a) We can see from the table that with the increase in value of x , there is decrease in value of y , hence we can say that coefficient of correlation is negative b) Equation of least squares is y = − 2.9866 x + 24.92 and correlation coefficient r = − 0.9995 Plot is c) Predicted value of y at x = 2.4 is 17.752 Given information: Five points x −4 −3 −1 3 5 y 3.7 33.7 27.5 16.4 9.8 Formula used: Slope, m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 Y axis intercept, b = Y ¯ − m X ¯ Correlation Coefficient, r = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 ∑ i = 1 n ( y i − Y ¯ ) 2 Where x i and y i are the i th entry of x and y ; X ¯ and Y ¯ are means of x and y Calculation: Step 1: Calculate the mean of x and y X ¯ = ( − 4 ) + ( − 3 ) + ( − 1 ) + 3 + 5 5 = 0 Y ¯ = 37.2 + 33.7 + 27.5 + 16.4 + 9.8 5 = 24.92 Step 2: Plot the table as shown i x i y i x i − X ¯ y i − Y ¯ ( x i − X ¯ ) 2 ( x i − X ¯ ) ( y i − Y ¯ ) ( y i − Y ¯ ) 2 1 -4 37.2 -4 12.28 16 -49.12 150.7984 2 -3 33.7 -3 8.78 9 -26.34 77.0884 3 -1 27.5 -1 2.58 1 -2.58 6.6564 4 3 16.4 3 -8.52 9 -25.56 72.5904 5 5 9.8 5 -15.12 25 -75.6 228.6144 ∑ i = 1 n ( x i − X ¯ ) 2 = 60 ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) = − 179.2 ∑ i = 1 n ( y i − Y ¯ ) 2 = 535.7480 Step 3: Calculate the slope m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 = − 179.2 60 ≈ − 2.9866 Step 4: Calculate y intercept b = Y ¯ − m X ¯ = 24.92 − ( − 2.9866 × 0 ) ≈ 24.92 The slope of the line is − 2.9866 and y intercept is 24.92 Using slope intercept form, y = m x + b ,equation is y = − 2.9866 x + 24.92 ; Step 5: Draw the scatter plot Step 6: Calculate the Correlation coefficient r = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 ∑ i = 1 n ( y i − Y ¯ ) 2 = − 179.2 60 × 535.7480 ≈ − 0.9995 Step 7: Prediction for x = 2.4 ; plug x = 2.4 in y = − 2.9866 x + 24.92 y = − 2.9866 ( 2.4 ) + 24.92 = 17.752
a) We can see from the table that with the increase in value of x , there is decrease in value of y , hence we can say that coefficient of correlation is negative b) Equation of least squares is y = − 2.9866 x + 24.92 and correlation coefficient r = − 0.9995 Plot is c) Predicted value of y at x = 2.4 is 17.752 Given information: Five points x −4 −3 −1 3 5 y 3.7 33.7 27.5 16.4 9.8 Formula used: Slope, m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 Y axis intercept, b = Y ¯ − m X ¯ Correlation Coefficient, r = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 ∑ i = 1 n ( y i − Y ¯ ) 2 Where x i and y i are the i th entry of x and y ; X ¯ and Y ¯ are means of x and y Calculation: Step 1: Calculate the mean of x and y X ¯ = ( − 4 ) + ( − 3 ) + ( − 1 ) + 3 + 5 5 = 0 Y ¯ = 37.2 + 33.7 + 27.5 + 16.4 + 9.8 5 = 24.92 Step 2: Plot the table as shown i x i y i x i − X ¯ y i − Y ¯ ( x i − X ¯ ) 2 ( x i − X ¯ ) ( y i − Y ¯ ) ( y i − Y ¯ ) 2 1 -4 37.2 -4 12.28 16 -49.12 150.7984 2 -3 33.7 -3 8.78 9 -26.34 77.0884 3 -1 27.5 -1 2.58 1 -2.58 6.6564 4 3 16.4 3 -8.52 9 -25.56 72.5904 5 5 9.8 5 -15.12 25 -75.6 228.6144 ∑ i = 1 n ( x i − X ¯ ) 2 = 60 ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) = − 179.2 ∑ i = 1 n ( y i − Y ¯ ) 2 = 535.7480 Step 3: Calculate the slope m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 = − 179.2 60 ≈ − 2.9866 Step 4: Calculate y intercept b = Y ¯ − m X ¯ = 24.92 − ( − 2.9866 × 0 ) ≈ 24.92 The slope of the line is − 2.9866 and y intercept is 24.92 Using slope intercept form, y = m x + b ,equation is y = − 2.9866 x + 24.92 ; Step 5: Draw the scatter plot Step 6: Calculate the Correlation coefficient r = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 ∑ i = 1 n ( y i − Y ¯ ) 2 = − 179.2 60 × 535.7480 ≈ − 0.9995 Step 7: Prediction for x = 2.4 ; plug x = 2.4 in y = − 2.9866 x + 24.92 y = − 2.9866 ( 2.4 ) + 24.92 = 17.752
Definition Definition Statistical measure used to assess the strength and direction of relationships between two variables. Correlation coefficients range between -1 and 1. A coefficient value of 0 indicates that there is no relationship between the variables, whereas a -1 or 1 indicates that there is a perfect negative or positive correlation.
Chapter 2.1, Problem 112E
To determine
To determine:
a) We can see from the table that with the increase in value of x
, there is decrease in value of y
, hence we can say that coefficient of correlation is negative
b) Equation of least squares is y=−2.9866x+24.92
and correlation coefficientr=−0.9995
Plot is
c) Predicted value of y at x=2.4
is 17.752
Given information: Five points
x
−4
−3
−1
3
5
y
3.7
33.7
27.5
16.4
9.8
Formula used: Slope, m=∑i=1n(xi−X¯)(yi−Y¯)∑i=1n(xi−X¯)2
1. Given that h(t) = -5t + 3 t². A tangent line H to the function h(t) passes through
the point (-7, B).
a. Determine the value of ẞ.
b. Derive an expression to represent the gradient of the tangent line H that is
passing through the point (-7. B).
c. Hence, derive the straight-line equation of the tangent line H
2. The function p(q) has factors of (q − 3) (2q + 5) (q) for the interval -3≤ q≤ 4.
a. Derive an expression for the function p(q).
b. Determine the stationary point(s) of the function p(q)
c. Classify the stationary point(s) from part b. above.
d. Identify the local maximum of the function p(q).
e. Identify the global minimum for the function p(q).
3. Given that m(q)
=
-3e-24-169 +9
(-39-7)(-In (30-755
a. State all the possible rules that should be used to differentiate the function
m(q). Next to the rule that has been stated, write the expression(s) of the
function m(q) for which that rule will be applied.
b. Determine the derivative of m(q)
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