Ch. 9

Why has the one-tailed hypothesis been criticized?

1. Ignores the possibility of an unexpected difference in the opposite direction. 2. More vulnerable in error if the distribution is not normal. 3. Might be used when a two-tailed might have been more appropriate.

2 Types of ALTERNATIVE hypothesis:

1. One-tailed (directional) 2. Two-tailed (nondirectional)

The hypothesis-testing process includes 6 steps:

1. State the hypothesis 2. Set a level of risk 3. Choose a sample size 4. Determine the critical value 5. Compute the statistic 6. Reject or accept the hypothesis

4 Possible outcomes when testing a research hypothesis:

1. The Null hypothesis is accepted when it's true (CORRECT decision) 2. The Null hypothesis is rejected when it's false (CORRECT decision) 3. The Null hypothesis is rejected when in reality it's true (Type I ERROR) 4. The Null hypothesis is accepted when in reality it's false (Type II ERROR)

The choice of a test statistic depends on several factors:

1. The hypothesis 2. The distribution of the target population 3. Sample size 4. Other sample characteristics

Normal distribution characteristics:

1. Unimodal (one mode at the center) 2. Symmetrical 3. Continuous (from one tail to the other) 4. Asymtpotic (curved line gets closer to the horizontal axis as it moves away from the center but does not meet the axis, most scores are in the middle)

Point estimates

A single number derived from a sample and used to estimate a population value. Present a problem because sample stats. reflect the population parameter but also contain sampling error. Thus, the calculations are not exactly correct.

Standard Unit

A single score in the distribution of scores can be represented by a standard unit of measure known as a z score. We can easily compare variables that have different scores or units of measurement,

Nonparametric test

A special class of inferential statistics that do not depend on the population conforming to a prescribed shape, because the raw data are replaced by their ranks. Common test: *Mann-Whitney U, aka: Wilcoxon rank rum test (Outside source:)Inferential statistical tests used when the data to be analyzed represent either a nominal or ordinal scale or when the assumptions for the other category of test have not been met.

(Step 5:) Test statistic

A standard score that indicate the numbers of standard errors the sample statistics is from the hypothesized parameter. (Other source:) A value, determined from sample information, used to determine whether to reject the null hypothesis.

What's a Two-tailed (nondirectional) hypothesis?

A type of alternative hypothesis. Does not predict the direction of results. For example, Treatment A is DIFFERENT from Treatment B

What's a One-tailed (directional) hypothesis?

A type of alternative hypothesis. Predicts direction of results; for example, hypothesizing that Treatment A will be BETTER than Treatment B

Step 6: Make a decision about the Null Hypothesis

After the test stat. is calculated, it's compared to the critical value. If the test statistic's absolute value EXCEEDS the critical value, the null is REJECTED. (If the null is rejected, the alternative hypothesis is SUPPORTED but not necessarily proven) If the test statistic's absolute value DOES NOT EXCEED the critical value, the null is ACCEPTED.

Standard normal distribution

All normal distributions can be converted to a common distribution with the same mean and standard deviation. This common distribution has a mean of zero and a standard deviation of one. It's important information for determining the probably of an outcome. Useful tool for describing a normally distributed variable.

The rejection region can be located in either tail ( + or -) specified by the ____ hypothesis.

Alternative

How are categorical variables displayed?

Bar graphs

When should the directional choice be decided: before, during or after a study begins?

Before. A hypothesis should be two-tailed unless there's compelling evidence that the expected result will be directional.

Interval estimation

Builds on point estimates to establish a range of values that we can say with confidence contain the population parameter; range of values is the confidence interval

The hypothesis-testing process step 3:

Choose a sample size. Sample size influences the power of a statistical test.

Constructing a confidence interval:

Critical value is chosen. Because distribution of sample means is assumed to be normally distributed, 95% of sample means are expected to fall within 2 standard errors of population mean.

Two major classes of statistics:

Descriptive and Inferential

Central Limit Theorem

Even if a variable is not normally distributed in the population, the sampling distribution will be approximately normal with a large sample (n>30). Thus, it justifies the use of normal -curve methods for solving many stat. problems.

Null hypothesis

Hypothesis that predicts NO relationship between variables and NO difference between two treatment methods. The aim of research is to reject this hypothesis (Ho)

Alternative Hypothesis

Hypothesis that we are trying to prove (opposite of the null hypothesis), A statement reflecting the possibility that there are differences between the objects and/or events that are being observed. (HA)

Estimating population parameters

If the characteristics of a target population are unknown, the population parameters can be estimated from a sample characteristics.

Effect size

Indicator of practical significance. In practice, clinicians are concerned with clinical relevance, and a statistically significant results may not be clinically relevant.

Relationship between alpha and beta

Inverse relationship; example: increase of beta will lead to a decrease of alpha

Z score

Is a measure of individual location. Tells us where individual scores are located within a distribution of scores. Tells us how many SD the corresponding values of x lies above or below the mean. Distribution is symmetrical. Often transformed because it originally includes negative values and decimals, making it difficult for interpretation. (Multiply each z score by 15 and add 100)

Nonparametric statistic

Is distribution free and may be more appropriate for populations that are not normally distributed. (Outside source:) Distribution-free tests: evaluate the shape of a distribution rather than the distribution's mean, variance, or other parameter. Include the chi-square tests, the *Mann-Whitney U, and the Wilcoxon matched-pairs test. Limitation: because they use less precise (nominal or ordinal) data, they are less powerful. Less likely to reject a false null hypothesis.

Is it practical to construct a distribution of sample means?

It's usually NOT practical to construct a distribution of sample means because the target population is often large in number, the number of possible means is unwiedly.

Student's t Distributions

Like normal distributions, these distributions are symmetrical , bell-shaped, and centered on the mean; however, distributions change as sample sizes change Research in CSD often involve small samples. Thus, researchers often depend on Student's t distributions to test hypotheses.

How are continuous variables displayed?

Line graphs or histograms

Inferential statistics

Make conclusions about populations from sample data.

Kurtosis:

Measure of peakedness for symmetrical distributions Pg. 200: Mesokurtic: moderate deg. of peakedness (normal curve) Leptokurtic: large proportion of scores located at the center Platykurtic: large proportion of scores in tails

Skewed Distributions

Not bell-shaped and symmetrical. Skewness is an indicator of symmetry or the lack of. Positive skewed distribution: scores with larger values toward the left tail Negative skewed distribution: has more scores with larger values toward the right tail *Tail point in the direction of the skew

Pearson coefficient of skewness (SK)

One of several tests for skewness; a simple statistic for comparing a distribution's mean and median to measure skewness.

If other factors are constant, a ____ - tailed test is more powerful.

One-tailed, because the rejection region is larger. (A two-tailed divides the rejection region in half)

Distribution:

Pattern of scores. All variables have distributions The distribution of a sample statistic indicates how often different values of that statistic should occur if samples of the same size are collected repeatedly from the same populations

Probability distributions

Provide info. about the chance occurrence of a particular outcome

What sample size does a Z statistic require?

Relatively large; n (sample size) > 30 A larger sample usually results in a more powerful test of the null hypothesis.

What sample size does a T statistic require?

Small; n (sample size) < 30

Standard error of the mean or SEM

Standard deviation of sample means. To determine this, we need to have the parameters of the population which we typically do not have available

The hypothesis-testing process step 1:

State the hypothesis. Contains: null & alternative hypothesis.

Descriptive statistics

Statistics that are used to organize and summarize data without inferring anything about the population.

Type II error

The Null hypothesis is accepted when in reality it's false. It's represented by beta. n<10 are at high risk.

Type I error

The Null hypothesis is rejected when in reality it's true. It's represented by alpha.

What is a critical value?

The cutoff point (required in statistical hypothesis testing) that separates sample results that should lead to the rejection or acceptance of the null hypothesis.

Distribution

The distribution of a variable provides information about individual cases as well as information about group of scores.

Definition of power?

The probability of rejecting the null hypothesis when it's false (CORRECT decision).

Goal of hypothesis testing is:

To accept or reject the hypothesis

Which type of error is considered more serious/more damaging? (Type I or II )

Type I (The Null hypothesis is rejected when in reality it's true)

A ______ sample usually results in a more powerful test of the null hypothesis.

larger

Transformed Standard Scores

z scores are transformed into a distribution of standard scores with a mean of 100 and SD equal to 15 units