Chapters 7&8

Question 1

Sampling Error

Answer 1

The natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
• a random sample may not be representative of the entire population
- extreme scores are not likely included in the sample
• difference between the sample statistic and population parameter is sampling error
• two samples from the same population will be different
- different individuals, scores, means

Question 2

Distribution of Sample Means

Answer 2

• a collection of all sample means (M) for all possible random samples of a specific size (n) from a population
- an example of a sampling distribution
- sampling distribution of M
- primary use is to find the probability associated with any specific sample
• to calculate the distribution we need to know all the possible samples
• How would we construct a distribution of sample means?
- select a random sample of a specific size (n), and calculate the mean
- place the sample mean in a frequency distribution
- repeat again, and again, and again, until you have the complete set of all the possible random samples

Question 3

Sampling Distribution

Answer 3

A distribution of statistics obtained by selecting all of the possible samples of a specific size from a population.

Question 4

Sample Means: Characteristics

Answer 4

(1) The sample means should pile up around the population mean.
(2) The pile of sample means should tend to form a normal-shaped distribution.
(3) In general, the larger the sample size, the closer the sample means should be to the population mean.

Question 5

Central Limit Theorum

Answer 5

For any population with mean (μ) and standard deviation (σ), the distribution of sample means for sample size (n) will have a mean of μ and a standard deviation of σ/ √n and will approach a normal distribution as n approaches infinity.
• For any population with a mean μ and a standard deviation σ
- expected value of M
- the sample mean distribution will be a normal distribution if: n ≥ 30 and/or the population has a normal distribution
- standard error of M
• The sample mean M will unlikely be equal to the population mean μ
- standard error = average distance between M and μ
• the larger the sample (n), the more likely M = μ
- the law of large numbers
• the larger the sample (n), the smaller the standard error

Question 6

Distribution of sample means is almost perfectly normal if either of the following two conditions is satisfied:

Answer 6

(1) The population from which the samples are selected is a normal distribution.
(2) The number of scores (n) in each sample is relatively large, around 30 or more.

Question 7

Expected Value of M

Answer 7

The mean of the distribution of sample means is equal to the mean of the population of scores, μ.
- distribution of sample means will have a mean of μ

Question 8

Sample Means are an example of:

Answer 8

an unbiased statistic

Question 9

Standard Error of M

Answer 9

σm = σ/√n = √σ2/√n = square root of σ2/n

• distribution of sample means will have a standard deviation of σ/ n
  The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ).
• The σ indicates that this value is a standard deviation, and the subscript m indicates that it is the standard deviation for the distribution of sample means.
• Is an extremely valuable measure because it specifies precisely how well a sample mean estimates its population mean—that is, how much error you
  should expect, on the average, between M and μ.

Question 10

The standard error serves the same two purposes for the distribution of sample means:

Answer 10

(1) The standard error describes the distribution of sample means.It provides a measure of how much difference is expected from one sample to another. When the standard error is small, then all of the sample means are close together and have similar values. If the standard error is large, then the sample means are
scattered over a wide range and there are big differences from one sample to another.
(2) Standard error measures how well an individual sample mean represents the entire distribution. Specifically, it provides a measure of how much distance is reasonable to expect between a sample mean and the overall mean for the distribution
of sample means. However, because the overall mean is equal to μ, the standard error also provides a measure of how much distance to expect between a sample mean (M) and the population mean (μ).

Question 11

The magnitude of the standard error is determined by two factors:

Answer 11

(1) the size of the sample
- law of large numbers
(2) the standard deviation of the population from which the sample is selected.
- When n = 1, σm = σ (standard error = standard deviation).

Question 12

Law of Large Numbers

Answer 12

The larger the sample size (n), the more probable it is that the sample mean is close to the population mean.
- the error between the sample mean and the population mean should decrease

Question 13

A z-Score for Sample Means

Answer 13

(1) we are finding the location for a sample mean (M) rather than a score (X).
(2) the standard deviation for the distribution of sample means is the standard error, σm.
With these changes, the z-score formula for locating a sample mean is:
z = M - μ / σm

Question 14

Hypothesis Test

Answer 14

A statistical method that uses sample data to evaluate a hypothesis about a population.
- inferential process

Question 15

The Unknown Population

Answer 15

The researcher begins with a known population. This is the set of individuals as they exist before treatment.
The unknown population, after treatment, is the focus of the research question.

Question 16

Null Hypothesis

Answer 16

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 17

Alternative Hypoethesis

Answer 17

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 18

The distribution of sample means is then divided into two sections:

Answer 18

(1) Sample means that are likely to be obtained if H0 is true; that is, sample means that are close to the null hypothesis
(2) Sample means that are very unlikely to be obtained if H0 is true; that is, sample means that are very different from the null hypothesis

Question 19

Alpha Level / Level of Significance

Answer 19

α
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 level for a hypothesis test is the probability that the test will lead to a Type I error if the null hypothesis is true.

Question 20

Critical Region

Answer 20

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.
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.
Can also define the critical region as sample values that provide convincing evidence that the treatment really does have an effect.

Question 21

The Boundaries of the Critical Region

Answer 21

To determine the exact location for the boundaries that define the critical region, we use the alpha-level probability and the unit normal table.
In most cases, the distribution of sample means is normal, and the unit normal table provides the precise z-score location for the critical region boundaries.

Question 22

The z-score formula for a sample mean

Answer 22

In the formula, the value of the sample mean (M) is obtained from the sample data, and the value of m is obtained from the null hypothesis. Thus, the z-score formula can be expressed in words as follows:
z = sample mean hypothesized - population mean / standard error between M and μ
a.k.a.
z = M - μ / σm

Question 23

Collect data and compute sample statistics

Answer 23

(1) The data are collected after the researcher has stated the hypotheses and established the criteria for a decision.
- helps to ensure that a researcher makes an honest, objective evaluation of the data and does not tamper with the decision criteria after the experimental outcome is known.
(2) Next, the raw data from the sample are summarized with the appropriate statistics
- Now it is possible for the researcher to compare the sample mean (the data) with the null hypothesis.
- The comparison is accomplished by computing a z-score that describes exactly where the sample mean is located relative to the hypothesized population mean from H0.

Question 24

Making a Decision

Answer 24

Comparing our treated sample with the distribution of sample means that would be obtained for untreated samples

Question 25

Possible Outcomes of Hypothesis Testing

Answer 25

(1) The sample data are located in the critical region.
- The sample data provide sufficient evidence to reject the null hypothesis and conclude that the treatment has an effect.
(2) The sample data are not in the critical region.
- The sample data do not provide enough evidence to reject the null hypothesis. In this case, you fail to reject H0 and conclude that the treatment does not appear to have an effect.

Question 26

Type 1 Error

Answer 26

(α)
• reject null hypothesis when it is actually true
• incorrectly conclude the treatment has an effect
• selecting an extreme sample by chance
• the alpha level determines the probability of this error occurring

Question 27

Type II Error

Answer 27

(ß)
• retain the null hypothesis when it is actually false
• the treatment effect really exists, but wasn’t detected
• sample mean not in the critical region even though treatment had an effect

Question 28

Primary Concern of Selecting an Alpha Level

Answer 28

Primary concern is to minimize the risk of a Type I error. Thus, alpha levels tend to be very small probability values.

Question 29

When you select a value for alpha at the beginning of a hypothesis test, your decision influences both of these functions:

Answer 29

(1) alpha level helps to determine the boundaries for the critical region by defining the concept of “very unlikely” outcomes.
(2) alpha level helps to determine the probability of a Type I error if the null hypothesis is true.

Question 30

Steps of Hypothesis Testing: Step 1

Answer 30

(1) State the two possible hypothesis
Null Hypothesis (H0): the treatment has no effect (no change)
- µ = 80 (no change)
Alternative Hypothesis (H1): the treatment has an effect
- µ ≠ 80 (treatment causes a change)

Question 31

Statistically Significant

Answer 31

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 32

The assumptions for hypothesis tests with z-scores

Answer 32

• Random Sampling
• Independent Observation
- two events (or observations) are independent if the occurrence of the first event has no effect on the probability of the second event.
• The value of σ is unchanged by the treatment
- we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population
> before treatment.
the effect of the treatment is to add a constant amount to (or subtract a constant amount from) every score in the population.
• Normal sampling distribution
- To evaluate hypotheses with z-scores, we have used the unit normal table to identify the critical region (only if distribution is normal)

Question 33

Factors that influence a hypothesis test

Answer 33

(1) The variability of the scores, which is measured by either the standard deviation or the variance. The variability influences the size of the standard error in the denominator of the z-score.
(2) The number of scores in the sample. This value also influences the size of the standard error in the denominator.

Question 34

Directional (One-Tailed) Hypothesis Test

Answer 34

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.
• the two hypotheses would state:
H0: Test scores are not increased. (The treatment does not work.)
H1: Test scores are increased. (The treatment works as predicted.)

Question 35

A directional (one-tailed) test requires changes in the first two steps of the step-by-step hypothesis-testing procedure:

Answer 35

(1) In the first step, the directional prediction is included in 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.

Question 36

One-Tailed VS Two-Tailed Tests

Answer 36

• The major distinction between one-tailed and two-tailed tests is the criteria that they use for rejecting H0.
• A one-tailed test allows you to reject the null hypothesis when the difference between the sample and the population is relatively small, provided that the difference is in the specified direction.
• A two-tailed test, on the other hand, requires a relatively large difference independent of direction.

Question 37

Concerns about Hypothesis Testing

Answer 37

There are two serious limitations with using a hypothesis test to establish the significance of a treatment effect.
Concerns:
(1) that the focus of a hypothesis test is on the data rather than the hypothesis. Specifically, when the null hypothesis is rejected, we are actually making a strong probability statement about the sample data, not about the null hypothesis.
(2) that demonstrating a significant treatment effect does not necessarily indicate a substantial treatment effect. In particular, statistical significance doesn’t provide any real information about the absolute size of a treatment effect. Instead, 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.

Question 38

Effect Size

Answer 38

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 39

Cohen’s d

Answer 39

One of the simplest and most direct methods for measuring effect size is Cohen’s d.
Cohen (1988) recommended that effect size can be standardized by measuring the mean difference in terms of the standard deviation. The resulting measure of effect size is computed as
Cohen’s d = mean difference / standard deviation = μ treatment - μ no treatment / σ

Question 40

estimated Cohen’s d

Answer 40

estimated Cohen’s d = mean difference / standard deviation = M treatment - μ no treatment / σ
• One of the simplest and most direct methods for measuring effect size
• Cohen (1988) recommended that effect size can be standardized by measuring the mean difference in terms of the standard deviation.
• The sample mean is expected to be representative of the population mean and provides the best measure of the treatment effect. , thus it is the estimated Cohen’s d
• Cohen’s d measures the distance between two means and is typically reported as a positive number even when the formula produces a negative value.

Question 41

Statistical Power

Answer 41

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.
• usually calculate the power of a hypothesis test before they actually conduct the research study. In this way, they can determine the probability that the results will be significant (reject H0) before investing time and effort in the actual research.

Question 42

Power and Effect Size

Answer 42

As the effect size increases, the probability of rejecting H0 also increases, which means that the power of the test increases.
Thus, measures of effect size such as Cohen’s d and measures of power both provide an indication of the strength or magnitude of a treatment effect.

Question 43

Factors that Affect Power

Answer 43

(1) Effect Size
(2) Sample Size
(3) Alpha Level
(4) One-Tailed versus Two-Tailed Tests

Question 44

Power and Sample Size

Answer 44

one of the primary reasons for computing power is to determine what sample size is necessary to achieve a reasonable probability for a successful research study.
If the probability (power) is too small, they always have the option of increasing sample size to increase power.

Question 45

Power and Alpha Level

Answer 45

• Reducing the alpha level for a hypothesis test also reduces the power of the test.
• Moving the critical boundary to the right means that a smaller portion of the treatment distribution (the distribution on the right-hand side) will be in the critical region.
- Thus, there would be a lower probability of rejecting the null hypothesis and a lower value for the power of the test.

Question 46

Power and One-Tailed versus Two-Tailed Tests

Answer 46

• 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.

Question 47

Increasing Sample Size

Answer 47

• increases the likelihood of rejecting the null hypothesis
• has no effect on Cohen’s d
• increases the power of the test
• produces a smaller standard error in the denominator, which produces a larger z-score.

Question 48

To Include in a Report Statement

Answer 48

(i) independent variable
(ii) significant effect (H1) vs no significant effect (H0)
(iii) dependent variable
(iv) sample
(v) calculated mean
(vi) calculated z-score
(vii) proportion
(vii) effect size

(i) The blueberry supplements had a (ii) significant effect on the (iii) cognitive function of (iv) adults (v) (M = 92) (iv) over 65 years of age, (vi) z = +3.00, (vii) p < 0.05. (viii) The effect size was small, d=0.22.

Question 49

Most Likely to Have a Significant Effect if

Answer 49

sample number is high
alpha level is high
have a one-tailed test

Question 50

Steps of Hypothesis Testing: Step 2

Answer 50

2) Set the criteria for the Alternative Hypothesis
• Null Hypothesis (H0): the treatment has no effect (no change)
- sample mean should be near M = 80
• Alternative Hypothesis (H1): the treatment has an effect
- sample mean should be very different from 80 (M ≠ 80)
- determine this exact value using the alpha value or level of significance
> critical region: z critical i.e. zc= +- (over top of each other) 1.96
> We determine the alpha value: α = 0.05 or 5%; α = 0.01 or 1%; α = 0.001 or .01%
> extreme 5% of sample means:
- the 5% is split between the two tails of the distribution (2.5% each tail)
- look up in table z-scores for the critical region (2.5% of each tail)

Question 51

Steps of Hypothesis Testing: Step 3

Answer 51

Collect data and compute sample statistics
• calculate the sample mean
• calculate a z-score to determine where the sample mean is located in the distribution of sample means

Question 52

Steps of Hypothesis Testing: Step 4

Answer 52

4) Make a decision

• reject the null hypothesis or fail to reject the null hypothesis

Question 53

Steps of Hypothesis Testing: Step 5

Answer 53

Report the results of the statistical test

Question 54

What happens to the population after a treatment is given?

Answer 54

• treatment is adding/subtracting a constant to each individual scores
• the mean changes
• the shape of the distribution does not change
• the standard deviation does not change

Question 55

Factors that Influence Whether to Reject the Null Hypothesis

Answer 55

(1) the size of the z-score
• the larger the z-score, the further M is from µ
(2) the variability of the individual scores
• variability influences the size of the standard error
(3) the size of the sample
• the larger the sample (n), the smaller the standard error

Question 56

Assumptions for Hypothesis Testing

Answer 56

• Subjects are obtained using random sampling
- the sample must be representative of the population
• Independent Observations
- each measurement is not influenced by any other measurement
- typically satisfied using random sampling
• The treatment does not influence the variance
• The shape of the distribution is normal
- use z-scores only with a normal distribution of sample mean

Question 57

Directional Hypothesis Testing

Answer 57

• the alternative hypothesis (H1) specifies the direction of the treatment
• predicts either an increase or decrease
• one-tailed test
- the prediction is in one tail of the distribution of sample means
• allows for rejection of H0 when the difference between M and µ are small

Question 58

Measuring Effect

Answer 58

• statistical significance does not describe the size of the effect
• if a sample is large enough, any treatment effect will appear significant
• Cohen’s d

Question 59

If The Alternative Hypothesis Proves Correct, What Does the Proportion Look Like?
What about if the Null Hypothesis is Correct?

Answer 59

Alternative Hypothesis (H1):
p < \_\_\_\_\_ (alpha value)

Null Hypothesis (H0):
p > \_\_\_\_\_ (alpha value)