Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.
The regression line equation that we calculate from the sample data gives the best fit line for our particular sample. We want to use this best fit line for the sample as an estimate of the best fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
The assumptions underlying the test of significance are:
- There is a linear relationship in the population that models the average value of y for varying values of x. In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
- The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) above implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
- The standard deviations of the population y values about the line are equal for each value of x. In other words, each of these normal distributions of y values has the same shape and spread about the line.
- The residual errors are mutually independent (no pattern).
**With contributions from Roberta Bloom
Kupima umuhimu wa coefficient uhusiano inahitaji kwamba dhana fulani kuhusu data ni kuridhika. Dhana ya mtihani huu ni kwamba data ni sampuli ya pointi kuchunguzwa kuchukuliwa kutoka idadi kubwa ya watu. Hatukuchunguza idadi nzima ya watu kwa sababu haiwezekani au haiwezekani kufanya hivyo. Kwa hivyo, tunachunguza sampuli ili kuamua ikiwa uhusiano wa linear kati ya x na y katika data ya sampuli hutoa ushahidi thabiti wa kutosha ili tuweze kuhitimisha kuwa kuna uhusiano wa linear kati ya x na y katika idadi ya watu. regression line equation kwamba sisi mahesabu kutoka data sampuli inatoa bora fit line kwa sampuli yetu maalum. Tunataka kutumia hii bora fit mstari kwa ajili ya sampuli kama makadirio ya bora fit mstari kwa ajili ya idadi ya watu. Kuchunguza scatterplot na kupima umuhimu wa coefficient uhusiano inatusaidia kuamua kama ni sahihi kufanya hivyo. Dhana msingi mtihani wa umuhimu ni: - Kuna uhusiano linear katika idadi ya watu kwamba mifano ya thamani ya wastani wa y kwa maadili tofauti ya x. Kwa maneno mengine, thamani ya kutarajiwa ya y kwa kila thamani maalum liko juu ya mstari wa moja kwa moja katika idadi ya watu. (Hatujui equation kwa ajili ya mstari kwa ajili ya idadi ya watu. Mstari wetu wa kurudi nyuma kutoka kwa sampuli ni makadirio yetu bora ya mstari huu katika idadi ya watu.) - Thamani ya y kwa thamani yoyote maalum ya x ni kawaida kusambazwa kuhusu mstari. Hii ina maana kwamba kuna zaidi ya maadili y kutawanyika karibu na mstari kuliko ni kutawanyika mbali zaidi. Dhana ya (1) hapo juu inaonyesha kwamba usambazaji huu wa kawaida umezungukwa kwenye mstari: wastani wa usambazaji huu wa kawaida wa maadili ya y uko kwenye mstari. - standard deviations ya idadi ya y maadili kuhusu mstari ni sawa kwa kila thamani ya x. Kwa maneno mengine, kila moja ya hizi usambazaji wa kawaida wa maadili ya y ina sura sawa na kuenea kuhusu mstari. - makosa ya mabaki ni huru kwa kila mmoja (hakuna muundo). ** Pamoja na michango kutoka Roberta Bloom