QMB FINALTopic 1 CHI-SQUARE TESTS:The chi-square test of independence is used to determine if two categorical variables areindependent of one anotherStep 1: Identify Null hypothesisStep 2: Calculate Expected FrequencyStep 3: Calculate chi square test statistic x2Step 4: Determine the Chi-Square test critical value x2∝Step 5: Compare the test statistic x2 with x2∝Step 6: State the ConclusionStep 1: Identify Null Hypothesis:H0 : The averages and class times are independent of one anotherH1 : The averages and class times are not independent of one anotherStep 2: Calculated Expected Frequency:fe=(RowTotal 09Column Total)Total Number of observationsStep 3: Calculate Chi-squared Test statistic:x2=∑(fo−fe)2feStep 4: Determine the Chi-Square critical valueDF=(r-1)(c-1)x2∝=x20.05, when ∝ = 0.05Step 5: Compare the statistic with the critical valuex2¿ x2∝ Rejectx2¿ x2∝ Fail¿ RejectStep 6: State the conclusionExample: We do not reject the null hypothesis so we conclude that the production shift and production quality are independent of one anotherTopic 2:CORRELATION ANALYSIS:Correlation analysis is used to measure both the strength and direction of a linear relationshipbetween two variables* A relationship is linear if the scatter plot of the independent and dependent variables has a straight-line pattern *Construct a Table to provide values for future calculations:The Six-Number Summary For correlation and Simple Regression (SNSCSR)∑X∑Y∑XY∑X2∑Y2NN, Represents the number of ordered pairs in the tableThe Correlation Coefficient for a random sample, r :r=n∑xy −(∑x )(∑y )√¿¿¿- indicates the strength and direction of the linear relationship between independentand dependent variables- Values range from -1.0 ( a strong negative relationship) to +1.0 (a strong positive relationship)- When r=0, there is no relationship between variables x and yThe Correlation Coefficient for a population, p :- Refers to the correlation between all values of two variables of interest in a population- Use a hypothesis test to determine if the population correlation coefficient is significantly different from zeroH0 : p = 0H1 : p ≠ 0Tests Statistic t,-t=r√1−r2n−2-df =n−2- Requires a two tail test with ∝/2 in each tailIf H0 : p = 0 H1 : p ≠ 0 then, one tail test ! -SIMPLE REGRESSION ANALYSIS:Used to determine a straight line that best fits a series of ordered pairs (x, y)- An independent variable (x) is a variable used to predict, explain, or forecast a dependent variable (y)- This technique is known as simple regression because we are using only one independent variable- multiple regression, which includes more than one independent variable, isdiscussed in the next chapterFormula for the equation describing a straight line through ordered pairs:- Regression equation : ^y=a+bxo^y , the predicted value of y given a value of xox , the independent variableoa , the y-intercept of the straight lineob ,the slope of the straight lineThe Residual, ei :-ei= yi−^yi- The difference between the actual data value and the predicted valueLEAST SQUARES METHODIdentifies the linear equation that best fits a set of ordered pairs- Used to find the values for a (the y-intercept) and b (the slope of the line)- The resulting best fit line is called the regression line*GOAL *: Minimize the total squared error between the values of y and the predicted values of ySum of squares error (SSE):SSE=∑i=1n( yi−^yi)2Regression Slope:b=n∑xy−(∑x)(∑y )n∑x2− ¿ ¿Example: Slope: On average, each additional TV ad increases the number of cars sold by 3.8947 per weeky-intercept:a=∑yn−b(∑xn)Total sum of squares (SST):- Measures he total variation in the dependent variableCoefficient of determination, R 2 : -R2=SSRSST- measures the percentage of the total variation of the dependent variable that is explained by the independent variable from a sample- Varies from 0% to 100%- Higher values are more desirable because we would like to explain as much of the variation in the dependent variable as possible-R2 is equal to the square of rThe Population Coefficient of determination, p 2 : - Unknown so we must use a hypothesis test to determine if p2 is significantly different than zero (based on R2)o H0 : p2 ¿ 0 (none of the variation is explained by x)o H1 : p2 > 0 (x does explain a significant portion of the variation in y)- F-test statistic is usedoF=SSR(SSEn−2)odf : D1=1∧D2=N−2oF>F∝REJECT (conclude that p2 is greater than 0)oF ¿ F∝FAIL¿ REJECTConstructing a confidence interval around the point estimate:Standard error of the estimate, se:-se=√SSEn−2- Measures the amount of dispersion of the observed data around a regression lineFormula for the confidence interval for an average value of y: Average value of x:-x=∑xn Prediction interval: Population Regression slope, β:- Unknown, must use hypothesis test to find out if it is significantly different than 0o H0 : β = 0 (There is no relationship between the independent and dependent variables)o H1 : β ≠ 0 (There is no relationship between x and y)- T-test statisticot=b− βsb osb=se√∑x2−n¿¿¿ ¿¿o Df = n-2o Two tailConfidence Interval for the slope of a regression: ASSUMPTIONS:• Assumption 1: The relationship between the independent and dependent variables is linearNot linear : For low and high values of x, the estimated ^y value will be too high; estimated ^y values for x s in the middle of the x range will be too low• Assumption 2: The residuals exhibit no patterns across values of the independent variable• Assumption 3: (homoscedasticity)Can view residual plot to evaluate this assumptionMultiple Regressions and Model BuildingThis chapter takes in consideration more than one independent variable Regression equation using k independent variables:^y=a+b1x1+b2x2+…+bkxkRegression Coefficient:Predicts the change in a dependent variable due to a one unit increase in an independent variable. All other variables held constant Slope Coefficients:-b1, b2, … , bkExplaining the variation of the dependent variable:*GOAL* : Determine the amount of the variation in y that is due to variation in the independentvariablesEquations for R2 are the same. Whatever R2 comes out too is multiplied by 100 to reveal percentwhich is equal to the variation in y. R2=SSRSSTTesting the Significance of the overall regression Model:*Goal* : decide if the relationship between