SOC/STAT/CSSS 221: Statistical Concepts and Methods for the Social Sciences Homework Assignment 7 Please use this form to record your answers and include any work that you would like to show the grader (you can paste in photos or scans of your hand-written work if you like). Save this document in Word, rtf, or pdf format and submit ONE document for grading via the course website. 1. A researcher collected the following data on years of education (X) and number of children (Y) for a sample of married adults: a. Draw a scatter plot of the data. DATA b. Calculate and interpret the correlation between education X Y and number of children. 12 2 c. Is the association between education and number of 14 1 children statistically significant? Explain the basis of your answer and the implications for the association in the 17 0 population. 10 3 d. Calculate and interpret the meaning of the regression slope. 8 5 e. Calculate and interpret the meaning of the Y-intercept for 9 3 the regression line. f. Predict the number of children for an adult with 11 years of 12 4 education. 14 2 g. Calculate and interpret the coefficient of determination. 18 0 16 2 2. The same researcher collected at random one child from each of the eight families in which there was at least one child. She calculated the following statistics to study the effects of number of siblings (X) on happiness of the child (Y), measured on an interval scale in which low scores indicate a low level of happiness and high scores indicate a high level of happiness. Happiness points (Y) Number of siblings (X) y =5.625 a. b. c. d. e. 3. x =1.750 s y =2.134 s x =1.282 r=.588 n=8 Is the association between number of siblings and happiness statistically significant? Explain the basis of your answer and the implications for the association in the population. Calculate and interpret the meaning of the regression slope. Calculate and interpret the meaning of the Y-intercept for the regression line. Predict the level of happiness for an only child and for a kid with two siblings. Calculate and interpret the coefficient of determination. A personnel specialist with a large accounting firm is interested in determining the effect of seniority (the number of years with the company) on hourly wages for administrative staff. She selects a random sample of 10 secretaries and collects the following data on their years with the company (X) and hourly wages (Y): Hourly Wage (Y) Years with company (X) y =$ 13.90 a. b. c. d. e. f. x =2.700 s y =$ 1.37 s x =1.889 r=.631 n=10 Is the association statistically significant? Explain your answer. Calculate and interpret the meaning of the regression slope. Calculate and interpret the meaning of the Y-intercept for the regression line. Predict the hourly wage of a randomly selected worker who has been with the company for 4 years. Calculate and interpret the coefficient of determination. Based on these results, what is the typical starting wage per hour, and what is the typical increase in wage for each additional year on the job? [not a trick question!] 60 50 40 Y-Values Column1 30 20 10 0 6 8 10 12 14 16 18 On the horizontal axis we have x values and y on the vertical axis. b) Correlation = covariance between x y standard deviation xstandard devition y Covariance=E (XY) - E(X) E(Y) E (XY) = 243 10 =24.3 E(X) = 130 10 = 13 E(Y) = 22 10 = 2.2 Covariance = 24.3- 2.2 *13= -4.3 STDEV X=3.399346 STDEV= 1.6193288 20 4.3 3.3993461619328 = -0.781156874 This means that the increase in the number of students leads to a decrease in the level of education d) Slope, = covariance between x y variance of x Where E(X) and E(y) are the respective means Slope, = 0.41346 The slope means that the number of children is inversely related to education since the slope is negative. An increase in the number of children reduces the level of education and vice verser e) Intercept= mean y- * mean x = 2.2 + (0.41346*13) = 7.575 f) The regression equation is y=7.575 + -0.41346X y=7.575 + -0.41346* 11 = 3.02694 g) Coefficient of determination is given by, the correlation squared and made into a percentage -0.7811568742 * 100= 61.0206061% Question 2 a) The association between number of siblings and happiness is statistically significant because the r=0.588 meaning that 58.8% 0f the level of happiness is determined by the number of children. b) Slope = 0.5882.1341.282 =1.6086 1.282 The slope is positive meaning that the level of happiness increases with the number of children c) y intercept= mean y- slope * mean x = 5.625- 1.6086* 1.750 = 2.8099 The y intercept represent the level of happiness when there are zero children d) The regression equation is y= 2.8099 + 1.6086X When one child the level of happiness= 2.8099 + 1.6086* 1 =4.4185 When 2 children the level of happiness is 2.8099 + 1.6086* 2= 6.0271 e) Coefficient of determination= correlation squared *100 = 0.5882 * 100= 34.5744% This means that 34.5744% of happiness is explained by the number of children. Question 3 a) The relationship between seniority (number of years with the company) and hourly wages for administrative staff is statistically significant because the correlation between the two is 0.631. covariance between X Y b) Slope= the variance of X = 0.6311.371.889 = 0.86447 1.889 The slope is positive meaning that an increase in the number of years in the company, is directly related to hourly wages. Therefore an increase in seniority leads to an increase in the hourly wage and vice verse. c) y intercept= mean y- slope * mean X = 13.90-(0.86447 *2.700) = 11.565931 This means that the hourly wage for a new employee is 11.565931 d) The regression equation is y= 11.565931 +0.86447X The hourly wage for one employed for 4 years y= 11.565931 + 0.86447 *4 y=15.023811 e) Coefficient of determination formula is xy x y x 2 n x ( 2) y r= 2 y 2 n n = 100.6311.8891.37 = 0.631 101.8891.37 Square the correlation and then convert to a percentage 0.631*0.631= 0.398161= 39.8161% This means that 39.81% of the hourly wage is determined by the number of years an employee has been working with the company f) The starting wage= y intercept= 15.023811 The increase in price for each additional year of a job is the slope= 0.86447