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Chapter 3

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

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David Medler

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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)
variabilitySXSY (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
meanρr = 0
estimated standard errorSr 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 sizeN = ?
-population parametersρ 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
PSYC 300B - Chapter 3: Hypothesis Testing of the Correlation Coefficient
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