1. For the following scores,
X Y
1 6
4 1
1 4
1 3
3 1
Sketch a scatter plot and estimate the value of the
Pearson correlation.
b. Compute the Pearson correlation
2. For the following set of scores,
X Y
6 4
3 1
5 0
6 7
4 2
6 4
a. Compute the Pearson correlation.
b. Add 2 points to each X value and compute the correlation for the modified scores. How does adding a constant to every score affect the value
of the correlation?
c. Multiply each of the original X values by 2 and compute the correlation for the modified scores. How does multiplying each score by a constant
affect the value of the correlation?
3. To simplify the weight variable, the women are classified into five categories that measure actual weight relative to height, from 1 � thinnest
to 5 � heaviest. Income figures are annual income (in thousands), rounded to the nearest $1,000.
a. Calculate the Pearson correlation for these data.
b. Is the correlation statistically significant? Use a two-tailed test with � � .05.
Weight (X) Income (Y)
1 125
2 78
4 49
3 63
5 35
2 84
5 38
3 51
1 93
4 44
4. Assume a two-tailed test with � � .05.
(Note: The table does not list all the possible df values. Use the sample size corresponding to the appropriate
df value that is listed in the table.)
a. A correlation of r � 0.30.
b. A correlation of r � 0.25.
c. A correlation of r � 0.20.
5. A professor obtains SAT scores and freshman grade point averages (GPAs) for a group of n � 15 college students. The SAT scores have a mean of M � 580
with SS � 22,400, and the GPAs have a mean of 3.10 with SS � 1.26, and SP � 84.
a. Find the regression equation for predicting GPA from SAT scores.
b. What percentage of the variance in GPAs is accounted for by the regression equation? (Compute the correlation, r, then find r2.)
c. Does the regression equation account for a significant portion of the variance in GPA? use a=.05 to evaluate the F-ratio.
6. a. One set of 20 pairs of scores, X and Y values, produces a correlation of r � 0.70. If SSY � 150, find the standard error of estimate for the regression line.
b. A second set of 20 pairs of X and Y values produces of correlation of r � 0.30. If SSY � 150, find the standard error of estimate for the regression line.
7. A researcher obtained the following multiple-regression equation using two predictor variables:
Yˆ � 0.5X1 � 4.5X2 � 9.6. Given that SSY � 210, the
SP value for X1 and Y is 40, and the SP value for X2 and Y is 9, find R2, the percentage of variance accounted for by the equation.
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5. A professor obtains SAT scores and freshman grade point averages (GPAs) for a group of n � 15 college students. The SAT scores have a mean of M � 580
with SS � 22,400, and the GPAs have a mean of 3.10 with SS � 1.26, and SP � 84.
a. Find the regression equation for predicting GPA from SAT scores.
b. What percentage of the variance in GPAs is accounted for by the regression equation? (Compute the correlation, r, then find r2.)
c. Does the regression equation account for a significant portion of the variance in GPA? use a=.05 to evaluate the F-ratio
1. To sketch a scatter plot, you would plot the values of X on the horizontal axis and the values of Y on the vertical axis. For example, for the first set of scores (1, 6), you would plot a point at (1, 6). Similarly, you would plot points for the remaining scores. Once you have all the points plotted, you can visually estimate the correlation by observing the overall trend of the points. If the points tend to form a straight line sloping upward from left to right, it indicates a positive correlation. If the points tend to form a straight line sloping downward from left to right, it indicates a negative correlation. If there is no clear pattern, it indicates no correlation.
To compute the Pearson correlation, you can use the following formula:
r = Σ((X - mean(X))(Y - mean(Y))) / √(Σ(X - mean(X))^2 * Σ(Y - mean(Y))^2)
where Σ represents the sum, X and Y are the individual scores, and mean(X) and mean(Y) are the mean values of X and Y respectively. Calculate the sums and means for X and Y, then plug the values into the formula to obtain the Pearson correlation coefficient (r).
2. To compute the Pearson correlation for the given set of scores, you can follow the same steps as described in step 1 above.
a. For the original set of scores, calculate the mean values of X and Y, then use the formula to compute the Pearson correlation coefficient (r).
b. To compute the correlation for the modified scores (adding 2 points to each X value), you would add 2 to each X value and calculate new mean values of X and Y. Then, use the formula to compute the Pearson correlation coefficient (r) for the modified scores. Compare this value to the correlation coefficient computed in part a to see how adding a constant affects the correlation.
c. To compute the correlation for the modified scores (multiplying each X value by 2), you would multiply each X value by 2 and calculate new mean values of X and Y. Then, use the formula to compute the Pearson correlation coefficient (r) for the modified scores. Compare this value to the correlation coefficient computed in part a to see how multiplying each score by a constant affects the correlation.
3. To calculate the Pearson correlation for the given data, you can follow the same steps as described in step 1 above.
a. Calculate the mean values of X and Y, then use the formula to compute the Pearson correlation coefficient (r).
b. To determine if the correlation is statistically significant, you would need to perform a hypothesis test. Using a two-tailed test with a significance level of α = 0.05, you can compare the computed correlation coefficient (r) to the critical values from the t-distribution with n-2 degrees of freedom. If the computed correlation coefficient falls outside the critical region, then the correlation is considered statistically significant.
4. To determine the critical values for various correlations, you would need to consult a table of critical values for the Pearson correlation coefficient. The critical values depend on the sample size (degrees of freedom). Find the appropriate degrees of freedom for the given sample size, then locate the critical value corresponding to a significance level of α = 0.05 in the table. Compare the computed correlation coefficient to the critical value to determine if the correlation is statistically significant.
5. To find the regression equation and evaluate its significance, you can follow these steps:
a. Find the regression equation for predicting GPA from SAT scores: The regression equation can be found using the formula:
Y = a + bX
where Y is the predicted GPA, X is the SAT score, a is the intercept, and b is the slope. The formula for b is:
b = SP / SSX
where SP is the sum of the cross-products of X and Y, and SSX is the sum of squares for X. The formula for a is:
a = mean(Y) - b * mean(X)
Compute the values of SP and SSX, then calculate b and a using the formulas. This will give you the regression equation.
b. To determine the percentage of variance accounted for by the regression equation, you can compute the correlation coefficient (r) using the formula:
r = √(SSR / SST)
where SSR is the sum of squares for regression (SSR = b * SP) and SST is the total sum of squares (SST = SSY). Once you have the correlation coefficient, square it (r^2) to find the proportion of variance accounted for by the regression equation.
c. To test the significance of the regression equation, you would need to perform an F-test. Calculate the F-ratio using the formula:
F = (SSR / p) / (SSE / (n - p - 1))
where SSR is the sum of squares for regression, SSE is the sum of squares for error (SSE = SST - SSR), p is the number of predictor variables (in this case, 1), and n is the sample size. Compare the computed F-ratio to the critical value from the F-distribution with p and (n - p - 1) degrees of freedom to evaluate the significance.
6. To find the standard error of estimate for a regression line, you can use the formula:
SE = √((SSY - r^2 * SSX) / (n - 2))
where SE is the standard error of estimate, SSY is the sum of squares for Y, SSX is the sum of squares for X, r is the correlation coefficient, and n is the sample size. Plug in the values and calculate the standard error of estimate for each set of 20 pairs of X and Y values.
7. To find R2, the percentage of variance accounted for by the multiple regression equation, you can use the formula:
R2 = SSR / SST
where SSR is the sum of squares for regression and SST is the total sum of squares. Calculate SSR by summing the cross-products of the predictor variables and their respective regression coefficients. Calculate SST by summing the squared deviations of Y from its mean. Once you have the value of R2, you can interpret it as the proportion of variance in Y that is accounted for by the multiple regression equation.