PSYCH248: Statistical Methods I, Unit 2 Study Guide

step one: state your hypothesis

*the hypotheses are in terms of μ and not M because we are attempting to draw conclusions about the population, not the sample the null hypothesis states that the treatment has no effect, and that the unknown population mean is equal to the known population mean the alternative hypothesis states that the treatment did have an effect, and that the unknown population mean is NOT equal to the known population mean

factors that influence hypothesis testing

1. treatment effect size. this is represented by the difference between the treatment and control group and can be found in the numerator of the test statistic. the larger the difference is, the more likely it is that we will obtain a significant result 2. variability. this is represented by the variance or standard deviation (SEM). greater variability decreases the probability of obtaining a significant result 3. the number of scores, aka sample size (n). the test statistic will become larger with a greater number of samples. the greater the sample, the greater the probability of obtaining a significant result.

sampling distribution

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

what does a hypothesis test evaluate?

a hypothesis test evaluates the statistical significance of the results from a research study

what is the difference between a one-tailed test and two-tailed test? how are the hypotheses written for a one-tailed test?

a one-tailed hypothesis specifies that one mean is larger than the other mean and we need to use a one-tailed test. a two-tailed hypothesis specifies that there is no difference between the means (non-directional)

what is a one-sample t-test? why is a t-test called an estimated z-score? when is t a good estimate of the z score?

a single- or one-sample t-test compares a sample mean (representing the unknown population's mean) and the population mean for the untreated (known) group. a t-test is often called an estimated z-score because the formula is very similar, and we can draw the same conclusions from the results a t-score is a good estimate of the z-score when we have a large sample size (n)

under either of what two conditions will the distribution of sample means be normal?

a.) the population is normally distributed, or: b.) if the population is skewed, n > 30

how does the z-score change as distance from the sample mean to the population mean increases/decreases?

as SEM increases, the z-score will decrease; as SEM decreases, the z-score will increase

how does the z-score change as n increases/decreases?

as n increases, SEM decreases, and so z-scores increase. as n decreases, SEM increases, and so z-scores decrease.

how does the z-score change as variability (standard deviation of the population) increases/decreases?

as variability increases, SEM will increase, and so the z-score will decrease. as variability decreases, SEM will decrease, and so the z-score will increase.

what is effect size?

effect size is a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample being used

define hypothesis testing and know what the goal is

hypothesis testing is a technique to help determine whether a specific treatment has an effect on the individuals in a population. the goal is to differentiate between real and random patterns.

what is the importance of the concept "significant ≠ large or important"?

hypothesis tests are influenced by two main factors: the size of the treatment effect (mean difference) and the size of the sample (standard error). for this reason, a very small effect can be significant in a very large sample, because standard error decreases as the sample size increases.

describe the critical region for a one-tailed test.

in a one-tailed test, the critical religion is in one tail of the distribution; upper or lower tail on the basis of how the hypothesis is stated

in the formula for the one-sample t-test, what is in the numerator? in the denominator?

numerator: sample mean - (hypothesized) population mean (M - u) denominator: estimated standard error of mean (eSEM)

step three: compute the test statistic

so i understand the formula, but how do i know what to plug in for μ? ?

step four: make a decision

so what are my options? 1. if you find evidence that the treatment works, you reject the null hypothesis 2. if your evidence is not convincing that the treatment works, you retain the null hypothesis why all this emphasis on the null hypothesis? we focus on the null hypothesis because of the limitations of inferential logic and because we make inferences about populations from samples. in other words, it's easier to demonstrate that a universal hypothesis is false than to prove that it is is true.

how is effect size using cohen's d calculated? what are the cutoffs for small, medium and large effects?

so what is cohen's d? cohen's d is a standardized measure of effect size. it measures the size of the mean difference in terms of the standard deviation. how do i calculate it? you can calculate cohen's d by subtracting the population mean from the sample mean, and dividing that by the standard deviation. and what are the cutoffs? d = 0.2 (small effect) d = 0.5 (moderate effect) d = 0.8 (large effect)

type ii error

so, what about type ii error? type ii errors happen when the sample does not appear to have been affected by the treatment when it, in fact, has. the researcher will fail to reject the null hypothesis. oh. so why does it happen? most of the time, type ii errors are the result of a very small treatment effect.

sampling error

the discrepancy between the statistics that a sample produces and the population parameter. different samples will have different amounts of sampling error.

distribution of sample means

the distribution of all possible sample means for all possible samples from a population (N); this is a type of sampling distribution

how is the distribution of sample means related to the null hypothesis?

the known original population mean supports the null hypothesis. if the difference between the treated sample and the known population mean is not significant, we retain the null hypothesis. if the difference between the sample mean and the population mean is significant, we know there was a treatment effect and we reject the null hypothesis.

describe zcritical for a one-tailed test (compared to two-tailed test). which z is lower or easier to pass?

the one-tailed cutoff is z = 1.65 -- lower than the z = 1.96 which is needed for a two-tailed test. therefore, it is easier to pass a one-tailed test!

what is the relationship between the treated sample and the unknown treated population?

the sample mean represents the unknown treated population mean

central limit theorem

the theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution

type i error

wait -- what's a type i error? a type i error occurs when the sample data appear to show a treatment effect when there isn't one, like giving a man a positive pregnancy test. in this case, the researcher will reject the null hypothesis and falsely conclude that the treatment has an effect. why do they happen? well, type i errors are usually caused by an unusual or unrepresentative sample. does this happen a lot? nope! hypothesis testing is structured so that type one errors are pretty unlikely. the probability of a type i error is actually equal to the alpha level!

when must we use a t-test (i.e., what information is missing)? what is a t-test?

we use a t-test when we're missing the exact population mean and the standard deviation (variance) (and thus can't compute a z-score). t-tests are a group of statistical tests that allow you to determine if two (population) means are significantly different.

as z increases, is a score or sample mean more or less likely? what do higher probabilities indicate?

well, as the test statistic increases (z-score, t-test, ANOVA, whatever) the probability of getting that score decreases. and the lower the probability, the harder it is to obtain that score and the more significant the difference between the two groups.

what is the mean of the distribution of sample means called? what is it equivalent to? what does it mean to say it is unbiased?

1. the mean of the distribution of sample means is called the expected value of M 2. it is equal to the mean of the population scores, μ 3. to say it is unbiased is to say that it is equal to the population parameter

how does the SEM change as σ increases/decreases? how does the SEM change as N increases/decreases? what is the law of large numbers?

1. as the standard deviation increases, SEM increases; as standard deviation decreases, SEM decreases 2. as n increases, SEM decreases. as n decreases, SEM increases. 3. the law of large numbers states that as n increases, the sampling error between the sample means and the population mean should decrease. in other words, the mean from a large sample should be more accurate than the mean from a small sample.

rules of hypothesis testing

1. data is always collected after the hypothesis is stated and the alpha level is chosen. this keeps the researcher honest in the scientific process making an objective decision about the outcome. 2. don't switch between a one-tailed test and a two-tailed test just to obtain statistical significance! keep it consistent!

the standard error of the mean (SEM) is the typical or average deviation between what two things? why does the SEM symbol have M for a subscript? what aspect of the sampling distribution of the means does it measure (hint: the variability or standard deviation)?

1. the SEM is the average distance between a sample mean and the population mean 2. i honestly dk lol 3. the standard error of the mean describes the variability of all sample means (i.e., larger SEM, greater variability)

according to the central limit theorem, what is the center, variability, and shape of the distribution of sample means? how is sample size related to the spread of the distribution of sample means?

central tendency: as n increases, the mean will get closer and closer to the population mean, μ shape: as n increases, the distribution will become more normal variability: as n increases, the means become closer to μ, and so variability decreases

step two: set your criteria for a decision

what is an alpha level? the alpha level (level of significance) defines the boundary(ies) that separate high probabilities (likely) from low probabilities (unlikely). the two most common alpha levels are .05 and .01, which means the most 5% or 1% of the outcomes are the most unlikely to occur cool. so what is the critical region? the critical region is composed of outcomes that are very unlikely to occur if the null hypothesis is true. when we say a mean is "almost impossible", we are talking about samples that have a probability (p) that are less than the alpha level. oh. so what happens if the sample data falls in the critical region? we reject the null hypothesis! but what happens if it doesn't fall in the critical region? if our data falls outside the critical region(s), we retain our null hypothesis. got it. so how do i get those critical values? you can find your critical t or z value by consulting either the unit table or the t-table. you will need the degrees of freedom and the alpha level to find it.

given μ, σ, and n, find the z-score for a given mean, find the probability of a given mean or range of means, etc.

z-score: z = the sample mean minus the population mean, divided by the standard error of the mean probability: *same as usual except use the unit normal table instead of the standard unit table