Statistics Exam 2

3 Things to Decrease to Increase Power

Beta Standard Deviation Standard Error

3 Things to Increase to Increase Power

Effect Size Sample Size Alpha

degrees of freedom

the number of scores in a sample that are free to vary (given a statistic). Always n-1

apa style conclusion (how to write one)

Results of a one-sample t-test suggest that (mean of our population of interest) (M= ) was significantly higher than/lower than/did not differ significantly from (comparison population (M= ), (t(df)=_________, p(>/<0.05)

2 Ways of Being Correct

Retaining the null hypothesis when it is true=1-alpha Rejecting the null hypothesis when it is false=1-beta

4 Steps of Hypothesis Testing

1. State the null and alternate hypotheses. 2. Set the criteria for a decision (decide on alpha or level of significance) 3. Collect data and compute the test statistic. 4. Make a decision and either retain or reject the null hypothesis.

t distribution

A t distribution is like a normal distribution, but with greater variability in the tails (a.k.a. fatter tails!)

Mean of Means

The mean of the sampling distribution of the means.

estimated standard error

an estimate of the standard deviation of a sampling distribution of sample means selected from a population whose variance is unknown.

one-sample t test

A One-Sample t Test is used to test whether a single sample mean is likely to have come from a population of interest with unknown variance (Ho); or, alternatively, is unlikely to have come from that population of interest (H1). Three assumptions are made: 1.Normality - assume data in the population being sampled is normally distributed 2.Random Sampling - assume that the data were obtained using a random sampling procedure 3.Independence - assume that probabilities of each measured outcome in a study are independent

Alternative Hypothesis

A statement that directly contradicts a null hypothesis by stating that the actual value of a population parameter, such as the mean, is less than, greater than, or not equal to the value stated in the null hypothesis. The hypothesis that states there is a difference.

Alpha

Level of significance or criterion of a hypothesis test (the cut off point beyond which we will reject the null). The significance level is the largest probability of committing a Type I error that we will allow and still decide to reject the null hypothesis. Usually set at 0.05.

Z statistic/Z obtained

The region in the distribution wherein the unknown data lies.

3 Ways of Being Wrong

Type I Error- rejecting a null hypothesis when it is actually true. False positive. Type II Error-retaining a null hypothesis when it is actually false. False negative. Type III Error- an error that occurs only in one-tailed tests. Happens when the researcher retains the null hypothesis because the rejection region was in the wrong tail.

t statistic/ obtained t

Used to determine the number of standard deviations in a t distribution that a sample mean deviates from the mean value or difference stated in the null. A t statistic (tobt) is the sample mean minus the population mean over the estimated standard error (which is just the sample standard deviation divided by the square root of the sample size)

One-tailed Test

When the researcher is testing in a specific direction, they have very strong evidence to believe that one group is much higher or much lower than the other. The critical region is only on one end of the distribution. This test is very rare! Only ever used when the researcher is certain the relationship or difference being tested can only run in one direction! (e.g., M can only be >μ, never <μ) Greater power! If value stated in null hypothesis is false, this test will make it easier to detect and reject. Difficult to justify - hard to publish. Vulnerable to Type III errors.

Sampling Distribution of the Means

a distribution of all sample means that could be obtained in samples of a given size from the same population. This distribution has it's own mean (likely equal to the population mean), and it's own narrower standard deviation.

Eta-squared

a measure of effect size in terms of the proportion or percent of variability in one variable (usually the dependent or outcome variable) that can be explained or accounted for by another variable (usually the independent or predictor variable).

Hypothesis Testing

a method for testing a hypothesis (an idea or prediction) about a parameter, or the true relationship between groups or variables, using data measured in a sample.

Null Hypothesis

a statement about a population parameter, such as the population mean, that is assumed to be true. Often in hypothesis testing, it is the statement that there is no difference between groups (because a difference hasn't been proven yet).

Estimated Cohen's D

a type of effect size that measures the number of standard deviations that a sample mean has shifted above or below the population mean stated by the null hypothesis. A positive Cohen's d indicates the mean of the sample is above the population mean stated by the null hypothesis. A negative Cohen's d indicates the mean of the sample is below the population mean stated by the null hypothesis.

Confidence intervals

provide a range used to estimate the true value of an unknown parameter. If the confidence interval is 95%, we can be 95% sure that the true population mean falls within our range. If the confidence interval is 99%, we can be 99% sure that the true population mean falls within our range.

Power

the probability of rejecting a false null hypothesis. The probability of a researcher being right in rejecting a null hypothesis. More specifically, it is the probability of a randomly selected sample will correctly be representative of the null hypothesis being false when in fact, it is false.

Statistical Significance

whether or a relationship between variables or difference between means is statistically meaningful. Only if a test statistic exceeds the level of significance set by alpha do we say the results are "statistically significant" (e.g., if p<.05 when alpha is set at .05)

Two-tailed test

Critical region lies on both ends of the distribution. More conservative & More common! Most studies in behavioral research are two-tailed tests ◦More difficult to reject null hypothesis Eliminates possibility of Type III error

t critical value

Found using the t-table, determines where the critical region is where you can reject the null hypothesis.

Z critical value

The value at which you can reject the null hypothesis, decided by alpha level.

Type III Error

Type of error for one-tailed tests: Occurs when we retain null hypothesis because rejection region was located in wrong tail.

Sample Distribution

a distribution of scores from a single sample.

Cohen's D

a type of effect size that measures the number of standard deviations that a sample mean has shifted above or below the population mean stated by the null hypothesis. Value for Cohen's d is 0 when there is no difference between the 2 means, and increases as differences get larger. A positive Cohen's d indicates the mean of the sample is above the population mean stated by the null hypothesis. A negative Cohen's d indicates the mean of the sample is below the population mean stated by the null hypothesis.

Unbiased Estimator

is any sample statistic obtained form a randomly selected sample that equals the value of its respective population parameter on average. The sample mean is an unbiased estimator because it equals the population mean on average.

Central Limit Theorem

states that regardless of the shape of the distribution of scores in a population, the sampling distribution of sample means selected at random from that population will approach the shape of a normal distribution as the number of samples in the sampling distribution increases.

One-sample Z test

statistical procedure used to test hypotheses concerning the mean in a single population with a known variance.

Effect Size

the magnitude of the relationship between variables or difference between means. Not affected by sample size. Used to only be reported when the relationship or difference is statistically significant (e.g., p<.05) .Starting to be reported more and more, even if the results are not statistically significant (e.g., p>.05). Measure by Cohen's D or eta squared.

Standard Error

the new standard deviation of the sampling distribution of the means.

P value

the probability of obtaining a sample outcome, given that the value stated in the null hypothesis is true. Compared to the alpha level for making a decision. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.