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**preview**shows half of the first page. to view the full**1 pages of the document.**CH15 – BIVARIATE CORRELATIONSHIP AND REGRESSION: HYPOTHESIS TESTING

Hypothesis Test

o First compute

and

o Then compute SCP and TSS

o Find Pearson’s r correlation coefficient, regression coefficient b

o Then use

and

and b to compute a

o Then specify regression line equation, plot it w/ lowest & highest X

values on the scatterplot

6 steps of statistical inference and 4 aspects of a relationship

o Requirements:

There is one representative sample from a single population

There are two interval/ratio variables

There are no restrictions on sample size, but generally, the larger

the n the better

Scatterplot of coordinates of the two variables fits a linear

pattern

o Existence of a Relationship

Does a linear relationship between X and Y truly exist in the

population or is the linear pattern in this sample the result of

sampling error?

ρ (rho) corresponding parameter of Pearson’s r statistic

measures the tightness of fit of coordinates around the

regression line for the population

if there’s no relationship in the population then ρ = 0

Pearson’s r will = 0 give or take sampling error

The effect of the hypothesis test is the difference between

observed sample statistic and the expected parameter when the

null hypothesis is true, hence:

Where: and

testing the significance of Pearson’s r bivariate correlation

coefficient: t-Test

Pearson’s r will center on zero as an approximately normal

t-distribution

Calculating Standard Error from a rho of zero

= estimated standard error of the t-distribution for

Pearson’s r

o Direction of the Relationship

Direction of a relationships between two interval/ratio is

ascertained by the sign of r and b, the slope of the regression line

o Strength of the Relationship

= proportion of the variation in Y explained by knowing that it

is related to X

When there’s a strong relationship btwn the two interval/ratio

variables, the X Y coordinates on the scatterplot will fit tightly

around the regression line

Tighter the fit, the larger the value of Pearson’s r and r2

For strength of relationship focus on r2 not r

r encourages an overestimation of the strength of the

relationships

r

r2

Strength of the Relationship

±1.00

1.00

Perfect positive/negative relationship

±0.90

0.81

Very strong positive/negative relationship

±0.70

0.49

Moderately strong positive/negative

relationship

±0.40

0.16

Moderately weak positive/negative

relationship

±0.10

0.01

Very weak positive/negative relationship

±0.00

0.00

No relationship

o t-statistic of the sample?

Larger slope and smaller standard error yield stronger evidence

against the null hypothesis

Standard Error of the slope ()

Estimates the degree of sample-to-sample variation if

regression slopes were calculated from many random

samples of size n

A small standard error implies a higher likelihood that

most of the sample slopes would be near the true

population slope

Larger standard error implies that the regression

coefficient estimate may not accurately reflect the true

relationship between X and Y in the population

Standard deviation of the residual ()

Careful Interpretations of Correlation and Regression Statistics

o Correlation apply to a population not an individual

o Careful interpretation of the slope, b

o Distinguishing statistical significance from practical significance

Tabular Presentation: Correlation tables

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