Question

The data below gives the gold medal times for every other Summer Olympics for the women's 100- meter freestyle (swimming). Can you predict the gold medal times by the year? Year Time (seconds) 1968 60.0 1976 55.65 1984 55.92 1992 54.64 2000 53.8 2008 53.1 Question 46 (2.63 points) What is the independent variable? gold medal times years Question 47 (2.63 points) What is the dependent variable? e gold medal times years Question 48 12.63 points) What is the y-intercept? Please round answer to one decimal place hp

Answer 1

46.The independent variable is "years." i.e. denoted by 'X'.

47.The dependent variable is" gola medal times".i.e. denoted by 'Y'.

X Values 1968 1976 1984 1992 2000 2008 Y Values 60.0 55.65 55.92 54.64 53.8 53.1 61 59 M: 1988 M: 55.5183 Y. Dependent 57 55 53 1968 1 2008 1978 1988 1998 X- Independent - Regression Line ® = -0,15X + 348 36] Estimate Calculation Summary Sum of X = 11928 Sum of Y = 333.11 Mean X-1988 Mean Y = 55.5183 Sum of squares (SSy) = 1120 Sum of products (SP) = -165.32 Regression Equation = ý = bx + a b = SPISSX = -165.32/1120 = -0.14761 a = My-bMx = 55.52 - (-0.15*1988) = 348.96133 y = -0,14761X + 348.96133 X-MX (X-MX)2 -20 - 12 -4 4 12 20 Y- My 4.4817 0.1317 0.4017 -0.8783 -1.7183 -2.4183 400 144 16 16 144 400 (X-MXXY - My -89.6333 -1.58 -1.6067 -3.5133 -20.62 -48.3667 SS: 1120 SP: - 165.32

8. 487) we know, The Regression cacation, ý = bxta where, we get eavation, ý = -0.15X + 348.96 b=slote a=Intercept we where shofe is b = -0.15 and ary Intereelt a = 349.0 [As one decimal Places know the 'Y' Intercept of this line is the valve of Y at the point where the line epossed the y axis. Therefore, the 'Y' Intercept value is 349.0, 55) we get, ý = -0.15x + 348.96 = -0.15x 1988 + 348.96 ( X = 1988 = – 288.2 + 348.96 50.8 Therefore, the predicted value of Y'is 50.8 when 1988 'x' is

X Values 1968 1970 1984 1992 2000 2008 Y Values 00.0 35.05 55.92 54.64 53.8 53.1 Vale x Values X-M - 20.000 - 12.000 -4.000 4.000 12.000 20.000 Y.MY 4.402 0.132 0.402 -0.870 -1.718 -2.418 (X - M₂² 400.000 144.000 10.000 10.000 144.000 400.000 (Y-M12 20.035 0.017 0.101 0.771 2.953 X-MMY-M) -39.633 -1.550 -1.007 -3,513 -20.020 -48.367 Mx: 1988.000 My: 55.510 Sum: 1120.000 Sum: 29.830 Sum: -103.320 Result Details & Calculation Key X Values = 11928 Mean = 1988 (X-MXP = 55,- 1120 XX Values Y: Y Values M: Mean of X Values Mi Mean of Y Values X-M, & V.MiDeviation scores (X-M, & /V-M? Deviation Squared (X-MXV.M): Product of Deviation Scores Y Values = 333.11 Mean = 55.518 ZLY - My)2 = 55y = 29.836 X and Y Combined N = 6 (X-MXMY - My) =-165,32 R Calculation r = EXCX - MyIY-M,))/ v(SSJSS r=-165.32 / 1120X29.836) -0.9044 Meta Numeris (cross-check) r=-0.9044 The value of Ris -0.9044 This is a strong negative correlation, which means that high X variable scores go with low Y variable scores (and vice versa).

style swim can be determined by the variation in the in the defendent variable that the Independent variables explain 5.) we get, the co-prebation coefficient ip is -0.9044. So, the co-efficient of determination, (r) = (-0.90 442 = 0.818 i Therefore, the the 'R' saware value is no. 818, 57) From this 'R' savare value, 80, 818 x 100 = = 81.8% We says that, Approximately 81.8% of the variation in the gold medal times for the women's 100 meter free- year. [As this statistic indicates the percentage of the variance collectively.