1. Which linear relationship has a stronger correlation, a relationship with r=−0.84 or a relationship with r=0.72? Explain. (1 point)
A. The linear relationship with r=0.72 because it has the smallest absolute value.
B. The linear relationship with r=0.72 because it is the closest to 1.
C. The linear relationship with r=−0.84 because it is the closest to −1.
D. The linear relationship with r=−0.84 because the absolute value of r is closer to 1.
2. A biology student is investigating the claim that the temperature can be predicted by counting cricket chirps. He collected some data and found that the correlation coefficient is 0.82. What kind of correlation exists between the variables? (1 point)
A. a strong negative correlation
B. a strong positive correlation
C. a weak positive correlation
D. a weak negative correlation
3. A researcher monitoring the power from the solar panels on a Mars lander has developed a regression model for the available power output in watts, P, against the number of days since landing, d, to help predict the change in power due to dust accumulation. The correlation coefficient is −0.92. What kind of correlation exists between the variables?
A. a weak positive correlation
B. a strong positive correlation
C. a weak negative correlation
D. a strong negative correlation
4. A real estate agent has developed a linear model for the price of a house, P, in dollars in terms of the area, A, in square feet for the homes in a certain neighborhood. The correlation coefficient for the relationship is 0.42. What kind of correlation exists between the variables? (1 point)
A. a strong positive correlation
B. a weak negative correlation
C. a weak positive correlation
D. a strong negative correlation
5. A fisherman has collected data about the length in inches, L, and weight in ounces, w, of fish he has caught. The data are in the table below. What is the correlation coefficient?
L 12 16 16 17 18 19 21 23
w 8 17 19 20 27 30 37 42
A. r=0.978
B. r=−33.274
C. r=3.283
D. r=0.989
D. r=0.989.
1. The correct answer is D. The linear relationship with r=−0.84 because the absolute value of r is closer to 1.
Explanation: The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. In this case, a correlation coefficient of -0.84 indicates a strong negative linear relationship. Absolute value represents the magnitude of a number without considering its sign. Since the absolute value of -0.84 is closer to 1 than the absolute value of 0.72, the linear relationship with r=-0.84 has a stronger correlation.
2. The correct answer is B. a strong positive correlation.
Explanation: A correlation coefficient of 0.82 indicates a strong positive linear correlation. This means that as the number of cricket chirps increases, the temperature also increases. The positive sign indicates a direct relationship between the two variables.
3. The correct answer is D. a strong negative correlation.
Explanation: A correlation coefficient of -0.92 indicates a strong negative linear correlation. This means that as the number of days since landing increases, the available power output from the solar panels decreases. The negative sign indicates an inverse relationship between the two variables.
4. The correct answer is C. a weak positive correlation.
Explanation: A correlation coefficient of 0.42 indicates a weak positive linear correlation. This means that as the area (A) of the house in square feet increases, the price (P) of the house in dollars also tends to increase. The positive sign indicates a direct relationship between the two variables.
5. The correct answer is A. r=0.978.
Explanation: To find the correlation coefficient, we can use the formula:
r = Σ((L - mean(L)) * (w - mean(w))) / sqrt(Σ(L - mean(L))^2 * Σ(w - mean(w))^2)
Using the provided data, we can calculate the mean length (mean(L)) and mean weight (mean(w)):
mean(L) = (12 + 16 + 16 + 17 + 18 + 19 + 21 + 23) / 8 = 18
mean(w) = (8 + 17 + 19 + 20 + 27 + 30 + 37 + 42) / 8 = 26
Next, we calculate the numerator and denominators of the formula:
numerator = (12 - 18) * (8 - 26) + (16 - 18) * (17 - 26) + ... + (23 - 18) * (42 - 26)
denominator(L) = (12 - 18)^2 + (16 - 18)^2 + ... + (23 - 18)^2
denominator(w) = (8 - 26)^2 + (17 - 26)^2 + ... + (42 - 26)^2
Plugging these values into the formula, we find:
r = numerator / sqrt(denominator(L) * denominator(w))
= -33.2 / sqrt(1688 * 4102)
≈ 0.978
Therefore, the correlation coefficient is approximately r=0.978.
1. To determine which linear relationship has a stronger correlation, we need to compare the correlation coefficients, r. The correlation coefficient measures the strength and direction of the relationship between two variables. In this case, we have r=-0.84 and r=0.72.
To compare the strength of the correlation, we look at the absolute value of r. The closer the absolute value of r is to 1, the stronger the correlation. In this case, the absolute value of -0.84 is 0.84, and the absolute value of 0.72 is 0.72. Since 0.84 is larger than 0.72, it means that the linear relationship with r=-0.84 has a stronger correlation.
Therefore, the correct answer is C. The linear relationship with r=-0.84 because it is the closest to -1.
2. In this question, the correlation coefficient is given as 0.82. To determine the type of correlation, we need to consider the sign and the magnitude of the correlation coefficient.
The sign of the correlation coefficient (+ or -) indicates the direction of the relationship. If it is positive, it means there is a positive relationship, and if it is negative, it means there is a negative relationship. In this case, the correlation coefficient is positive (0.82), indicating a positive relationship between the variables.
The magnitude of the correlation coefficient (between 0 and 1) indicates the strength of the relationship. The closer the absolute value of the correlation coefficient is to 1, the stronger the correlation. In this case, the absolute value of 0.82 is 0.82, which is relatively close to 1.
Based on this, we can conclude that there is a relatively strong positive correlation between the temperature and the number of cricket chirps.
Therefore, the correct answer is B. a strong positive correlation.
3. In this question, the correlation coefficient is given as -0.92. To determine the type of correlation, we follow the same process as in question 2.
The sign of the correlation coefficient (-) indicates a negative relationship between the variables.
The magnitude of the correlation coefficient (between 0 and 1) indicates the strength of the relationship. In this case, the absolute value of -0.92 is 0.92, which is relatively close to 1.
Based on this, we can conclude that there is a relatively strong negative correlation between the available power output and the number of days since landing.
Therefore, the correct answer is D. a strong negative correlation.
4. In this question, the correlation coefficient is given as 0.42. Again, we follow the same process as in questions 2 and 3 to determine the type of correlation.
The sign of the correlation coefficient (+) indicates a positive relationship between the variables.
The magnitude of the correlation coefficient (between 0 and 1) indicates the strength of the relationship. In this case, the absolute value of 0.42 is 0.42, which is relatively closer to 0 than to 1.
Based on this, we can conclude that there is a relatively weak positive correlation between the house price and the area.
Therefore, the correct answer is C. a weak positive correlation.
5. To find the correlation coefficient in this question, we need to calculate it based on the given data.
First, we calculate the mean (average) of the lengths (L) and the weights (w).
Mean of L = (12 + 16 + 16 + 17 + 18 + 19 + 21 + 23) / 8 = 18
Mean of w = (8 + 17 + 19 + 20 + 27 + 30 + 37 + 42) / 8 = 25.5
Next, we calculate the deviations of each value from the mean for both L and w.
Deviation of L = L - Mean of L
Deviation of w = w - Mean of w
The deviations are:
L: -6, -2, -2, -1, 0, 1, 3, 5
w: -17.5, -8.5, -6.5, -5.5, 1.5, 4.5, 11.5, 16.5
Then, we calculate the product of the deviations for each data point and sum them up.
Sum of (Deviation of L * Deviation of w) = (-6 * -17.5) + (-2 * -8.5) + (-2 * -6.5) + (-1 * -5.5) + (0 * 1.5) + (1 * 4.5) + (3 * 11.5) + (5 * 16.5) = 331
Next, we calculate the sum of the squares of the deviations for each variable.
Sum of (Deviation of L)^2 = (-6)^2 + (-2)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 3^2 + 5^2 = 70
Sum of (Deviation of w)^2 = (-17.5)^2 + (-8.5)^2 + (-6.5)^2 + (-5.5)^2 + 1.5^2 + 4.5^2 + 11.5^2 + 16.5^2 = 1015
Finally, we can calculate the correlation coefficient using the formula:
r = (Sum of (Deviation of L * Deviation of w)) / sqrt((Sum of (Deviation of L)^2) * (Sum of (Deviation of w)^2))
r = 331 / sqrt(70 * 1015) ≈ 0.978
Therefore, the correct answer is A. r=0.978.