# The test statistic used to test our hypothesis is a

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The test statistic used to test our hypothesis is a chi-square statistic , represented by X 2 :
X 2 = (Observed Expected) 2 Expected Σ
What happens to X 2 is the observed and expected values are very similar? What happens to X 2 if the observed and expected values are very different? To find the appropriate p-value for our hypothesis test, we must compare our test statistic to a chi- square distribution , χ 2 . 12.1 Chi-Square ( χ 2 ) Distribution The χ 2 Distribution is a skewed distribution, whose shape depends on degrees of freedom . The degrees of freedom is equal to df = ( r 1) × ( c 1). The p-value is always calculated as p-value = P ( χ 2 X 2 )
We can use software to find the area under a χ 2 distribution, with the Excel command CHISQ.DIST.RT : CHISQ.DIST.RT(x, deg freedom)
EXAMPLE: Return to our data above, regarding company size and Facebook page. Our interest is in determining whether there is a relationship between these two variables - that is, does a company’s size influence whether or not it will have a Facebook page? Company Size and Social Media Facebook Yes No Page Company Size Large Small 30 76 364 130 Total 106 494 Total 394 206 600
It is much faster to conduct this analysis in software. We can use the function CHISQ.TEST , which will return the p-value for our test for independence. To use this function, we need to have a table of our observed and expected values. The Excel command CHISQ.TEST(G3:H4,G7:H8) yields the following results: Assumptions Required for a Valid Chi-Square Test : In order for our chi-square analysis to be valid, the following assumptions must be satisfied: Our sample must be a simple random sample For a 2 × 2 table, the expected values for each cell must be at least 5 If our table is larger than 2 2, then the mean of all expected values must be at least 5, and each individual expected value count must be at least 1 ×
13 Introduction to Regression & Correlation In regression analysis , we explore the possible relationship between a quantitative response vari- able and one (or more) explanatory variables. The explanatory variable(s) can be a quantitative or qualitative random variable, but for now we will only consider quantitative variables. When we have one explanatory variable: When we have more than one explanatory variable: Regression analysis allows us to describe the relationship between X and Y with a model . 13.1 The Linear Regression Model To motivate our regression model, reconsider the following example which we first saw when we intro- duced the idea of correlation: (Adapted from Question 13, page 675 in Business Statistics , 2 nd Canadian Ed. (2014). Sharpe, N.R., DeVeaux, R.D., Velleman, P.F., Wright, D. Pearson Toronto). Data on the number of sales associates working, and the number of sales (in \$1000s), were recorded for 10 randomly selected small book stores. The objective of the study was to determine if there was a linear relationship between the number of sales associates on the floor (explanatory variable) and the amount of business done in sales (response variable). A partial table of the data is presented below.