1. Construct a scatter plot using excel for the given data. Determine whether there is a positive linear correlation, negative linear correlation, or no linear correlation. Complete the table and find the correlation coefficient r.
a. The data below are the ages and systolic blood pressure (measured in millimeters of mercury) of 9 randomly selected adults.
Age, x / 38 / 41 / 45 / 48 / 51 / 53 / 57 / 61 / 65Pressure, y / 116 / 120 / 123 / 131 / 142 / 145 / 148 / 150 / 152
Part 1: Scatter plot (2.5 points)
Part 2: Type of correlation (positive linear correlation, negative linear correlation, or no linear correlation) (2 points)
The scatter plot shows that as age increases pressure also increases. Thus there exist a positive linear correlation between age and pressure.
Part 3: Complete the table and find the correlation coefficient r.
x / y / xy / x2 / y238 / 116 / 4408 / 1444 / 13456
41 / 120 / 4920 / 1681 / 14400
45 / 123 / 5535 / 2025 / 15129
48 / 131 / 6288 / 2304 / 17161
51 / 142 / 7242 / 2601 / 20164
53 / 145 / 7685 / 2809 / 21025
57 / 148 / 8436 / 3249 / 21904
61 / 150 / 9150 / 3721 / 22500
65 / 152 / 9880 / 4225 / 23104
459 / 1227 / 63544 / 24059 / 168843
Use the last row of the table to show the column totals.
From the given data we have
=9, = 459, =1227, =63544, = 24059 and = 168843
We have,
=
= 0.95969
2. Construct a scatter plot using excel for the given data. Determine whether there is a positive linear correlation, negative linear correlation, or no linear correlation. Complete the table and find the correlation coefficient r. The data for x and y is shown below.
x / 11 / -6 / 8 / -3 / -2 / 1 / 5 / -5 / 6 / 7y / -5 / -3 / 4 / 1 / -1 / -2 / 0 / 2 / 3 / -4
Part 1: Scatter plot
Part 2: Type of correlation (positive linear correlation, negative linear correlation, or no linear correlation)
Here the scatter plot does not show any strong pattern. Thus we may conclude that x and y are uncorrelated.
Part 3: Complete the table and find the correlation coefficient r.
Answer
x / y / xy / x2 / y211 / -5 / -55 / 121 / 25
-6 / -3 / 18 / 36 / 9
8 / 4 / 32 / 64 / 16
-3 / 1 / -3 / 9 / 1
-2 / -1 / 2 / 4 / 1
1 / -2 / -2 / 1 / 4
5 / 0 / 0 / 25 / 0
-5 / 2 / -10 / 25 / 4
6 / 3 / 18 / 36 / 9
7 / -4 / -28 / 49 / 16
22 / -5 / -28 / 370 / 85
Use the last row of the table to show the column totals.
From the given data we have
=10, = 22, = -5, = -28, = 370 and = 85
We have,
=
= -0.10437
3. Using the r calculated in problem 1 test the significance of the correlation coefficient using a = 0.01 and the claim k = 0. Use the 7-steps hypothesis test shown at the end of this project. (References: example 7 page 505; end of section exercises 23 - 28 pages 510 - 511)
Answer:
Let k be the population correlation coefficient.
1. H0: k = 0
Ha: k ≠ 0
2. a = 0.01
3. We have,
=
= 9.03355
4. Since a =0.01, from Student’s t table with (n-2) = 7 degrees of freedom, the critical value is t0 = 3.499
5. Rejection region:
Reject Ho if t < -3.499 or t > 3.499
6. Decision:
Here, t = 9.03355 > 3.499
So we reject the null hypothesis Ho.
7. Interpretation:
Thus the correlation between age and pressure is statistically significant at 1% level of significance.
[If the alternative hypothesis is Ha: k > 0, use the following
Let k be the population correlation coefficient.
1. H0: k = 0
Ha: k > 0
2. a = 0.01
3. We have,
=
= 9.03355
4. Since a =0.01, from Student’s t table with (n-2) = 7 degrees of freedom, the critical value is t0 = 2.998
5. Rejection region:
Reject Ho if t > 2.998
6. Decision:
Here, t = 9.03355 > 2.998
So we reject the null hypothesis Ho.
7. Interpretation:
Thus there exist a positive correlation between age and pressure at 1% level of significance.]
Section 9.2: Linear Regression
(References: example 1 - 3 pages 514 - 516; end of section exercises 13 -22 pages 518 - 520)
4. The data below are the ages and systolic blood pressure (measured in millimeters of mercury) of 9 randomly selected adults.
Age, x / 38 / 41 / 45 / 48 / 51 / 53 / 57 / 61 / 65Pressure, y / 116 / 120 / 123 / 131 / 142 / 145 / 148 / 150 / 152
a. Find the equation of the regression line for the given data. Round the line values to the nearest two decimal places
Let y denote the pressure and x denote the age. Assume that x and y are linearly related. Let be the suggested linear relationship.
By the method of least squares the estimates of and are given by,
and
From the given data we have
=9, = 459, =1227, =63544, = 24059 and = 168843
We have,
=
= 1.48769
= 1.49
=
= 60.46103
= 60.46
Thus the fitted regression equation is
Pressure = 60.46 + 1.49 Age
b. Using the equation found in part a, predict the pressure when the age is 50. Round to the nearest year
If age = 50, the predicted value of pressure is
Pressure = 60.46 + 1.49*50
= 60.46 + 74.5
= 143.96