Psychological Statistics Exam #2 S.G.

X

An individual person’s raw score

M

Sample mean

µ

Population mean

s

Sample standard deviation; average distance scores in a sample are from the mean of the sample.

σ

Population standard deviation

• Average distance scores in a population are from the mean of the population.

n

Number of people (scores) in sample

N

Number of people (scores) in population

z

Z-Score

Can be obtained for…

• A single person’s score within a sample or population.

• OR a whole sample mean pulled from a population.

p

Proportion (or probability) of something

σ_M

Standard error of the mean

• Average distance all random samples (of a given n) are from the mean of the population

What is a difference between a Raw Score & a Z-Score?

• Raw Score: a score on its original metric (aka scale)

• Z-Score: standard scores that tell you how many standard deviation units a score was away from the mean.

What are the advantages of a Z-Score over a raw score? (2)

1. You can compare a score relative to everyone else’s score in the distribution; see its exact location within the distribution.

2. You can compare scores from different distributions (SAT & ACT example)

How do you convert a Raw Score into a Z-Score?

Population: z = (x - µ) / σ

Sample: z = (x - M) / s

What is the Distribution of Sample Means?

• When samples differ from each other (given 2 random samples, it’s unlikely that they would always be the same).

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

  • It is obtained by selecting all the possible random samples of a specific size (n) from a population & calculating the mean for each. These samples will form a distribution.

What is the Expected Value of the Mean? What can you predict its value to be?

• Is the mean of the Distribution of Sample Means

• Represented by μM and will ALWAYS match the mean of population scores (μ)… it is an unbiased statistic b/c they match

• In short, the mean of the sample means will match the population mean, but that does not mean that any one particular sample mean will match the population mean (one particular sample may be an outlier).

What is the Standard Error of the Mean?

• The standard deviation of the distribution of sample means. When it is large, the means are widely scattered.

• Provides a measure

• Represented by: σ_M (or SE)

How is Standard Error of the Mean calculated?

σ_M = σ / √ n

As Sample Size _______, Standard Error (of M) _______

Increases; Decreases

How does the Population Standard Deviation (σ) affect the Standard Error?

• A larger population standard deviation directly leads to a larger standard error.

• As the variation within a population increases, the standard error of the sample mean also increases.

• A wider spread in the population data results in a larger potential error when estimating the population mean from a sample.

What are the principles of the Central Limit Theorem? What does it say about Distribution of Sample Means? (2)

The Distribution of Sample Means will be almost perfectly normal in either of 2 conditions:

1. The population from which the samples are selected is a normal distribution.

2. The number of scores (n) in each sample is relatively large (about 30).

What is the primary purpose of visualizing the Distribution of Sample Means?

So we can know when it will be normal and therefore use the table to find the probability associated with a specific sample mean (next chapter).

• It is not like researchers actually take a million random samples. Usually they just take one and want to know the probability of obtaining it.

Can you obtain a Z-Score for a sample mean?

Yes

z = ( M - µ ) / σ_M

What is Hypothesis Testing?

A statistical method that uses sample data to evaluate the validity of a hypothesis about an unknown population.

What are the steps in Hypothesis Testing? (4)

1. State null & alternative hypothesis

2. Set the critical value (z) associated w/ the probability test

3. Collect data, calculate various pieces, & plug into z formula for one sample z test (sample mean - population mean ÷ standard error of the mean).

4. See if the z is in the critical regions. If it is… reject the null. If not… fail to reject the null (retain it).

What is a null hypothesis?

• H₀

• States that the treatment has NO effect

• µ1 = µ2

What is an alternative hypothesis?

• H₁

• States that the treatment DOES have an effect

• µ1 ≠ µ2

What is the critical region?

• The pink areas on the graph

• In order to reject the null (state that there is a statistically significant effect), your calculated z-score needs to be 1.96 (or more) extreme, or -1.96 (or more) extreme.

What is an alpha region?

The level of risk you are willing to accept of wrongfully concluding there is an effect.

• If we set our alpha level to 0.5 for the p value, that means we want a 0.05 (5%) or less chance of obtaining the results if the null were true.

How does an alpha level relate to the critical region & z-scores?

• If sample statistic (z) is located in the critical region, the null hypothesis is rejected.

• If the sample statistic (z) is not located in the critical region, the researcher fails to reject the null hypothesis. Sort of like saying “retains it.”

What is the difference b/w a directional (one-tailed) & non-directional (two tailed) alternative hypothesis?

…

What are the 3 measures of Central Tendency? What is the goal of them as a class of measures?

1. Mean

2. Median

3. Mode

GOAL: to identify a single value that represents the entire set of data.

What the 3 measures of Variability? What is the goal of them as a class of measures?

1. Range

2. Variance

3. Standard Deviation

GOAL: to quantify how spread out scores are or how different they are.