Given a one-tailed test, lower tail critical with α = .01 and z = 4.04
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.)
Should H0 be retained or rejected?
Given a one-tailed test, uppertail critical with α = .05 and z = 2.00
Reject H0 at the .05 level of significance because z = 2.00 is more positive than 1.65.
Should H0 be retained or rejected?
Given a one-tailed test, uppertail critical with α = .05 and z = –1.80
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.)
Should H0 be retained or rejected?
Given a one-tailed test, uppertail critical with α = .05 and z = 1.61
Retain H0 at the .05 level of significance because z = 1.61 is less positive than 1.65.
Specify the decision rule for the following situation:
a two-tailed test with α = .05
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.
Specify the decision rule for the following situation:
a one-tailed test, upper tail critical, with α = .01
Reject H0 at the .01 level of significance if z equals or is more positive than 2.33.
Specify the decision rule for the following situation:
a one-tailed test, lower tail critical, with α = .05
Reject H0 at the .05 level of significance if z equals or is more negative than –1.65.
Specify the decision rule for the following situation:
a two-tailed test with α = .01
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.
Type I Error
Rejecting a true null hypothesis.
Type II Error
Retaining a false null hypothesis.
List the four possible outcomes for any hypothesis.
Correct decision (True H0 is retained)
Type I Error
Correct decision (False H0 is rejected)
Type II Error
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.
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)
Alpha (α)
The probability of a type I error, that is, the probability of rejecting a true null hypothesis.
In order to eliminate the type I error, someone decides to use the .00 level of significance. What’s wrong with this procedure?
A false H0 will never be rejected.
Effect
Any difference between a true and a hypothesized population mean.
Hypothesized Sampling Distribution
Centered about the hypothesized population mean, this distribution is used to generate the decision rule.
True Sampling Distribution
Centered about the true population mean, this distribution produces the one observed mean (or z).
Beta (β)
The probability of a type II error, that is, the probability of retaining a false null hypothesis.
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.
Because of the small sample size, only very large effects will be detected.
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.
Because of the large sample size, even small, unimportant effects will be detected.
Power (1 – β)
The probability of detecting a particular effect.
Power
Shows how the likelihood of detecting any possible effect varies for a fixed sample size.
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?
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.
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?
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.
