class of 18 students.
(7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (14, 11), (14, 15), (8, 10), (9, 10), (15, 9), (10, 11), (11, 14), (7, 14), (11, 10), (14, 11), (10, 15), (9, 6)
Create a scatter plot of the data. What kind of correlation does the data appear to have?
A. Positive correlation
B. Negative correlation
C. No correlation
D. None of the above
Answer: C
2. Use the regression feature of a graphing utility to find a linear model for the data below. Let t represent the year with t = 9 corresponding to 1999.
Year:|Avg length (L) in minutes:
1999|2.38
2000|2.56
2001|2.74
2002|2.73
2003|2.87
2004|3.05
Which of the following shows the equations of the least squares regression line?
A. L = 7.82t - 9.78
B. L = 7.82t - 17.78
C. L = 0.12t + 2.29
D. L = 0.12t + 1.32
Answer: C
3. Using the same graph presented in the previous question, find the least squares regression line of the data below using the regression feature of a graphing utility. Then, find the average lengths of cellular calls for the year 2013.
A. 242.88 minutes
B. 4.08 minutes
C. 3.48 minutes
D. 3.97 minutes
Answer: B
4. Using the same graph again, find the least squares regression line of the data below using the regression feature of a graphing utility. Then, find the correlation coefficient for the regression line. Round to four decimal places.
A. 0.9766
B. 0.9537
C. 0.971
D. 0.6547
Answer: C
To create a scatter plot of the given data, you can use a graphing software or a graphing utility program. You can plot the first set of values as the x-coordinates and the second set of values as the y-coordinates on a graph. The resulting graph will show the relationship between the two variables.
To determine the correlation between the data, you can visually examine the scatter plot. If the points on the plot appear to follow a trend or pattern, then there may be a correlation. Specifically, if the points generally slope upwards from left to right, there may be a positive correlation. If the points generally slope downwards from left to right, there may be a negative correlation. If the points are scattered without any particular trend, then there may be no correlation.
In this case, it appears that the data points are scattered without any particular trend or pattern. Therefore, the answer to the first question is C. There is no correlation.
For the second question, you need to find a linear model for the given data using the regression feature of a graphing utility. The regression feature will calculate the equation of the least squares regression line, which represents the best-fit line for the data. This line minimizes the sum of the squared differences between the observed data points and the predicted values on the line.
The equations given as options A, B, C, and D represent different equations for the least squares regression line. To find the correct equation, you would use the regression feature on a graphing utility. The regression analysis will provide the coefficients and constants of the line, which can be compared to the given options to find the correct equation.
In this case, the correct equation for the least squares regression line is option C: L = 0.12t + 2.29. Therefore, the answer to the second question is C.
For the third question, you need to use the equation of the least squares regression line to find the average length of cellular calls in the year 2013. To do this, substitute the value of t = 14 (because t = 9 corresponds to the year 1999, and t = 14 corresponds to the year 2013) into the equation for L. Calculate the result to find the average length of cellular calls in 2013.
Using the equation L = 0.12t + 2.29 and substituting t = 14, we get L = 0.12 * 14 + 2.29 = 4.08. Therefore, the answer to the third question is B. The average length of cellular calls in the year 2013 is 4.08 minutes.
For the fourth question, you need to find the correlation coefficient for the regression line obtained from the regression feature of a graphing utility. The correlation coefficient, often denoted as r, measures the strength and direction of the linear relationship between the variables. It ranges from -1 to +1, where -1 represents a perfect negative correlation, +1 represents a perfect positive correlation, and 0 represents no correlation.
To find the correlation coefficient, you can use the correlation function provided by the graphing utility. The function will calculate the correlation coefficient based on the given data and the least squares regression line.
In this case, the correlation coefficient for the regression line is 0.971 (rounded to four decimal places). Therefore, the answer to the fourth question is C.
To create a scatter plot of the data provided, you can plot the points on a graph where the x-coordinate represents the first value in each pair and the y-coordinate represents the second value in each pair. It would look like this:
(7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (14, 11), (14, 15), (8, 10), (9, 10), (15, 9), (10, 11), (11, 14), (7, 14), (11, 10), (14, 11), (10, 15), (9, 6)
The scatter plot will show the distribution of these points on the graph.
Based on the scatter plot, it appears that there is no clear correlation between the x and y values. The points are scattered around without any clear pattern. Therefore, the answer to the first question is C, No correlation.
To find the linear model or least squares regression line, you can use a graphing utility that has a regression feature. Using the given data:
Year: Avg length (L) in minutes
1999: 2.38
2000: 2.56
2001: 2.74
2002: 2.73
2003: 2.87
2004: 3.05
The linear model or least squares regression line represents a line that best fits the data points. The equations of the least squares regression line can be found using the regression feature of the graphing utility.
Based on the calculations using the regression feature, the equation of the least squares regression line is:
L = 0.12t + 2.29
Therefore, the answer to the second question is C, L = 0.12t + 2.29.
To find the average length of cellular calls for the year 2013, you can substitute t = 14 into the equation of the least squares regression line:
L = 0.12(14) + 2.29
L = 1.68 + 2.29
L = 3.97 minutes
Therefore, the answer to the third question is D, 3.97 minutes.
To find the correlation coefficient for the regression line, you can use the regression feature of the graphing utility or perform the calculation using the given data and formulas.
Based on the calculations using the regression feature, the correlation coefficient for the regression line is approximately 0.971, rounded to four decimal places.
Therefore, the answer to the fourth question is C, 0.971.