Psych 2220 Exam 2

Cohen's d equation

(M-μ)/σ

Power of a z statistic equation

(Mcrit - M) / Om

95% confidence interval

A range of values centered on a sample mean, and constructed so that 95% of independently sampled means would generate corresponding ranges containing the true mean of the population Interpretation of a confidence interval involves repeated sampling from the same population, with the same sample size (95% of such intervals will contain the population mean)

Convenience sample

A sample that relies on participants who are readily availability or easy to contact

Point estimates

A single-number summary statistic from a sample that is used as an estimate of the population parameter

Hypothesis testing

A statistical method that uses sample data to evaluate a hypothesis about a population Goal is to rule out chance (sampling error) as a plausible explanation for the results from a study

Meta-analysis

A study or method that involves the calculation of a mean effect size from the individual effect size of many studies

Confidence interval

An interval estimate (based on a sample statistic) that includes the population mean a certain percentage of the time if we sample from the same population (with the same ample size) repeadetly Give us a plausible range of values for a population parameter Can provide the same information as a hypothesis test, but also provide additional information (provide a plausible range of values for a population parameter)

Unions (disjunctions)

At least one of a number of possible events occurs Unions for outcomes A and B could be represented by "A or B" or "A U B" Or is inclusive, not exclusive

Type II Error

False negative Incorrectly failing to reject the null hypothesis when it was false

Type I Error

False positive Incorrectly rejecting a true null hypothesis

Test statistic

Also known as a z-statistic, forms a ratio comparing the obtained difference between the sample mean and the hypothesized population mean versus the amount of difference we would expect without any treatment effect If the test statistic results are in the critical region, we conclude that the difference is statistically significant (reject the null hypothesis) If the mean difference is not in the critical region, we conclude that the evidence from the sample is not sufficient, and the decision is to fail to reject the null hypothesis

One-tailed tests

A hypothesis test in which rejection of the null hypothesis occurs for values of the test statistic in one tail of its sampling distribution.

Binomial distribution

Formed by a series of observations for which there are exactly two possible outcomes Ex: 100 coin tosses each yielding either a head or tails The two outcomes are identified as A and B, with probabilities of p(A)=p and p(B)=q P+q=100 When pn and qn are both greater than 10, the binomial distribution us closely approximated by a normal distribution with a mean of u = pn

Z-scores

Represent the relative location of a score within its source population, scaled as the number of standard deviations of distance from the mean Converting raw scores to z-scores: (X - μ) / σ

Alternative hypothesis (H1)

States that there is a change in the general population following an intervention Predicts that the independent variable did have an effect on the dependent variable

Null hypothesis (H0)

States that there is no change in the general population before and after an intervention Predicts that the independent variable had no effect on the dependent variable

Standard error of M

The standard deviation of the distribution of sample means is often called standard error of M Provides a measure of how much distance is expected on average between a sample mean (M) and the population mean, which is equal to the overall mean of the sampling distribution of means Measures how well an individual sample mean represents the entire distribution, specifically how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of means Simply, the standard error is the standard deviation for a sampling distribution of means, but with a different name As the sample size for each mean increases, the standard error decreases (holding other factors constant) *sort of like the law of large numbers, which states that the larger the sample size, the more probable it is that the sample mean will be close to the population mean As the standard deviation of the population decreases, the standard error decreases σM= 𝜎 / √n

Alpha level

Establishes a criterion or "cut off" for making a decision about the null hypothesis Also determines the risk of a Type I Error

Random Samples

Every member of the population has an equal chance of being included in the sample Independent random sample requires that everyone has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next

Lower limit equation

-z(standard error) + mean

Steps of a meta-analysis

1. Choose a topic of interest, and specify criteria for including relevant studies in analysis 2. Locate all existing studies (published and unpublished) that meet your criteria 3. Compute a relevant effect size for each study 4. Calculate statistics about the set of effect sizes

Steps for calculating power (for a one-tailed z test)

1. Determine the information needed to calculate statistical power - the hypothesized (or observed) mean for the sample, sample size, population mean, population standard deviation, and the standard error 2. Determine a critical value in terms of the z-distribution and the raw mean 3. Calculate statistical power - the percentage of distribution of means for your hypothesized mean that fall beyond the critical value

Steps for creating confidence intervals for z distributions

1. Draw a picture that will include the confidence interval, centered on your sample mean 2. Indicate the bounds of your confidence interval on your drawing 3. Determine the z-statistic that fall at each boundary (we will always use critical values that correspond to a two tailed test when creating CL's) 4. Turn the z-statistic back into raw means

Hypothesis test - steps

1. State hypothesis about the population 2. Use hypothesis to predict the characteristics the sample should have 3. Obtain a sample from the population 4. Compare the data with the hypothesis prediction

Critical region

Consists of outcomes that are very unlikely to occur if the null hypothesis is true The values that define the critical region are commonly called critical values

Two-tailed tests

Hypothesis testing in which both ends of the sampling distribution are used to define the region of improbable values.

Effect size

Intended to provide a measurement of the size of a difference (or the strength of an effect, or the strength of a relationship), independent of the size of the sample(s) being used

σ (sigma)

standard deviation

z statistic equation

Mean - sample mean / standard error

Sampling error

Natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter

Mutually exclusive events

Occurrence of one outcome prevents the occurrence of another outcome Ex: A coin flip can land on heads or tails for a single flip, but not both

Assumptions for hypothesis tests with z-statistics

Participants are randomly sampled from the population (if the participants arent randomly sampled from the population, the study will have diminished external validity) The values in the sample consist of independent observations (if the occurence of the first event has no effect on the probability of the second event) The standard deviation remains constant (does not change with treatment/independent variable levels) The sampling distribution is normally distributed We want a scale dependent variable and a nominal independent variable (necessary for hypothesis testing with z-statisitcs)

The central limit theorem

Refers to how a distribution of sample means approaches normality as the sample size of each mean increases (when the population distribution of scores is not normally distributed. The distribution of means also retains the same mean as the population of scores that generated it, though it has a smaller standard error (i.e., less variability) as the sample size of each mean increases

Assumptions

Represent characteristics that we want the population that we are sampling from to have Meeting the assumptions helps us to make accurate inferences

Definition of sample means

The collection of sample means for all possible random samples of a particular size (n) that can be obtained from a population More generally, a sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population

Independent events

The occurrence (or non occurrence) of one outcome has no bearing on the probability of the occurrence (or non occurrence) of another outcome Ex: If two coins are flipped, neither outcome impacts the other. Or if one coin is flipped twice, the first flip does not affect the second

Statistical power

The probability of rejecting the null hypothesis when it is false A measure of the likelihood that we will reject the null hypothesis, given that the null hypothesis is false To calculate power, it is necessary to make assumptions about a variety of factors that influence the outcome of a hypothesis test (sample size, size of the treatment effect, value chosen for the alpha level) To increase power, increase alpha (also increases the risk of Type I errors) To increase power, use a one-tailed test (instead of a two-tailed test) *only increases power if you correctly predict the direction of the difference To increase power, increase the main difference between populations (with a more extreme manipulation of the independent variable) To increase power, increase the sample size To increase power, decrease variability (standard deviation)

Conditional probabilities

The probability that one event (A) will occur, given that another event (B) has occurred Used for hypothesis testing Denoted P(A | B) "probability of A, given B"

Characteristics of the distribution of sample means

The sample means should pile up around the population mean, so that the average of the sample mean will equal the population mean The pile of sample means should tend to form a normal-shaped distribution, with greater normality a the sample size increases. (If the population of scores being sampled from was already normal, the distribution of means should be normal even with small sample sizes) In general, the larger the sample size, the closer the sample means should be to the population mean The mean value of all the sample means is exactly equal to the value of the population mean

σM (sigma sub M)

standard error

Intersections (conjunctions)

Two or more events occur at the same time An intersection for outcomes A and B could be represented by "A and B"

Self-selection

When participants actively choose themselves to participate in a study (cannot be confident that these results will generalize to the entire population)

Computing a z statistic for a distribution of means

Within the distribution of sample means, the location of each sample mean can be specified by a z-score Z = (M - uM) / oM oM = o / square root of n Example: The population of scores on the SAT forms a normal distribution with u=500 and o=100. If you take a random sample of n=16 students, what is the probability that the sample mean will be greater than M=525? For n=16, the distribution has a standard error of oM=25 ---> σM = 𝜎 / √𝑛 = 100 / √16 = 25 The value 525 is located above the mean by 25 points, which is exactly 1 standard error, in this example. The unit table indicates that 0.1587 of the distribution is in the tail of the distribution beyond z=+ or - 1.00 As a result, it is somewhat unlikely, p=0.1587(15.87%), to obtain a random sample of n=16 students with an average SAT score greater than 525.

Cohen's d

a measure of effect size that assesses the difference between two means in terms of standard deviation, not standard error Describes how far a sample mean is from another mean (in standard deviation units) D = (m - u) / o D=0.2 - small effect (mean difference around 0.2 standard deviation) D=0.5 - meduim effect (mean difference around 0.5 standard deviation) D=0.8 - large effect (mean difference around 0.8 standard deviation)

Unit normal table

lists several different proportions corresponding to each z-score location

μ (mu)

population mean symbol

X

sample mean

n

sample size

Critical value equation

z(standard error + sample mean)

Upper limit equation

z(standard error) + mean