The systolic blood pressure of individuals is thought to be related to both age
ID: 3350904 • Letter: T
Question
The systolic blood pressure of individuals is thought to be related to both age and weight. For a random sample of 11 men, the following data were obtained.
(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)
Relative to its mean, which variable has the greatest spread of data values? Which variable has the smallest spread of data values relative to its mean?
(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)
Which variable (other than x1) has the greatest influence (by itself) on x1? Would you say that both variables x2and x3 show a strong influence on x1? Explain your answer.
What percent of the variation in x1 can be explained by the corresponding variation in x2? Answer the same question for x3. (Use 1 decimal place.)
(c) Perform a regression analysis with x1 as the response variable. Use x2 and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2 and x3 taken together? (Use 1 decimal place.)
%
(d) Look at the coefficients of the regression equation. Write out the regression equation. (Use 3 decimal places.)
Explain how each coefficient can be thought of as a slope.
If age were held fixed, but a person put on 11 pounds, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)
If a person kept the same weight but got 11 years older, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)
(e) Test each coefficient to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)
(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)
Systolic Blood pressuex1 Age (years)
x2 Weight (pounds)
x3 132 52 173 143 59 184 153 67 194 162 73 211 154 64 196 168 74 220 137 54 188 149 61 188 159 65 207 128 46 167 166 72 217
Explanation / Answer
Result:
(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)
x s CV
Descriptive statistics
x1
x2
x3
n
11
11
11
mean
150.09
62.45
195.00
sample standard deviation
13.63
9.11
17.31
coefficient of variation (CV)
9.08%
14.59%
8.88%
Relative to its mean, which variable has the greatest spread of data values? Which variable has the smallest spread of data values relative to its mean?
greatest spread : x2
smallest spread : x3
(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)
r r2
r
r2
x1,x2
0.979
0.958
x1,x3
0.971
0.943
x2,x3
0.946
0.895
Which variable (other than x1) has the greatest influence (by itself) on x1? Would you say that both variables x2and x3 show a strong influence on x1? Explain your answer.
x2; Yes, both have r2 values close to 1.
What percent of the variation in x1 can be explained by the corresponding variation in x2? Answer the same question for x3. (Use 1 decimal place.)
x2 Incorrect: Your answer is incorrect. 95.8%
x3 Incorrect: Your answer is incorrect. 94.3%
(c) Perform a regression analysis with x1 as the response variable. Use x2 and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2 and x3 taken together? (Use 1 decimal place.)
97.7 %
(d) Look at the coefficients of the regression equation. Write out the regression equation. (Use 3 decimal places.)
x1 = 30.994 + 0.861 x2 + 0.335 x3
Explain how each coefficient can be thought of as a slope.
"If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."
If age were held fixed, but a person put on 11 pounds, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)
11*0.8614 = 9.47
If a person kept the same weight but got 11 years older, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)
11*0.3349 =3.68
(e) Test each coefficient to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)
t P value
2 3.47 0.008
3 2.56 0.034
(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)
lower limit upper limit
2 0.40 1.32
3 0.09 0.58
(g) Suppose Michael is 68 years old and weighs 192 pounds. Predict his systolic blood pressure, and find a 90% confidence range for your prediction. (Use 1 decimal place.)
prediction
lower limit 148.3
upper limit 159.4
Can someone please explain to me how to establish if something is significant and what the relationship is between the r-value and the p-value? The questions are below the chart as a guide for the what the questions are asking for.
Regression Analysis
R²
0.977
Adjusted R²
0.971
n
11
R
0.988
k
2
Std. Error
2.318
Dep. Var.
x1
ANOVA table
Source
SS
df
MS
F
p-value
Regression
1,813.9163
2
906.9581
168.76
2.87E-07
Residual
42.9928
8
5.3741
Total
1,856.9091
10
Regression output
confidence interval
variables
coefficients
std. error
t (df=8)
p-value
90% lower
90% upper
Intercept
30.9941
11.9438
2.595
.0319
8.7841
53.2041
x2
0.8614
0.2482
3.470
.0084
0.3998
1.3230
x3
0.3349
0.1307
2.563
.0335
0.0919
0.5778
Predicted values for: x1
90% Confidence Interval
90% Prediction Interval
x2
x3
Predicted
lower
upper
lower
upper
68
192
153.863
150.356
157.371
148.306
159.421
Descriptive statistics
x1
x2
x3
n
11
11
11
mean
150.09
62.45
195.00
sample standard deviation
13.63
9.11
17.31
coefficient of variation (CV)
9.08%
14.59%
8.88%