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.

Systolic Blood pressue x1	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

(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
x1			%
x2			%
x3			%

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?

x₂; x₁

x₁; x₃     

x₂; x₃

x₃; x₂

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r². (Use 3 decimal places.)

	r	r2
x1, x2
x1, x3
x2, x3

Which variable (other than x₁) has the greatest influence (by itself) on x₁? Would you say that both variables x₂ and x₃ show a strong influence on x₁? Explain your answer.

x₂; No, both have r² values far from 1.

x₃; Yes, both have r² values close to 1.     

x₂; Yes, both have r² values close to 1.

x₃; No, both have r² values far from 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	%
x3	%

(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.)

x1 =

+  x2

+  x3

Explain how each coefficient can be thought of as a slope.

If we look at all coefficients together, the sum of them can be thought of as the overall "slope" of the regression line.

If we look at all coefficients together, each one can be thought of as a "slope."     

If we hold all explanatory variables as fixed constants, the intercept 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 14 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 14 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.)

	t	P value
β2
β3

Conclusion

We reject both null hypotheses, there is sufficient evidence that β₂ and β₃ differ from 0.

We fail to reject both null hypotheses, there is insufficient evidence that β₂ and β₃ differ from 0.     

We fail to reject both null hypotheses, there is sufficient evidence that β₂ and β₃ differ from 0.

We reject both null hypotheses, there is insufficient evidence that β₂ and β₃ differ from 0.

Why would the outcome of each test help us determine whether or not a given variable should be used in the regression model?

If a coefficient is found to be not different from 0, then it does not contribute to the regression equation.

If a coefficient is found to be different from 0, then it contributes to the regression equation.     

If a coefficient is found to be different from 0, then it does not contribute to the regression equation.

If a coefficient is found to be not different from 0, then it contributes to the regression equation.

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

	lower limit	upper limit
β2
β3

(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
upper limit

Answer 1