Question

Here are two sets of bivariate data with the same response variable. The first contains the variables x & y. The second contains the variables w & y х У y 47.1 56.4 68.8 26.8 63.1 77.6 52.6 39.4 68.5 75.3 36 88.5 60.1 78.2 60.2 44.3 37.6 66.2 83.2 -10.2 98.9 93.5 70.5 17.7 86.2 87.7 36.6 80.4 63.5 54.8 50.1 78.8 50.4 57 54.4 42.5 66.3 69.5 64.8 32.2 63.6 63.6 51.8 45.9 47.5 58.4 54 51.6 57.6 26.2 58 56.6 70.7 62.9 37.6 75.4 42.3 49 35.6 80.4 57.8 68.1 67.8 22.3 29.6 43.1 57.5 67.1 59.5 46.8 59.9 40.9 71.8 62.7 79.3 8.5 71.9 82.8 49.4 66.6 63.9 73.5 49.3 64 43.4 58.2 53.4 58.4 65.9 72.7 43.2 72.2 54.6 61.7 57.6 48.7 Which explanatory variable has a stronger relationship with the response variable y? O The first variable (x) has a stronger relationship with the response variable y. O The second variable (w) has a stronger relationship with the response variable y.

Answer 1

Let us find correlation for both the explanatory variable

First for x and y

X - Mx	Y - My	(X - Mx)2	(Y - My)2	(X - Mx)(Y - My)
-10.758	-10.262	115.742	105.319	110.407
5.242	10.938	27.475	119.629	57.331
10.642	8.638	113.245	74.606	91.917
2.242	11.538	5.025	133.114	25.863
-20.258	-0.462	410.4	0.214	9.369
41.042	26.838	1684.418	720.251	1101.456
28.342	21.038	803.25	442.576	596.238
5.642	-11.862	31.828	140.719	-66.924
-7.458	-9.662	55.627	93.364	72.066
8.442	2.838	71.262	8.051	23.953
5.742	-3.062	32.967	9.379	-17.584
-10.358	-8.262	107.295	68.269	85.586
-31.658	-8.662	1002.25	75.039	274.24
12.842	-3.762	164.908	14.156	-48.317
-15.558	-17.662	242.062	311.964	274.799
-0.058	1.438	0.003	2.066	-0.084
-28.258	-23.562	798.533	555.191	665.837
-11.058	-6.762	122.287	45.731	74.782
4.842	5.138	23.442	26.394	24.874
14.042	16.138	197.168	260.419	226.597
6.042	6.838	36.502	46.751	41.31
-14.458	-8.462	209.043	71.614	122.354
8.042	6.038	64.668	36.451	48.552
-3.258	-4.962	10.617	24.626	16.169
Mx: 57.858	My: 66.662	Sum: 6330.018	Sum: 3385.896	Sum: 3810.793

X Values
∑ = 1388.6
Mean = 57.858
∑(X - M_x)² = SS_x = 6330.018

Y Values
∑ = 1599.9
Mean = 66.662
∑(Y - M_y)² = SS_y = 3385.896

X and Y Combined
N = 24
∑(X - M_x)(Y - M_y) = 3810.793

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 3810.793 / √((6330.018)(3385.896)) = 0.8231

Now for w and y

W - Mw	Y - My	(W - Mw)2	(Y - My)2	(W - Mw)(Y - My)
13.392	-23.2	179.337	538.24	-310.687
-2.808	-10.6	7.887	112.36	29.768
-19.408	38.5	376.683	1482.25	-747.221
4.792	-5.7	22.96	32.49	-27.312
27.792	-60.2	772.377	3624.04	-1673.058
15.092	-32.3	227.758	1043.29	-487.461
-18.808	30.4	353.753	924.16	-571.773
-5.308	28.8	28.178	829.44	-152.88
-1.008	-7.5	1.017	56.25	7.563
9.392	-17.8	88.203	316.84	-167.172
-3.608	-4.1	13.02	16.81	14.794
-1.408	1.6	1.983	2.56	-2.253
1.192	7.6	1.42	57.76	9.057
-17.808	25.4	317.137	645.16	-452.332
-19.808	30.4	392.37	924.16	-602.173
12.392	-27.7	153.553	767.29	-343.249
2.092	17.1	4.375	292.41	35.767
4.092	-9.1	16.742	82.81	-37.234
23.892	-41.5	570.812	1722.25	-991.504
-6.008	16.6	36.1	275.56	-99.738
-6.108	14	37.312	196	-85.517
-2.008	8.4	4.033	70.56	-16.87
-12.208	22.2	149.043	492.84	-271.025
2.192	-1.3	4.803	1.69	-2.849
Mx: 55.408	My: 50.000	Sum: 3760.858	Sum: 14507.220	Sum: -6945.360

X Values
∑ = 1329.8
Mean = 55.408
∑(X - M_x)² = SS_x = 3760.858

Y Values
∑ = 1200
Mean = 50
∑(Y - M_y)² = SS_y = 14507.22

X and Y Combined
N = 24
∑(X - M_x)(Y - M_y) = -6945.36

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -6945.36 / √((3760.858)(14507.22)) = -0.9403

As value for w and y is more close to -1, hence there is strong correlation between w and y than between x and y

So answer here is

The second variable (W) has a stronger relationship with the response variable y.