Psych 252 exam 2

Central Limit Theorem

- A mathematical proposition that provides a description about the sample mean distribution of all possible random samples

Definition of hypothesis testing

- Researchers usually collect data from a sample and then use the sample data to make inferences about the population → inferential statistics - A hypothesis test: a statistical method that uses sample data to evaluate a hypothesis about a population

4 steps of hypothesis testing

- State hypothesis about a population - Set the decision criteria by locating the crticial lregion - Collect data and compute statstic - Compare the obtained sample data (a statistic from Step 3) with the prediciton that was made from (Step 2) to make a decision about a population

Shape

- The distribution of sample means will be normally distributed if either of the below two conditions meets - the population in normally distributed - a sample size of n> 30 even if the population distribution isn't normally distributed

Mean

- The mean of the distribution of the sample mean is: - same as the population mean - called expected value of sample mean - an unbiased estimate of the population mean - an average, the sample mean produces a value that exactly matches the population mean

Probability of type II error

- impossible to determine a single exact probability - can be calculated based on statistical power

Sampling Distribution

- inferential statistics generalize findings from a Sample to its population - distribution of sample means - distribution of sample variances - however, multiple samples to the same size from a population are likely to give different results - a particular sample won't reflect the population exactly - due to a random chance because samples vary by nature

When can we make a directional hypothesis?

Past research, theory... should decide before testing the hypothesis

Standardization

Putting raw scores into SD units First, we standradized X scores to z scores Z distribution has a mean of 0 and a SD of 1 Also, we standardized X scores to other standardized scores Standardized distributions can have any pre-determined mean and SD

Population Scores

Z = X- μ / σ

Transforming z-scores to raw scores

Z = X- μ / σ → X = (z)(σ ) + μ

Meaning of numerator and denominator of z-test statistic

Z = sample mean - hypothesized population mean ----------------------------------------------------------- Standard error between the sample and population means Z = actual obtained differene between M and μ -------------------------------------------------------- Standard difference between M and μ that is expected from only random error

Sample Scores

Z= X-M/ s

would researchers want to get a small or large p value?

a small p value! so that they can reject H0 and to support H0

Type II

retain a null hypothesis - actually there was a real effect, but we erroneously concluded that their was no effect False negative= false non-significant finding

SD

the standard deviation of sampling distribution - is known as the standard error - the standard/ average/ expected amount of difference between M and μ - due to random chance

effect size

A concern about hypothesis testing - a statistically significant treatment effort (size of z) does NOT necessarily mean a substantial treatment effect (size of nominator) - A small treatment effect can be statistically significant, if SD is very small and/or sample size is very big a measurement of the absolute magnitude of a tx effect, independent of the sample(s) being used

Standard error

A measure of "standard" "average" distance from the population mean and a sample mean by chance Standard deviation of the sampl mean distribution Magnitude of SE is determined by two factors Sample size Large sample → less error → smaller SE Population SD Larger SD of population → larger SE Formula σM = √σ^2/ n

Cohns d for z tests

D= mean difference/ standard deviation= M - μ / σ

Probability and the normal distribution

Different portions of the normal distribution are indicated by z-score units (number of SDs from the mean) A z-score corresponds to a specific proportion under the normal curve The sum of all proportions is 1 A proportion of .5 lies to the left of the mean, and a proportion of .5 lies to the right The proportion of the left side is exactly the same as the proportion as the right side

Alternative Hypothesis

Does state a relationship, change, or effect of the IV on the DV Also called the specific hypothesis

Are standardized scores and z-scores the same thing?

No, there are many standardized scores other than z-scores!

Directional versus non-directional alternative hypotheses

Non-directional: H1: states difference but no direction μ w/banana ≠ 100 Directional: H1: states difference with a specific direction μ w/banana > 100

Type I

Reject a true null hypothesis Actually, it was due to sampling error, but we erroneously concluded that there was a tx effect False-positive = false significant finding

Null Hypothesis

States that in the population there is - NO change - NO difference - NO relationship - The IV has No effect on the DV

Sampling error

The discrepancy (amount of error) that exist by chance between statistics and parameters

A random sample of n = 36 is selected from a population which of the following distributors definitely will be normal?

The distribution of sample mean

Law of large numbers

The more indudivuals you obtain from the population, the more representitive the sample is Each sampling distribution has a same mean, but standard error changes depending on sample sizes Obtained from a sample population mean = 80

Properties of normal distribution

The most commonly obsereved frequency ( = probability) distribution in populations Bell-shaped; sometimes reffered to as "bell curve" Defining characteristics Symmetrical Uni-modal Mean = median = mode Extends to +/ - infinite numbers

Meaning of p values of z-test statistic

The p-value or significance level, of your obtained test statistics, tells the strength of the evidence against H0 - the probability of obtaining the sample mean if H0 is true - the smaller the p-value, the stronger the evidence against H0

The probability of Type I error = alpha level

The probability that a sample mean happen to be in a critical region by chance not because of a tx effect = alpha level

Critical region

The region of the sample mean distribution where the sample means that are extremly UN-likely to occur (a very low probability) if H0 is true If the obtained M doesnt fall in the critical region, The M is similar to the μ We conclude that H0 is true: we retain H0 If the obtained M falls in the crtical region, The M is very different from μ Thus, we conclude that HO is not true: we reject HO

z-scores versus raw scores

Through transforming raw test scores to z-scores you can easily tell which test score represents the better performance relative to the whole class Now you have seen the two usages of z-scores To know the exact location of the raw score within the distribution To compare scores from different distributions (with different means and SDs)

Definition and unit of z-scores

Transform X values (raw scores) into z scores (one type of standardized scores) to... Place scores within their distributions Compare RELATIVE positons of scores from different distributions How to transform raw scores to z scores? A deviation score, divided by the SD

One-tailed versus two-tailed testing

Two-tailed testing is non directional testing Critical region z > +1.96 and z < -1.96 One tailed testing is directional testing Critical region of z > +1.64

properties of a z distribution

When you transform all X values into z-scores, you get the z-score distribution The z-distribution of any dataset of scores will always have Sum of z scores = 0 Mean = 0 Standard deviation = 1 Transforming an entire distribution of scores into z-scores will not change the shape of the distribution The standardization process involves mere "re-labeling" of each score They are in the same distribution but each induviudal is labebled with a z-score instead of an x-value

Cohn (1988)

effect size can be standardized by mean difference in terms of SD

Cohn's d

mean differences/ Standard deviation