The data given below indicate the existence of a linear relationship between the x and y variables. Suppose an analyst who prepared the solutions and carried out the RI measurements was not skilled and as a result of poor technique, allowed intermediate errors to appear. The results are the following: Concentration of solution in percent (x) 10 26 33 50 61 Refractive indices (y) 1.497 1.493 1.485 1.478 1.477 Step 1. Carefully plot the given x and y values (from the table) on a regular graphing paper. Label then connect the points to observe a zigzag plot due to the scattered points. Step 2: Copy and fill the table given below: x (x - x̄) (x - x̄) 2 y (y - ȳ) (y - ȳ) 2 (x - x̄) (y - ȳ) 10 1.497 26 1.493 33 1.485 50 1.478 61 1.477 ∑ = ∑ = ∑ = ∑ = ∑ = ∑ = ∑ = x̄= ∑xi ÷ N x̄= ȳ = ∑yi ÷ N ȳ = Step 3. After completing the table, present following computations and the interpretation. a. Calculate the correlation coefficient (r), using the working formula: r = Σ (x − x ) (y − ȳ) √(Σ(x − x ) 2)(Σ(y − ȳ) 2) o The sign or integer before the value just interprets the value as either directly or inversely proportional. Just consider the rounded-off value if it is equal to 1 or 0. o If the computed r > 0, it means that there is a perfect correlation between the variables. Therefore, there will be a need to continue solving for the slope and y intercept. No further calculations will be made if r < 0 because two variables are completely independent. o HINT: At this point, the answer is -0.97.... The negative sign means that the x and y values are inversely proportional. Only consider “0.97”, rounding it off to the nearest whole number, it is equal to 1. Therefore, there will be a need to continue solving for the slope and y intercept. b. The slope (aka regression coefficient) can be computed using the formula: b = Σ (x − x ) (y − ȳ) Σ(x − x ) 2 o The sign or integer confirms the relationship of the variables either directly or indirectly proportional. c. The y intercept can be solved using the formula: ȳ = y + b (x - x̄) o Note that you are solving for the value of y. o Substitute the values of ȳ and x̄ from the table and the value of x as 0. d. Complete the statement for the interpretation: The plotted x and y values formed a scattered plot. The correlation coefficient was solved as _____, with a _____ (positive or negative) integer signifying a/an ______ (directly or indirectly) proportional relationship of the values. Continue your interpretation if r < 0: Since the correlation coefficient is greater than 0, there is a need to solve for the regression coefficient and y- intercept. The regression coefficient is solved as _____ and the y-intercept as ____.
The data given below indicate the existence of a linear relationship between the x and y variables. Suppose an analyst who prepared the solutions and carried out the RI measurements was not skilled and as a result of poor technique, allowed intermediate errors to appear. The results are the following:
Concentration of solution in percent (x) 10 26 33 50 61
Refractive indices (y) 1.497 1.493 1.485 1.478 1.477
Step 1. Carefully plot the given x and y values (from the table) on a regular graphing paper. Label then connect the points to observe a zigzag plot due to the scattered points.
Step 2: Copy and fill the table given below:
x (x - x̄) (x - x̄)
2 y (y - ȳ) (y - ȳ)
2 (x - x̄) (y - ȳ)
10 1.497
26 1.493
33 1.485
50 1.478
61 1.477
∑ = ∑ = ∑ = ∑ = ∑ = ∑ = ∑ =
x̄= ∑xi ÷ N
x̄=
ȳ = ∑yi ÷ N
ȳ =
Step 3. After completing the table, present following computations and the interpretation.
a. Calculate the
Σ (x − x ) (y − ȳ)
√(Σ(x − x )
2)(Σ(y − ȳ)
2)
o The sign or integer before the value just interprets the value as either directly or inversely proportional. Just consider the rounded-off value if it is
equal to 1 or 0.
o If the computed r > 0, it means that there is a perfect correlation between the variables. Therefore, there will be a need to continue solving for
the slope and y intercept. No further calculations will be made if r < 0 because two variables are completely independent.
o HINT: At this point, the answer is -0.97.... The negative sign means that the x and y values are inversely proportional. Only consider “0.97”, rounding
it off to the nearest whole number, it is equal to 1. Therefore, there will be a need to continue solving for the slope and y intercept.
b. The slope (aka regression coefficient) can be computed using the formula: b = Σ (x − x ) (y − ȳ)
Σ(x − x )
2
o The sign or integer confirms the relationship of the variables either directly or indirectly proportional.
c. The y intercept can be solved using the formula: ȳ = y + b (x - x̄)
o Note that you are solving for the value of y.
o Substitute the values of ȳ and x̄ from the table and the value of x as 0.
d. Complete the statement for the interpretation:
The plotted x and y values formed a
signifying a/an ______ (directly or indirectly) proportional relationship of the values.
Continue your interpretation if r < 0: Since the correlation coefficient is greater than 0, there is a need to solve for the regression coefficient and y-
intercept. The regression coefficient is solved as _____ and the y-intercept as ____.
Trending now
This is a popular solution!
Step by step
Solved in 6 steps with 1 images
