# PSYC 300B Chapter Notes - Chapter 3: Joint Probability Distribution, Sampling Distribution, Descriptive Statistics

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**preview**shows half of the first page. to view the full**1 pages of the document.**•hypothesis testing — applying to correlation coefﬁcients

-Pearson r: a descriptive measure of the relationship between 2 variables

•has both magnitude & direction

-apply hypothesis testing in order to determine whether r represents

sampling error or an actual meaningful association between 2 variables

-use the random sampling method, which has the 3 distributions:

(i) population distribution:

•data — bivariate distribution of X & Y scores

•parameter of interest (mean) — rho (ρ) (not µ)

-range of ρ is -1.00 to +1.00

-assume there is no linear correlation, so ρ = 0

•variability — product of the variability of X & Y (σXσY = ?)

•shape — values assumed to be a normally distributed

“mountain” which becomes a single distribution with ρ = 0

(ii) sample distribution:

•data — distribution of XY scores for ﬁxed N

•statistic for analysis (mean) — Pearson r (an estimate of ρ)

•variability — SXSY (product of standard deviation for X & Y)

-make df adjustment when estimating σXσY from SXSY

-(N - 1 for each variable) in computation of SXSY!

•shape — linear on a scatterplot

(iii) sampling distribution:

•the distribution of all possible values for r given df = …

•data — correlation (r)

•shape — both variables are normally distributed

•mean — ρr = 0

•estimated standard error — Sr computed as

•2 different options:

(a) Table of Critical Values of Pearson r —

Sampling Distribution of Correlations for df = …

-use the table of critical values of Pearson r to

deﬁne the cut-off region based on the value for

df & of p(𝛼) 1- or 2-tailed

(b) Critical Values from Student t-table — use the

t-table to determine the statistical signiﬁcance of r

-transform r to a t-statistic

-critical value for t is determined from the

distribution of values for t-statistic, based on

p(𝛼) & df of N - 2

•steps in random sampling hypothesis testing — Pearson r:

(i) state hypotheses in verbal & symbolic form

-deﬁne H1 & H0

-H1: r ≠ ρr or H1: ρr ≠ 0

-H0: r = ρr or H0: ρr = 0

(ii) select & uniquely deﬁne the sampling distribution —

Distribution of Pearson Correlations for df = ?

-hypothesis testing — random sampling model

-data — scores for X & Y variables

-research design — correlation

-statistic for analysis — Pearson Correlation, r

-sample size — N = ?

-population parameters — ρ is assumed to be 0

•σXσY is estimated by SXSY

(iii) determine p(𝜶)

-tcrit or rcrit is determined from the t-table or Pearson r table based

on the df & 𝜶, 1- or 2-tailed

(iv) gather data, compute descriptive statistics, & the test ratio

-compute r

•look up rcrit on special table if using option (a)

•apply test ratio if using option (b)

(v) compare p(obs) to p(𝜶)

-p(obs) > p(𝛼) — retain H0 or fail to reject H0

•suspend judgement on H1 until further data is collected,

because the outcome is inconclusive & we cannot reject H1

•nothing systematic is happening in the data & the data

represents a random occurrence

-p(obs) < p(𝛼) — reject H0 & accept H1

•conclude that the data represents a systematic change in the

behaviour & that the outcome is statically signiﬁcant

(vi) interpret & report outcome

-less than (p <) always implies statistical signiﬁcance

-greater than (p >) always implies no statistical signiﬁcance

•interpreting a correlation value — a strong correlation doesn’t always mean

signiﬁcance, because we have 4 things to consider:

(i) the qualitative nature of the correlation — strong, moderate, weak,

or non-existence

(ii) the quantitative magnitude of the correlation — its value

(iii) the statistical signiﬁcance of the correlation value

(iv) the practical signiﬁcance of the correlation

•assumptions — when applying hypothesis testing to Pearson r:

(i) participants were randomly sampled from the population

(ii) data are scores (because we are testing the correlation coefﬁcient)

(iii) each variable is independently normally distributed in the population

(iv) bivariate distribution is normally distributed in the population

(v) N ≥ 7 when estimating ρ from r

•key points to remember:

-just because you have statistical signiﬁcance, that does not mean that you

have practical signiﬁcance

•statistical signiﬁcance of a Pearson r does not mean that the

correlation is meaningful, it just means that you sample has

potentially captured the true underlying relationship of the

population when compared to the Null Hypothesis

-what we are doing with hypothesis testing is just seeing if something ≠ 0

-the size of N does not affect the the size of r

-although, the larger the N, the smaller the variance in the data, resulting

in greater chance of signiﬁcance

•a large N means a smaller value for r will be statistically signiﬁcant

•a smaller N means a larger value for r will be judged to be statistically

signiﬁcant

PSYC 300B - Chapter 3: Hypothesis Testing of the Correlation Coefﬁcient

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