Psyc 208 Exam 2 (Ch. 5-8)

Sampling Distribution

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

Central Limit Theory

•A mathematical proposition known as the central limit theory provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and serves as a cornerstone for much of inferential statistics.

Effect Size

•A measure of the 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.

Significant

•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.

Independent Random Sample

•A second requirement, necessary for many statistical applications, states that if more than one individual is being selected, the probabilities must stay constant from one selection to the next. •Adding this second requirement produces what technically is called independent random sampling. •The term independent refers to the fact that the probability of selecting any particular individual is independent of those individuals who have already been selected for the sample.

Hypothesis Test

•A statistical method that uses sample data to evaluate a hypothesis about a population.

Standardized Distribution

•Composed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable.

Probability

•For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. If the possible outcomes are identified as A, B, C, D, and so on, then (FORMULA).

Random Sample

•For the preceding definition of probability to be accurate, it is necessary that the outcomes be obtained by a process called random sampling. •A simple random sample requires that each individual in the population has an equal chance of being selected.

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. •For a hypothesis test is the probability that the test will lead to a Type I error if the null hypothesis is true. That is, the alpha level determines the probability of obtaining sample data in the critical region even though there is no treatment effect.

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.

Distribution of Sample Means

•Is the collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population.

Standardized Score

•It is common to standardize a distribution by transforming the scores into a new distribution with a predetermined mean and standard deviation that are whole round numbers. •The goal is to create a new (standardized) distribution that has "simple" values for the mean and standard deviation but does not change any individual's location within the distribution. •The procedure for standardizing a distribution to create new values for μ and σ involves two steps: 1)The original raw scores are transformed into z-scores. 2)The z-scores are then transformed into new X values so that the specific μ and σ are attained. •The Standardized Procedure: 1)Transform each of the original scores into z-scores. Remember: The values of μ and σ are for the distribution from which X was taken. 2)Change each z-score into an X value in the new standardized distribution that has a mean of μ = 50 and a standard deviation of σ = 10.

Directional Test -- Or a One-Tailed Test

•It is possible to state the statistical hypotheses in a manner that incorporates the directional prediction into the statement of H0 and H1. The result is a directional test, or what commonly is called one-tailed test.

z-Score Transformation

•It is possible to transform every X value in a distribution into a corresponding z-score. The result of this process is that the entire distribution of X values is transformed into a distribution of z-scores. •The new distribution of z-scores has characteristics that make the z-score transformation a very useful tool. Specifically, if every X value is transformed into a z-score, then the distribution of z-scores will have the following properties: 1)Shape: The distribution of z-scores will have exactly the same shape as the original distribution of scores. 2)The Mean: The z-score distribution will always have a mean of zero. 3)The Standard Deviation: The distribution of z-scores will always have a standard deviation of 1. •There is no need to create a whole new distribution. Instead, you can think of the z-score transformation as simply relabeling the values along the X-axis. That is, after a z-score transformation, you still have the same distribution, but now each individual is labeled with a z-score instead of an X value. •When any distribution, (with any mean or standard deviation) is transformed into z-scores, the resulting distribution will always have a mean of μ = 0 and a standard deviation of σ = 1. •Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a standardized distribution.

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.

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 that the researcher concludes that a treatment does have an effect when, in fact, it has no effect.

Cohen's d

•One of the samples and most direct methods for measuring effect size is Cohen's d. Cohen 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 (FORMULA). •Remember, the sample mean is expected to be representative of the population mean and provides the best measure of the treatment effect. Thus, the actual calculations are really estimating the value of Cohen's d as follows (FORMULA).

Percentile Rank

•Specifically, the percentile rank for a specific score is defined as the percentage of the individuals in the distribution who have scores that are less than or equal to the specific score.

z-Score

•Specifies the precise location of each X value within a distribution. •The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative). •The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ. •Note: A z-score always consists of two parts— oA sign (+ or -) and a magnitude oBoth parts are necessary to describe completely where a raw score is located within a distribution.

Null Hypothesis

•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.

Law of Large Numbers

•States that the larger the sample size (n), the more probable it is that the sample mean is close to the population mean.

Alternative Hypothesis

•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.

Expected Value of M

•The mean of the distribution of sample means is equal to the mean of the population of scores, μ, and is called the expected value of M.

Sampling Error

•The natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.

Deviation Score

•The numerator of the equation, X and μ, is a deviation score, which measures the distance in points between X and μ and indicates whether X is located above or below the mean. •The deviation score is then divided by a σ because we want the z-score to measure distance in terms of standard deviation units. •The formula performs exactly the same arithmetic that is used with the z-score definition, and it provides a structured equation to organize the calculations when the numbers are more difficult.

Power

•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.

Beta

•The probability of a Type II error.

Standard Error of M

•The standard deviation of the distribution of sample means, σM, is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ).

Test Statistic

•The z-score statistic that is used in the hypothesis test is the first specific example of what is called test statistic. •The term test statistic simply indicates that the sample data are converted into a single, specific statistic that is used to test the hypotheses.

Unit Normal Table

•This table lists proportions of the normal distribution for a full range of possible z-score values. oThe first column (A): oYou should also realize that a vertical line separates the distribution into two selections: ♦A larger section called the body ♦A smaller section called the tail oColumns B and C in the table identify the proportion of the distribution in each of the two sections. oColumn (B): Represents the proportion in the body (the larger portion) oColumn (C): Represents the proportion in the tail oColumn (D): Identifies the proportion of the distribution that is located between the mean and the z-score.

Raw Score

•To demonstrate a score by itself does not necessarily provide much information about its position within a distribution. These original, unchanged scores that are the direct result of measurement are called raw scores. •To make raw scores more meaningful, they are often transformed into new values that contain more information. This transformation is one purpose for z-scores. In particular, we transform X values into z-scores so that the resulting z-scores tell exactly where the original scores are located.

Level of Significance

•To find the boundaries that separate the high-probability samples from the low-probability samples, we must define exactly what is meant by "low" probability and "high" probability. This is accomplished by selecting a specific probability value, which is known as level of significance.

Sampling With Replacement

•To keep the probabilities from changing from one selection to the next, it is necessary to return each individual to the population before you make the next selection. This process is called sampling with replacement. •The second requirement for random samples (constant probability) demands that you sample with replacement.

Percentile

•When a score is referred to by its percentile rank, the score is called a percentile.