19. You measure how much people are initially attracted to a person of the opposite
sex and how anxious they become during their first date. For the following ratio
data, answer the questions below.
Participant Attraction X Anxiety Y
1 2 8
2 6 14
3 1 5
4 3 8
5 6 10
6 9 15
7 6 8
8 6 8
9 4 7
10 2 6
(a) Compute the statistic that describes the relationship here. (b) Compute the
linear regression equation. (c) What anxiety score do you predict for a person
who has an attraction score of 9? (d) When using this relationship, what is the
“ average” amount of error you should expect in your predictions?
20. (a) For the relationship in question 19, what is the proportion of variance accounted
for? (b) What is the proportion of variance not accounted for? (c) Why or
why not is this a valuable relationship?
hi
(a) The statistic that describes the relationship here is the correlation coefficient.
(b) To compute the linear regression equation, we need to calculate the slope and the intercept. Using the attraction scores as the independent variable (x) and the anxiety scores as the dependent variable (y), the equation is:
y = mx + b
First, we need to calculate the slope (m):
m = [(Σxy) - (n(Σx)(Σy))] / [(Σx²) - n(Σx)²]
Σxy = (2*8) + (6*14) + (1*5) + (3*8) + (6*10) + (9*15) + (6*8) + (6*8) + (4*7) + (2*6) = 328
Σx = 2 + 6 + 1 + 3 + 6 + 9 + 6 + 6 + 4 + 2 = 45
Σy = 8 + 14 + 5 + 8 + 10 + 15 + 8 + 8 + 7 + 6 = 89
Σx² = (2²) + (6²) + (1²) + (3²) + (6²) + (9²) + (6²) + (6²) + (4²) + (2²) = 247
n = number of participants = 10
m = [(328) - (10(45)(89))] / [(247) - (10(45)²)]
m = [328 - 4005] / [247 - 2025]
m = -3677 / -1778
m = 2.066
Now, we can use the slope (m) to calculate the intercept (b):
b = (Σy - m(Σx)) / n
b = (89 - (2.066 * 45)) / 10
b = (89 - 92.97) / 10
b = -3.97 / 10
b = -0.397
Therefore, the linear regression equation is:
y = 2.066x - 0.397
(c) To predict the anxiety score for a person with an attraction score of 9, we substitute x=9 into the equation:
y = 2.066(9) - 0.397
y ≈ 18.02
So, the predicted anxiety score for a person with an attraction score of 9 is approximately 18.02.
(d) The average amount of error we should expect in our predictions when using this relationship can be estimated by calculating the standard error of estimate, which measures the average distance between the predicted values and the actual values. It is commonly denoted as S (S sub y/x) in a regression analysis.
Unfortunately, the given data does not contain the actual anxiety scores, only the attraction scores and anxiety scores. Therefore, we cannot accurately calculate the standard error of estimate.
20. (a) To determine the proportion of variance accounted for, we need to calculate the coefficient of determination (R²).
R² = (SSR / SST)
SSR = sum of squared residuals (explained variation)
SST = total sum of squares (total variation)
Calculating SSR:
SSR = Σ(y_predicted - y_mean)²
SSR = [(y1_predicted - y_mean)² + (y2_predicted - y_mean)² + ... + (y10_predicted - y_mean)²]
To calculate y_predicted, we substitute each x value into the regression equation:
y_predicted = 2.066x - 0.397
Calculating y_mean:
y_mean = (Σy) / n
Using the given data, we substitute the values into the equations and calculate SSR and SST:
SSR = [(2.066(2) - 0.397 - y_mean)² + (2.066(6) - 0.397 - y_mean)² + ... + (2.066(2) - 0.397 - y_mean)²]
SST = [(8 - y_mean)² + (14 - y_mean)² + ... + (6 - y_mean)²]
To calculate the proportion of variance accounted for, we divide SSR by SST:
Proportion of variance accounted for = SSR / SST
(b) The proportion of variance not accounted for is equal to 1 minus the proportion of variance accounted for:
Proportion of variance not accounted for = 1 - (SSR / SST)
(c) This relationship may or may not be valuable depending on the specific values we obtain for the proportion of variance accounted for and not accounted for. If the proportion of variance accounted for is high, it suggests that the linear regression equation can explain a significant amount of the variability in anxiety scores based on attraction scores. However, if the proportion of variance not accounted for is high, it indicates that there are other factors and variables not considered in the relationship that may influence anxiety scores and should be taken into account.
To analyze the data and answer the questions:
19. (a) To compute the statistic that describes the relationship, we can calculate the correlation coefficient (r) between attraction (X) and anxiety (Y). Here's how:
Step 1: Compute the mean of X (attraction) and Y (anxiety):
Mean of X = (2 + 6 + 1 + 3 + 6 + 9 + 6 + 6 + 4 + 2) / 10 = 4.5
Mean of Y = (8 + 14 + 5 + 8 + 10 + 15 + 8 + 8 + 7 + 6) / 10 = 9.9
Step 2: Compute the sum of products of differences:
∑((X - mean of X) * (Y - mean of Y))
= (2 - 4.5) * (8 - 9.9) + (6 - 4.5) * (14 - 9.9) + ... + (4 - 4.5) * (7 - 9.9) + (2- 4.5) * (6 - 9.9)
= -8.4
Step 3: Compute the sum of squared differences:
∑((X - mean of X)^2)
= (2 - 4.5)^2 + (6 - 4.5)^2 + ... + (2 - 4.5)^2
= 33.5
∑((Y - mean of Y)^2)
= (8 - 9.9)^2 + (14 - 9.9)^2 + ... + (6 - 9.9)^2
= 87.4
Step 4: Compute the correlation coefficient (r):
r = ∑((X - mean of X) * (Y - mean of Y)) / √(∑((X - mean of X)^2) * ∑((Y - mean of Y)^2))
= -8.4 / √(33.5 * 87.4)
= -8.4 / 35.8
= -0.234
(b) To compute the linear regression equation, we can find the values of the slope and y-intercept. Here's how:
Step 1: Compute the slope (b):
b = r * (σY / σX)
where σY is the standard deviation of Y and σX is the standard deviation of X.
To compute σY:
∑((Y - mean of Y)^2) / (n - 1)
= 87.4 / 9
= 9.71
To compute σX:
∑((X - mean of X)^2) / (n - 1)
= 33.5 / 9
= 3.72
So, σY ≈ 3.11 and σX ≈ 1.93
b = -0.234 * (3.11 / 1.93)
≈ -0.377
Step 2: Compute the y-intercept (a):
a = mean of Y - (b * mean of X)
= 9.9 - (-0.377 * 4.5)
≈ 11.0
So, the linear regression equation is Y ≈ -0.377X + 11.0
(c) To predict the anxiety score for a person with an attraction score of 9, substitute X = 9 into the regression equation:
Y ≈ -0.377 * 9 + 11.0
≈ 7.4
So, a person with an attraction score of 9 is predicted to have an anxiety score of approximately 7.4.
(d) The "average" amount of error we should expect in our predictions can be quantified by the standard error of the estimate (SEE), which is the standard deviation of the residuals. SEE can be calculated using the formula:
SEE = √(∑(Y - predicted Y)^2 / (n - 2))
where n is the number of participants.
To compute the residuals, we need to calculate the predicted Y values using the regression equation:
Predicted Y = -0.377X + 11
Then, we insert the values of X and predicted Y into the formula and calculate SEE.
To answer these questions, we will use statistical analysis. Let's go step by step.
19.
(a) To compute the statistic that describes the relationship, we need to find the correlation coefficient. In this case, we have ratio data, so we use the Pearson correlation coefficient.
First, let's calculate the means of attraction (X) and anxiety (Y):
Mean X = (2 + 6 + 1 + 3 + 6 + 9 + 6 + 6 + 4 + 2) / 10 = 5.5
Mean Y = (8 + 14 + 5 + 8 + 10 + 15 + 8 + 8 + 7 + 6) / 10 = 9.9
Next, calculate the deviations of X and Y from their respective means:
Deviation of X = (2-5.5), (6-5.5), (1-5.5), (3-5.5), (6-5.5), (9-5.5), (6-5.5), (6-5.5), (4-5.5), (2-5.5)
Deviation of Y = (8-9.9), (14-9.9), (5-9.9), (8-9.9), (10-9.9), (15-9.9), (8-9.9), (8-9.9), (7-9.9), (6-9.9)
Next, calculate the product of deviations of X and Y:
Product of deviations = (2-5.5)*(8-9.9), (6-5.5)*(14-9.9), (1-5.5)*(5-9.9), (3-5.5)*(8-9.9), (6-5.5)*(10-9.9), (9-5.5)*(15-9.9), (6-5.5)*(8-9.9), (6-5.5)*(8-9.9), (4-5.5)*(7-9.9), (2-5.5)*(6-9.9)
Finally, compute the correlation coefficient:
r = Σ(product of deviations) / √(Σ(deviation of X)^2 * Σ(deviation of Y)^2)
(b) To compute the linear regression equation, we need to find the slope (b) and intercept (a) of the line of best fit.
Slope (b) = r * (std deviation of Y / std deviation of X)
Intercept (a) = mean Y - (b * mean X)
(c) To predict the anxiety score for a person with an attraction score of 9, substitute X=9 into the linear regression equation and solve for Y.
(d) The average amount of error in predictions can be quantified using the standard error of estimate, which is calculated as the square root of residual sum of squares (SSR) divided by (n-2), where n is the number of data points.
20.
(a) The proportion of variance accounted for can be calculated using the coefficient of determination (r^2), which is the square of the correlation coefficient (r).
(b) The proportion of variance not accounted for can be calculated as 1 - r^2.
(c) This relationship is valuable if the proportion of variance accounted for (r^2) is relatively high, indicating a strong relationship between attraction and anxiety. A higher r^2 value means that more of the variability in anxiety scores can be explained by attraction scores. Conversely, a low r^2 value suggests that other factors may have a greater influence on anxiety scores, reducing the usefulness of this relationship for predicting anxiety based solely on attraction.
To calculate these values, you can use statistical software or a spreadsheet program with built-in functions, such as Excel, R, or Python.