What is the Slope and Y-intercept of this chart? x y 4 10.96 5 10.1 6 9.94 7 9.68 8 9.42 9 9.16 10 10.8 11 9.64 12 11.58 13 10.22 14 12.36 15 13

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Answer to a math question What is the Slope and Y-intercept of this chart? x y 4 10.96 5 10.1 6 9.94 7 9.68 8 9.42 9 9.16 10 10.8 11 9.64 12 11.58 13 10.22 14 12.36 15 13

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Adonis

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104 Answers

To find the slope and y-intercept of the line that best fits the given data, we will use the least squares regression method.

Step 1: Calculate the means of x and y:

\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i=\frac{4+5+6+7+8+9+10+11+12+13+14+15}{12}=9.5

\bar{y}=\frac{1}{n}\sum_{i=1}^ny_i=\frac{10.96+10.1+9.94+9.68+9.42+9.16+10.8+9.64+11.58+10.22+12.36+13}{12}=10.5717

Step 2: Calculate the slope:

m = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}

\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})=(4-9.5)(10.96-10.5717)+\cdots+(15-9.5)(13-10.5717)=28.27

\sum_{i=1}^n(x_i-\bar{x})^2=(4-9.5)^2+(5-9.5)^2+\cdots+(15-9.5)^2=143

m=\frac{28.27}{143}\approx0.1977

Step 3: Calculate the y-intercept:

b=\bar{y}-m\bar{x}=10.5717-(0.1977\times9.5)\approx8.6936

Therefore, the equation of the line that best fits the data is

y=0.1977x+8.6936

.

Answer: The slope is 0.1977 and the y-intercept is 8.6936.

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