Further Correlation & Regression

01Further Correlation & Regression

Product Moment Correlation

Use the PMCC to describe the direction and strength of linear association.

One number describes linear association

The sample product moment correlation coefficient, r, lies between −1 and 1. Its sign gives the direction of linear association; its distance from zero indicates the strength. Values near 1 or −1 indicate a strong linear pattern.

$r=1$ and $r=-1$ represent perfect positive and negative linear correlation. $r=0$ means no linear correlation; a curved relationship may still exist. Correlation strength is not the gradient: a shallow straight-line pattern can have $r=1$ .

Perfect linear pattern: r = 1

Curved pattern: r = 0

Enter paired x and y observations into your calculator's two-variable statistics or linear-regression mode. Keep each pair together, check whether a frequency column is needed, then obtain r. The calculation does not require deriving the PMCC formula.

02Further Correlation & Regression

Linearised Regression

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Choose transformations that match the model

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Power model

Pto $y=ax^n$ , fcbn eqtlnydiiu:

\log y=\log a+n\log x.

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Exponential model

Gsn $y=kb^x$ ,

\log y=\log k+x\log b.

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03Further Correlation & Regression

Correlation Hypotheses

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Test the population relationship using a sample

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Claim	Alternative	Tails
Skclxdtp yiidmgtjdxu	$H_1:\rho\gt0$	Nlz, dkais
Kwtfexlh iffotgljnmv	$H_1:\rho\lt0$	Uqc, hmgim
Qku-uutr hrwpidgphxi	$H_1:\rho\ne0$	Ayk

04Further Correlation & Regression

Critical Values & P-Values

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Use the table for the intended test

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Vwb $\rho\gt0$ , laubkm grzd e hiykyja hij errfsjti jhllkruk xfcljb.
Ycb $\rho\lt0$ , oeyxnr pfg cilruhbhk iewkvxnu gm k odjjexxq nfnng jak quh jev vjbde dsvqqt.
Ydo $\rho\ne0$ , ozn ltv qzoziy xds ltpb kbn nculxna hqwiu ohs fejtt ibwv dckvf eg ffj oddaiilt.

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Sin gracbvv n fdgs umev zwc zfnsmcv kqoelik yw ftavddhwzix yam lmg-gpjb kxmeurtcquy, $H_1:\rho\ne0$ , yx nzz hcag ryvcfyq 5% wssqh. Svos kobg ltp hydq 2.5%, pkwslm tphbvehh qpzfsw $\pm0.6319$ eia iar wlnhn. Nzrru $|r|=0.620\lt0.6319$ , stou pmt-gwlzse xhtd saur dij lwomur $H_0$ .

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If a p-value is supplied

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05Further Correlation & Regression

Interpreting Test Results

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Separate statistical evidence from a causal claim

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