Chapter 8 Intro to Hypothesis Testing
Define a Type 2 error.
Type 2 error occurs when the failure to reject a false null hypothesis. In terms of a research study, a Type II error occurs when a study fails to detect a treatment effect that really exists.
If the alpha level is increased from a= .01 to a=.05, then the boundaries for the critical region move farther away from the center of the distribution. (True or false?)
False. A larger alpha means that the boundaries for the critical region move closer to the center of the distribution.
If a sample mean is in the critical region w/a = 05, it would still (always) be in the critical region if alpha were changed to a=.01. (True or false?)
False. W/a=.01, the boundaries for the critical region move farther out into the tails of the distribution. It is possible that a sample mean could be beyond the .05 boundary but not beyond the .01 boundary.
State the steps in a directional/one-tailed test.
First Incorporate the directional prediction into the hypotheses. Then locate the critical region and determine what kind of data would refute the null hypothesis by demonstrating that the treatment worked as predicted.
Name the steps to determine the power for a hypothesis test.
First identify the treatment and full distributions and specify the magnitude of the treatment effect. Second sketch the distribution of sample means predicted by the null hypothesis and the distribution predicted by the specified treatment effect. Third locate the critical region in the null hypothesis distribution and then determine the proportion of the treated distribution that is beyond the critical boundaries.
What are the 2 types of errors that can be committed when making an incorrect decision on the hypothesis? Define these 2 errors.
Type I error rejects a true H0- A serious error b/c it results in falsely reporting a treatment effect. Type II error is the failure to reject a false H0- The experiment fails to detect an effect that actually occurred.
What does the effect of size and power do?
Tells the whether an effect is significan't but can't share the size of effect nor real world settings/clinical trials
If a researcher conducted a hypothesis test with an alpha level of a=.02, what z-score values would form the boundaries for the critical region?
The .02 would be split between the two tails, with .01 in each tail. The z-score boundaries would be z= 2.33 and z= -2.33.
True/False is Cohen's d calculated the same for 1-Talied and 2-tailed tests and are they concrete rules?
True. Cohen's d is calculated the same type 1-Tailed & 2-tailed tests but arennot concrete rules are used as guidelines
In a research report, the results of a hypothesis test include the phrase "z = 1.63, p > .05." This means that the test failed to reject the null hypothesis. (True or false?)
True. The probability is greater than .05, which means there is a reasonable likelihood that the result occurred without any treatment effect.
If a sample mean is in the critical region w/a=.01, it would still (always) be in the critical region if alpha were changed to a=.05. (True or false?)
True. With a=.01, the boundaries for the critical region are farther out into the tails of the distribution than for a=.05. If a sample mean is beyond the .01 boundary it is definitely beyond the .05 boundary.
In directional or one-tailed testing, what percentage are the outcomes located entirely in one tail of the distribution for a critical region?
5%, 1%, or 0.1% depending on a (alpha)
Under what circumstances is a Type II error likely to occur
A Type II error is likely to occur when the treatment effect is very small. In this case, a research study is more likely to fail to detect the effect.
What factors in statistical power influenced by?
A large sample results in more power than a small sample. Increasing the alpha level increases power. A one-tailed test has greater power than a two-tailed test.
As the power of a test increases, what happens to the probability of a Type II error?
As power increases, the probability of a Type II error decreases.
How is Type II error specified?
Cannot be specified as a single value and depends in part on the size of the treatment effect. It is identified by the symbol B (beta).
In addition to using a hypothesis test to evaluate the significance of a treatment effect, what else can we use to also measure and report the effect size?
Cohen's d= Mean difference/standard deviation
What is a directional, or one-tailed test, and when should it be used?
Directional, or one-tailed test are used when researchers expect that a treatment will change scores in a particular direction (increase/decrease).
What does statistical power imply?
Implies the more power assist the more powerful a statistical test is the more readily it will detect a treatment effect when one exists
Critical region
composed of the extreme sample outcomes that are very unlikely to occur if the null hypothesis is true
Explain how increasing the sample size influences the outcome of a hypothesis test and how it influences the value of Cohen's d.
Increasing sample size increases the likelihood of rejecting the null hypothesis but has no effect on Cohen's d.
A researcher selects a sample from a population with u=45 and sigma= 8. A treat- ment is administered to the sample and, after treatment, the sample mean is found to be M =47. Compute Cohen's d to measure the size of the treatment effect.
d= 2/8 = 0.25
Does cohen's d effect size dependent on or otherwise influence by whether the hypothesis test is one-tailed or two-tailed?
No either way the calculations are the same
What are some of the values of Cohen's d?
d=0.2-Small Effect(mean difference = 0.2 Standard deviation) D=0.5 Medium Effect (mean difference = 0.5 standard deviation) D=0.8- Large effect (mean difference = 0..8)
Alternative hypothesis
The hypothesis stating what the researcher is seeking evidence of. A statement of inequality. It can be written looking for the difference or change in one direction from the null hypothesis or both.
Level of significance
The probability of making a Type I error when the null hypothesis is true as an equality.
How does Type I error occur?
The risk of a Type I error is determined by the alpha level and, therefore, is under the experiment's control.
A researcher selects a sample from a population with u=70 and sigma=12. After administering a treatment to the individuals in the sample, the researcher computes Cohen's d =0.25. What is the mean for the sample?
There is a 3-point difference between the sample mean and u=70, so the sample mean is either 73 or 67.
In a research report, the term significant is used when the null hypothesis is rejected. (True or false?)
True.
If other factors are held constant, increasing the size of the sample increases the likelihood of rejecting the null hypothesis. (True or false?)
True. A larger sample produces a smaller standard error, which leads to a larger z-score.
A small value (near zero) for the z-score statistic is evidence that the sample data are consistent with the null hypothesis. (True or false?)
True. A z-score near zero indicates that the data support the null hypothesis.
statistical power
a measure of the likelihood that we correctly will reject the null hypothesis, given that the null hypothesis is false.
If other factors are held constant, are you more likely to reject the null hypothesis with a standard deviation sigma=2 or with sigma=10?
sigma= 2. A smaller standard deviation produces a smaller standard error, which leads to a larger z-score.
Alpha level
the probability level used by researchers to indicate the cutoff probability level (highest value) that allows them to reject the null hypothesis
Define Type 1 error.
type 1 error is rejecting a true null hypothesis- that is, saying that the treatment has an effect when, in fact, it does not.
A z-score value in the critical region means that you should reject the null hypothesis. (True or false?)
True. A z-score value in the critical region means that the sample is not consistent with the null hypothesis.
Null Hypothesis (H0)
a statement or idea that can be falsified, or proved wrong
hypothesis test
a statistical method that uses sample data to evaluate a hypothesis about a population
What is the 4-step process that is used in testing hypothesis?
a) State the null hypothesis (H0), and select an alpha level (a level defines "very unlikely"). Also state an alternative hypothesis (H1) which is exactly opposite of the nulll hypothesis. B) Locate the critical region. C) Collect the data, and compute the test statistic. D) Make a decision.
For a 5-point treatment effect, a researcher computes power of p = 0.50 for a two-tailed hypothesis test with a= .05. a. Will the power increase or decrease for a 10-point treatment effect? b. Will the power increase or decrease if alpha is changed to a= .01? c. Will the power increase or decrease if the researcher changes to a one-tailed test? 3. How does sample size influence the power of a hypothesis test?
a. The hypothesis test is more likely to detect a 10-point effect, so power will be greater. b. Decreasing the alpha level also decreases the power of the test. c. Switching to a one-tailed test will increase the power.