Question

What's the regression line and write an equation in y=ax+b form. Round a and b to the nearest tenth.

What's is the correlation coefficient for the data? Use a DiagnosticOn
r = round to the nearest thousandth

Can someone please help me figure this out please and thank you

Answer 1

The calculated regression equation is y = 0.10x + 50.9 and the correlation coefficient for the data is 0.96

Finding the regression equation of the table of values

From the question, we have the following parameters that can be used in our computation:

The table of values

Using a graphing tool, we have the following summary

• Sum of X = 1386
• Sum of Y = 653
• Mean X = 138.6
• Mean Y = 65.3
• Sum of squares (SSX) = 8176.4
• Sum of products (SP) = 848.2
• r = 848.2 / √((8176.4)(96.1)) = 0.9569

The regression equation is represented as

y = bx + a

Where

b = SP/SSX = 848.2/8176.4 = 0.10374

a = MY - bMX = 65.3 - (0.1*138.6) = 50.92197

So, we have

y = 0.10x + 50.9

Hence, the regression equation is y = 0.10x + 50.9 and the correlation coefficient is 0.96

Answer 2

The data is plotted in the scatter graph shown below. The regression line shows a positive correlation. The equation of a straight line could normally be formed by reading the y-intercept and working out gradient, but in this graph, it would be tricky to read the y-intercept, so we will have to use the Least Square method.

We need the value of the slope (m) and the value of y-intercept (b). The formula to find both values are shown in picture 2 and picture 3 below
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These following steps are for finding the slope

STEP 1: Find the mean of height and the mean of weight

Mean of height = ∑height ÷ number of data
Mean of height =
`(61+61+62+64+65+68+66+67+69+70)/(10)=65.3`

Mean of weight = ∑weight ÷ number of data
Mean of weight =
`(99+104+110+133+130+142+146+153+184+185)/(10)=124`

STEP 2: Subtract each value of height by its mean and subtract each value of weight by its mean

Weight Height
99 - 124 = -25 61 - 65.3 = -4.3
104 - 124 = -20 61 - 65.3 = -4.3
110 - 124 = -14 62 - 65.3 = -3.3
133 - 124 = 9 64 - 65.3 = -1.3
130 - 124 = 6 65 - 65.3 = -0.3
142 - 124 = 18 68 - 65.3 = 2.7
146 - 124 = 22 66 - 65.3 = 0.7
153 - 124 = 29 67 - 65.3 = 1.7
184 - 124 = 60 69 - 65.3 = 3.7
185 - 124 = 61 70 - 65.3 = 4.7

STEP 3: Multiply each pair of weight and height from STEP 2 and total the answers

(-25×-4.3) + (-20×-4.3) + (-14×-3.3) + (9×-1.3) + (6×-0.3) + (18×2.7) + (22×0.7) + (29×1.7) + (60×3.7) + (61×4.7)

107.5 + 86 + 46.2 + 11.7 + 1.8 + 48.6 + 15.4 + 49.3 + 222 + 286.7 = 875.2

STEP 4: Find (Value of weight - mean of weight)²
(-25)² + (-20)² + (-14)² + (9)² + (6)² + (18)² + (22)² + (29)² + (60)² + (61)² = 10308

The value of m = STEP 3÷STEP 4 = 875.2 ÷ 10308 = 0.085

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To find the y-intercept, refer to formula in picture 3
We have,
The mean of weight (x) = 124
The mean of height (y) = 65.3
The slope (m) = 0.104

b = 65.3 - (0.085)(124) = 54.76

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The equation of line best fit is
`y=0.085x+5476`

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To find coefficient of correlation, we are going to need these following values
A ∑(Weight × Height) ⇒ Multiply each pair of weight and height and total the numbers
B ∑(Weight - mean of weight)² ⇒ Subtract each weight by the mean of weight, square each answer then add up
C ∑(Height - mean of height)² ⇒ Subtract each height by the mean of height, square each answer then add up

After calculation, we have
∑(Weight - Mean of weight) × (Height - Mean of height) = 875.2
∑(Weight - Mean of weight)² = 10308
∑(Height - Mean of height)² = 96.1

Coefficient of correlation is given by

`r= (875.2)/( √(10308*96.1) )= 0.879`

The value 0.879 shows a strong positive correlation