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OLEMISS / Psychology / PSY 202 / What their shape tells us about the bivariate distribution?

# What their shape tells us about the bivariate distribution? Description

##### Description: These notes cover what will be on Exam 2 which includes chapters 6-10.
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FA16 PSY 202 Section 1 Exam 2 Review Sheet

## What their shape tells us about the bivariate distribution?

Chapter 6

I. CORRELATION

Comparison with regression

• Correlation Model – neither variable is dependent on the other; the two can be  related but that does not mean one causes the other

- Correlation ≠ causation

- Lowest possible value for correlation = -1.00

o Means there is a perfect negative correlation

 as one variable increases, the other decreases

- Highest possible value for a correlation = +1.00

o Means there is a perfect positive correlation

 As one variable increases, the other increases

- A value of 0 means there is no correlation at all

o The closer the value is to 0, whether that value is negative or positive,  means the correlation is not as strong

## What line best summarizes a bivariate distribution by just looking at its scatterplot?

Don't forget about the age old question of What is the best economic activity in nordic countries?
We also discuss several other topics like How much elements does matter consist of?

• Regression Model – predicts cause and effect between variables - Used for quasi-experimental designs

- Uses independent and dependent variables

II. SCATTERPLOTS

How they’re created

- ONLY SCATTERPLOTS show all scores in a distribution

o Y-axis and X-axis

 Plot each pair of the variables on the graph

∙ Every single pair

 This helps with correlation coefficients

- Magnitude tells us how strong the correlation between the two variables is  o how far away from |1| determines how strong

What their shape tells us about the bivariate distribution

## What goes on one side must go to the other?

- Shape tells us the strength of the linear association

o Positive vs. negative correlation

o If our scatterplot is not linear, the information given to us by the  correlation coefficient is underestimated If you want to learn more check out What is considered as the information center of the cell?
We also discuss several other topics like What is the formula for energy in a planet?

III. INTERPRETING THE CORRELATION COEFFICIENT (PEARSON’S R) Other Considerations

- Restrictions in range

- Presence of outliers

- Presence of subgroups

- Causality

To use Pearson product-moment correlation our data must be:

- continuous and linear

- Interval and ratio scales

IV. ALTERNATIVE APPROACHES TO CORRELATION

Which approaches fall into each subcategory & Characteristics of data for which  each is used Don't forget about the age old question of What is another name for the spermatic cord?

- Other Product-moment correlation coefficients

o Spearman Rank Correlation, Point-Biserial Correlation, and The Phi  Coefficient

 Still must be continuous and linear with interval and ratio scales - Non-product-moment with quantitative and dichotomous variables o The Biserial Correlation Coefficient and the Tetrachoric Correlation  - Non-product-moment for categorical values

o Contingency Coefficient and Cramer’s V

Chapter 7

I. STATISTICAL EQUATION OF A LINE If you want to learn more check out What does remain dwells on love and public rituals?

• Y-hat –> predicted value of Y

• b0 –> Y-intercept

• b1 –> slope of the line

II. CHOOSING LINE OF BEST FIT

Difficulties

• We cannot know what line best summarizes a bivariate distribution by just looking at its scatterplot

- Because many lines can be found to summarize most scatterplots  - Must find the line that does best job of including every score

Sum of squared error

• Estimation for all data points

- Helps predict Y from X and also squares the terms in error of estimation to  find the best line possible

o Sum of squared errors helps us figure out which line best summarizes a scatterplot

 We want a lower sum because this means the line fits the

distribution best with least amount of error

• Errors of estimation (residuals)

- The difference between the predicted scores and the actual scores o Vertical distance between Y1 and Yhat

o The amount of variance not shared

• Equations that minimize Sum of Squared Errors

- Because we have more than one population parameter, we lose 2 degrees of  freedom for estimates of 2 means

o Deviations taken from regression line which have 2 statistics: X and Y

Proportion of variance accounted for

• This is how well X predicts Y

- Aka how much variance in Y is shared with X

- Calculated by dividing the sum of squares between groups by the sum of  squares total

Reversing the roles of the IV and DV

- How relation of one slope predicts the other

o Geometric mean

Standardized regression solution

- How to use the z-score for X to predict the z-score for Y

o Predicting Y using X

 Y-intercept where X = 0 intersects to where Y = 0

∙ Y intercept always equal to 0

 Slope

∙ Slope is always = rxy

Standard error of the estimate

- Treats error across entire regression line as equal

o Why it is not a good predictor for the population

- Can also differ in terms of y-intercept and slope

Standard error of the prediction

• Confidence Interval – to predict the average value of Y at a particular X • Prediction Interval – to predict the individual values of Y at a particular X

Cook’s distance (D)

• Sometimes one point will pull entire regression line towards it - If D is larger than 1, there is a substantial influence

• Cook’s distance tells us whether its pulling the regression line but NOT how it is  pulling it

Chapter 8

I. THREE VIEWS OF PROBABILITY

Definitions of each

• Personal/Subjective View

- Used most often

- Humans are naturally terrible at probability, so this is not the one we should  us for statistical inference

- Biggest issue: not based on imperical probability

o Based on how a person feels, which can change from moment to  moment and person-person

 Predictions one person might make will be very different than  predictions of someone else

• Classical/Logical View

- How nature distributes events

• Empirical/Relative Frequency View

- nA –> number of outcomes satisfying condition A

- n –> total number of observed outcomes

o if you have a large enough sample, the probability that you get from  that sample should mimic that of the population from which the  sample came from

Why the personal/subjective view can’t be used

• It is based on how a person feels

- Means it can change from moment-moment and person-person

How the empirical/relative frequency view and classical/logical view are related • As sample size increases, empirical/relative frequency view should start to  converge with the classical/logical view

II. DEFINITIONS

• Experiment – some probabilistic event you make some predictions based off of  and then the observe the outcome to see how solid your predictions were • Event

- Simple event – one outcome can satisfy it

- Compound event – multiple outcomes could satisfy it

• Sample space – how many possible events there are

• Probability function – defines the probability that any individual event will occur

III. RULES FOR FINDING PROBABILITIES

Meanings of terms

• P(A∪B) –> probability of A or B

• P(A∩B) –> probability of A and B

When we simplify formulas

• Simplify formula only when A and B are mutually exclusive

P(A∪B) = P(A) + P(B) – P(A∩B)

- Subtract probability of both events happening together to eliminate the  probability of anything being counted twice.

P(A∪B) = P(A) + P(B)

- If A and B are mutually exclusive, we do not have to subtract that probability

Multiplicative/intersection rule

P(A∪B) = P(A)* P(B|A)

- Multiply probabilities of each B and A

o P(A|B) –> probability of B given A

P(A∪B) = P(A)* P(B)

IV. RULES FOR COUNTING

• Fundamental counting rule

• Permutations – Pnn = n!

- n = number of permutations of n objects

• Combinations – number of combinations of an object using r < n  - combinations do not care about the orders

- How they relate to permutations

Chapter 9

Probability distributions

• Definition

- Probability distribution – tells us the probability that the random variable  will take on each of its possible values

o Classical/logical side of probability

• Random variables – variable whose value is determined by probability  • Sampling distributions – probability distribution based on repeated samples of  a statistic

- Empirical/relative frequency side of probability

- If you have a large enough sample, sampling distribution should resemble  probability distribution because the 2 should converge

o like why empirical/relative frequency converges with classical/logical  when there is a large enough sample size

Bernoulli trials

• Five characteristics

1) only 2 possible outcomes

o success and failure

2) p + q = 1.00

o probability success –> p

o probability failure –> q

3) specified number of trials

o how many trials you are going to do

4) those trials have to be independent of each other

5) random variable

The binomial expansion

•P(Y = r) = Cnr pr q(n-r)

- C = # of combinations

o C42–> 4 coins; 2 land on heads

- r –> how many heads you wanted

- Cnr = n! / (4 – 2)!2! = 6

- = 6/16

The binomial distribution

• Know how to read a Pascal’s triangle

Expected value of a random variable

• Definition

- Expected Value – measure of central tendency; average for a random value  over an infinite set of scores

• How it’s calculated

- E(Y) = ∑[Yj • P(Yj)] = µy

- Get products of the possible values first first and sum them

• Relationship to the population mean

- µy = mean

Variance of a random variable

• Use expected value to get variance of infinite set of scores

Probability and area

• For the binomial distribution

• Why it can’t be calculated the same way for continuous distributions - Must figure out area under curve instead

Chapter 10

Normal distributions

• Characteristics

- Bell-curve – shape of distribution

o Infinite number of possible values

o Frequency polygon that never touches X value

- Kurtosis & Skewness both = 0

o Helpful for lots of reasons

- 3 major measure of central tendency: mean, median, mode are all equal o What goes on one side must go to the other

o 50% above and below median

- Points of inflection at -1σ and +1σ

o 34.1% distribution between the mean and that 1 σ away

o mean of unit normal distribution always = 0 and standard deviation  always =1

• Reasons for importance

- Many values in nature are already normally distributed, certain distributions  are pretty close approximations to normal distribution

o AKA even if distribution is not normal, you can calculate the normal  distribution for the original distribution

- All distributions can be connected by deriving info from the formula we use  for normal distribution.

• Transforming to the unit-normal distribution

- Compare to other normal distributions by converting a unit-normal  distribution to a normal distribution

o mean of 0

o standard deviation of 1

o use the table of probabilities for that normal distribution.

• Areas under the curve

- Above

o Convert the raw scores into Z-scores

- Below

o Convert the raw scores into Z-scores

- Between

o Convert Y into z-scores and subtract lowest value from the highest.

• Approximating the binomial distribution

- Ex: coin flips – 20 flips –> n = 20

- Want percent of distribution higher than 9

o Convert 9 into a z-score after adjusting for continuity

 Adjust because these are discrete variables

o The more times you flip the coin, the more the distribution becomes  normal

 More trials –> more normal

Sampling distribution of the mean

• Definition

- Sampling Distribution of the mean – distributes the averages of the raw  scores

o these averages will create the sampling distribution, which comes out  to be approximately normal

o based on repeated samples.

• Purpose

• Central limit theorem

- Tells us no matter what the raw score distribution looks like, sample  distribution is still going to come out normal

o means will still be equal

o as sample size gets bigger, standard error of mean gets smaller which  means sample means get closer to grand mean

 grand mean always equals mean for the raw scores of the

population.

o hypothetically, if N equals infinite size, standard error would = 0

- Standard error of the mean – quantifies how precise we know the true  mean of a population

• Three possible explanations for extreme scores

1) Sampling error (can’t really avoid)

- took too many of one type in your raw scores just by chance  2) Sampling bias

- Sample differs from population due to the way you drew the sample - Did not use simple random sampling (our preferred way)

3) Sample comes from a different population

- Worry about the most; key for inferential statistic

- “Phantom distributions” around the one we think we get our sample from

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