1. What is the correlation coefficient for the following data set? Round to three decimal places.
x 3.0 7.3 4.5 3.3 7.6
y 9.8 8.3 8.6 9.8 8.8
r=
2.What is the correlation coefficient for the following data set? Round to three decimal places.
x 1.5 3.9 4.6 4.9 6.5 6.6 7.3 7.7
y −11 −1 −12 −5 −9 −28 −15 −29
r=
3 .A student is investigating if the amount of free time per week affects grades. He randomly surveys the students at his school and finds a linear relationship between the variables with a correlation coefficient of 0.62. Can the student conclude that more free time causes an improvement in
grades? Explain.(1 point)
A. Yes. The student controlled the grades, so the correlation implies causation.
B. No. The student did not control the grades, so the data only show correlation.
C.Yes. The student controlled the amount of free time, so the correlation implies causation.
D. No. The student did not control the amount of free time, so the data only show correlation.
4.What are the requirements for an experiment to be able to show that changes in one variable cause changes in another?(4 points)
1. To calculate the correlation coefficient, we can use the formula:
r = [(n * Σxy) - (Σx * Σy)] / [√((n * Σx^2) - (Σx)^2) * √((n * Σy^2) - (Σy)^2)]
Here, n represents the number of data points, Σxy represents the sum of the product of x and y values, Σx represents the sum of x values, and Σy represents the sum of y values.
Plugging in the values from the given data set:
n = 5
Σxy = (3.0 * 9.8) + (7.3 * 8.3) + (4.5 * 8.6) + (3.3 * 9.8) + (7.6 * 8.8) = 231.67
Σx = 3.0 + 7.3 + 4.5 + 3.3 + 7.6 = 25.7
Σy = 9.8 + 8.3 + 8.6 + 9.8 + 8.8 = 45.3
Using these values in the formula, we get:
r = [(5 * 231.67) - (25.7 * 45.3)] / [√((5 * Σx^2) - (25.7)^2) * √((5 * Σy^2) - (45.3)^2)]
Now, we need to calculate Σx^2 and Σy^2:
Σx^2 = (3.0)^2 + (7.3)^2 + (4.5)^2 + (3.3)^2 + (7.6)^2 = 134.99
Σy^2 = (9.8)^2 + (8.3)^2 + (8.6)^2 + (9.8)^2 + (8.8)^2 = 396.79
Substituting these values into the formula, we get:
r = [(5 * 231.67) - (25.7 * 45.3)] / [√((5 * 134.99) - (25.7)^2) * √((5 * 396.79) - (45.3)^2)]
Solving this equation will give the correlation coefficient value.
Similarly, for the second data set, you can follow the same steps to calculate the correlation coefficient.
2. I would need the values of x and y for the second data set to calculate the correlation coefficient.
3. The student cannot conclude that more free time causes an improvement in grades based solely on the correlation coefficient.
The correlation coefficient only indicates the strength and direction of the linear relationship between the variables, in this case, the amount of free time and the grades. It does not prove causation.
To establish causation, the student would need to conduct a controlled experiment where they manipulate the amount of free time students have and measure its impact on the grades. Without controlling the grades or randomly assigning students to different free time conditions, it is not possible to infer causation based solely on the correlation coefficient.
Therefore, the correct answer is B. No. The student did not control the grades, so the data only show correlation.
4. To be able to show that changes in one variable cause changes in another, experimental studies should adhere to the following requirements:
- Control group: There should be a control group that does not receive the experimental treatment or intervention. This allows for a comparison between the experimental group and the control group to assess the impact of the independent variable.
- Random assignment: Participants or subjects should be randomly assigned to either the experimental or control group. Random assignment helps ensure that any differences observed between the groups are due to the independent variable and not pre-existing differences between the individuals.
- Manipulation of the independent variable: The researcher must actively manipulate or change the independent variable to observe its impact on the dependent variable. This allows for causal conclusions about the relationship between the variables.
- Measurement of the dependent variable: The dependent variable, which is the variable being impacted or influenced by the independent variable, should be measured accurately and reliably.
By meeting these requirements, an experiment can provide evidence for causal relationships between variables.
1. To find the correlation coefficient for the given data set, you can use the formula for Pearson's correlation coefficient (r).
First, you need to calculate the mean (average) for both x and y. For x, add up all the values and divide by the number of values (5 in this case). For y, do the same.
Then, for each value of x, subtract the mean of x and for each corresponding value of y, subtract the mean of y. Multiply these differences together and sum them all up. This will give you the numerator of the correlation coefficient formula.
Next, calculate the standard deviation for both x and y. To calculate the standard deviation, sum the squares of the differences between each value and the mean, divide by the number of values, and take the square root.
Finally, divide the numerator by the product of the standard deviations for x and y.
The result will be the correlation coefficient (r), round it to three decimal places.
2. To find the correlation coefficient for the given data set, you can follow the same steps as described above for the first question.
3. The student cannot conclude that more free time causes an improvement in grades based solely on the correlation coefficient of 0.62. The correlation coefficient only measures the strength and direction of the linear relationship between the variables. It does not establish causation.
To establish causation, additional experimental design and control are necessary. The student would need to conduct a controlled experiment where they manipulate the amount of free time given to different groups of students and measure the effect on their grades. By controlling other variables and randomly assigning students to groups, the student can minimize confounding factors and establish a causal relationship between free time and grades.
Therefore, the correct answer is D. No. The student did not control the amount of free time, so the data only show correlation.
4. To show that changes in one variable cause changes in another, an experiment needs to meet the following requirements:
a) Control: The experiment must have control over the variables involved. This means that the experimenter can manipulate and control the independent variable (the variable being changed) while keeping all other relevant variables constant.
b) Randomization: Participants or subjects should be randomly assigned to different groups or conditions. This helps to minimize biases and ensure that any observed effects are not due to preexisting differences between groups.
c) Replication: The experiment should be replicated multiple times to ensure that the findings are consistent and not due to chance. Replication also helps to establish the reliability of the results.
d) Measurement: The experiment should have reliable and valid measures for both the independent variable and the dependent variable (the variable being measured or observed for change). This helps to accurately track and quantify any changes.
By meeting these requirements, an experiment can provide evidence to support the claim that changes in one variable cause changes in another.
1. r = -0.426
2. r = -0.915
3. D. No. The student did not control the amount of free time, so the data only show correlation. Correlation does not imply causation, and in this case, there could be other factors affecting both free time and grades. Without controlling for those factors, it is not possible to conclude that more free time causes improved grades.
4. (1) Manipulation of the independent variable, (2) control of extraneous variables, (3) random assignment to conditions, and (4) measurement of the dependent variable.