Chapter 8

Question 1

hypothesis test

Answer 1

a hypothesis test first determines the probability that the pattern could have been produced by chance alone. If this probability is large enough, we conclude that the pattern can reasonably be explained by chance. However, if the probability is extremely small, we can rule out chance as a plausible explanation and conclude that some meaningful, systematic force has created the pattern.

• A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis about a population.
• using a sample mean to test a hypothesis about a population mean.

Question 2

steps for hypothesis test

Answer 2

1. state a hypothesis about a population. Usually the hypothesis concerns the value of a population parameter. (there is the null hypothesis and the alternative hypothesis)
2. Before we select a sample, we use the hypothesis to predict the characteristics that the sample should have.
3. Next, we obtain a random sample from the population.
4. Finally, we compare the obtained sample data with the prediction that was made from the hypothesis. If the sample mean is consistent with the prediction, we conclude that the hypothesis is reasonable.

Question 3

hypothesis testing - unknown population

Answer 3

• we are assuming that the original set of scores forms a normal distribution
• The purpose of the research is to determine the effect of a treatment on the individuals in the population. That is, the goal is to determine what happens to the population after the treatment is administered.
• To simplify the hypothesis-testing situation, one basic assumption is made about the effect of the treatment: if the treatment has any effect, it is simply to add a constant amount to (or subtract a constant amount from) each individual’s score.
• we cannot administer the treatment to the entire population so the actual research study is conducted using a sample
• Note that the unknown population is actually hypothetical (the treatment is never administered to the entire population). Instead, we are asking what would happen if the treatment were administered to the entire population.

Question 4

null hypothesis

Answer 4

The null hypothesis (H0) states that in the general population there is no change, no difference, or no relationship. In the context of an experiment, H0 predicts that the independent variable (treatment) has no effect on the dependent variable (scores) for the population.

Question 5

alternative hypothesis

Answer 5

The alternative hypothesis (H1) states that there is a change, a difference, or a relationship for the general population. In the context of an experiment, (H1) predicts that the independent variable (treatment) does have an effect on the dependent variable.

Question 6

alpha level, or the level of significance

Answer 6

The alpha level, or the level of significance, is a probability value that is used to define the concept of “very unlikely” in a hypothesis test.

The alpha value is a small probability that is used to identify the low-probability samples. By convention, commonly used alpha levels a=0.05

a = 0.05. we separate the most unlikely 5% of the sample means (extreme values) from the most likely 95% (central values)

Question 7

critical region

Answer 7

The critical region is composed of the extreme sample values that are very unlikely (as defined by the alpha level) to be obtained if the null hypothesis is true (very unlikely to occur if the treatment has no effect). The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected.

• To determine the exact location for the boundaries that define the critical region, we use the alpha-level probability and the unit normal table.

Question 8

test statistic

Answer 8

A statistic that summarizes the sample data in a hypothesis test. The test statistic is used to determine whether or not the data are in the critical region.

Question 9

z-score formula as a recipe

Answer 9

1. Make a hypothesis about the amount of flour. For example, hypothesize that the correct amount is 2 cups.
2. To test your hypothesis, add the rest of the ingredients along with the hypothesized flour and bake the cake.
3. If the cake turns out to be good, you can reasonably conclude that your hypothesis was correct. But if the cake is terrible, you conclude that your hypothesis was wrong.
4. Make a hypothesis about the value of μ. This is the null hypothesis.
5. Plug the hypothesized value in the formula along with the other values (ingredients).
6. If the formula produces a z-score near zero (which is where z-scores are supposed to be), we conclude that the hypothesis was correct. On the other hand, if the formula produces an extreme value (a very unlikely result), we conclude that the hypothesis was wrong.

Question 10

z-score formula as a ratio

Answer 10

z-score = actual difference between the sample (M) and the hypothesis (mu) / Standard difference between M and mu with no treatment effect

Thus, for example, a z-score of z = 3.00 means that the obtained difference between the sample and the hypothesis is 3 times bigger than would be expected if the treatment had no effect.

Question 11

Type 1 error

Answer 11

A Type I error occurs when a researcher rejects a null hypothesis that is actually true. In a typical research situation, a Type I error means the researcher concludes that a treatment does have an effect when in fact it has no effect.

• The problem is that the information from the sample is misleading.
• A Type I error occurs when a researcher unknowingly obtains an extreme, nonrepresentative sample.

Question 12

alpha level

Answer 12

The alpha level for a hypothesis test is the probability that the test will lead to a Type I error. That is, the alpha level determines the probability of obtaining sample data in the critical region even though the null hypothesis is true.

• The primary concern when selecting an alpha level is to minimize the risk of a Type I error. Thus, alpha levels tend to be very small probability values.

Question 13

Type II error

Answer 13

A Type II error occurs when a researcher fails to reject a null hypothesis that is really false. In a typical research situation, a Type II error means that the hypothesis test has failed to detect a real treatment effect.

• A Type II error occurs when the sample mean is not in the critical region even though the treatment has an effect on the sample. Often this happens when the effect of the treatment is relatively small.

Question 14

Beta

Answer 14

Beta is the probability of a Type II error.

1. The sample data provide sufficient evidence to reject the null hypothesis and conclude that the treatment has an effect.
2. The sample data do not provide enough evidence to reject the null hypothesis. In this case, you fail to reject and conclude that the treatment does not appear to have an effect.

Question 15

significant or statistically significant

Answer 15

A result is said to be significant or statistically significant if it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. Thus, a treatment has a significant effect if the decision from the hypothesis test is to reject H0.

Question 16

factors that influence the outcome of a hypothesis test

Answer 16

the variability of scores - high variability can make it very difficult to see any clear patterns in the results from a research study

the number of scores in a sample - In general, increasing the number of scores in the sample produces a smaller standard error and a larger value for the z-score

Question 17

assumption for a hypothesis test with z-scores

Answer 17

random sampling - It is assumed that the participants used in the study were selected randomly.

independent observations - The values in the sample must consist of independent observations. In everyday terms, two observations are independent if there is no consistent, predictable relationship between the first observation and the second.

Question 18

Directional hypothesis or a one-tailed test

Answer 18

In a directional hypothesis test, or a one-tailed test, the statistical hypotheses ( H0 and H1 ) specify either an increase or a decrease in the population mean. That is, they make a statement about the direction of the effect.

1. The first step (and the most critical step) is to state the statistical hypotheses. (H0 and H1). In the first step of the hypothesis test, the directional prediction is incorporated into the statement of the hypotheses.
2. In the second step of the process, the critical region is located entirely in one tail of the distribution.

After these two changes, a one-tailed test continues exactly the same as a regular two-tailed test. Specifically, you calculate the z-score statistic and then make a decision about depending on whether the z-score is in the critical region.

Question 19

There are two serious limitations with using a hypothesis test to establish the significance of a treatment effect.

Answer 19

The first concern is that the focus of a hypothesis test is on the data rather than the hypothesis.

A second concern is that demonstrating a significant treatment effect does not necessarily indicate a substantial treatment effect. the hypothesis test has simply established that the results obtained in the research study are very unlikely to have occurred if there is no treatment effect. The hypothesis test reaches this conclusion by

(1) calculating the standard error, which measures how much difference is reasonable to expect between M and
μ
(2) demonstrating that the obtained mean difference is substantially bigger than the standard error.

Question 20

effect size

Answer 20

A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used.

Question 21

Cohen’s d

Answer 21

A standard measure of effect size computed by dividing the sample mean difference by the sample standard deviation.

Cohen’s d = mean difference / standard deviation = mu treatment - mu no treatment / standard deviation

Question 22

power

Answer 22

The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one really exists.

Researchers typically calculate power as a means of determining whether a research study is likely to be successful. Thus, researchers usually calculate the power of a hypothesis test before they actually conduct the research study.

Question 23

steps to find the power

Answer 23

To calculate the exact value for the power of the test we must determine what portion of the distribution on the right-hand side is shaded. Thus, we must locate the exact boundary for the critical region, then find the probability value in the unit normal table

This distance is equal to
z score * standard error

Next, we determine what proportion of the treated samples are greater than: z = M-mu / standard deviation

Finally, look up the z-score in the unit normal table and determine that the shaded area corresponds to . Thus, if the treatment has an effect, all the possible sample means will be in the critical region and we will reject the null hypothesis.

Question 24

alpha level and power

Answer 24

Reducing the alpha level for a hypothesis test also reduces the power of the test.

the boundaries for the critical region are drawn using a = 0.05 . Specifically, the critical region on the right-hand side begins at z = 1.96. If a were changed to a = .01, the boundary would be moved farther to the right, out to z = 2.58

Question 25

one tailed and two tailed and power

Answer 25

If the treatment effect is in the predicted direction, then changing from a regular two-tailed test to a one-tailed test increases the power of the hypothesis test.

Moving the boundary to the left would cause a larger proportion of the treatment distribution to be in the critical region and, therefore, would increase the power of the test.