Question

1. Box, Hunter, and Hunter have analyzed the dispersion of an aerosol spray as a function of its age. The dispersion y is measured as the reciprocal of the number of particles in a unit volume. Age x is measured in minutes. The experiments were performed in random order, but their results are listed here in increasing order of x:

Age, x	Dispersion, y
8	6.16
22	9.88
35	14.35
40	24.06
57	30.34
73	32.17
78	42.18
87	43.23
98	48.76

1. Obtain a scatter plot of the observations and determine whether the scatter plot indicates that a linear relationship may exist between the two variables

2. Obtain the least-squares regression equation.

3. What proportion of the variation in the dependent variable, y, is shared with or explained by the independent variable, x, in the regression model?

4. What is the standard error of the estimate?

5. Determine the aptness of the fit of the model.

6. At the 0.05 level of significance, is there evidence of a linear relationship between age and dispersion?

7. Estimate the mean response at age 90.

8. Find a 95% confidence interval estimate of the population average dispersion for an aerosol spray of age 90.

1. Find a 95% prediction interval estimate of the dispersion for an individual aerosol

spray of age 90.

10. Which of the interval in h) and i) is the widest? Explain why.

11. Find the correlation coefficient.

Answer 1

a)

60 50 40 30 20 10 20 40 60 80 100 120

A linear relationship does exist between the two variables.

b) The linear regression model is given by:-

Y = 0.838666143927522 + 0.489120491374804 * X

c)

proportion of the variation in the dependent variable, y, is shared with or explained by the independent variable, x, in the regression model is: 0.966581113039793

d) the standard error of the estimate is

3.00698427253601

The model run in excel is:

1 Age, X Dispersion, y 6.16 9.88 14.35 24.06 30.34 32.17 42.18 43.23 48.76 Scatterplot 4 35 60 50 57 30 20 10 78 87 10 12 SUMMARY OUTPUT 13 14 15 Multiple R 16 R Square 17 Adjusted R Square 18 Standard Error 19 Observations 20 21 ANOVA 20 60 80 100 120 Regression Statistics 0.983148571 0.966581113 0.961806986 3.006984273 Significance F MS 23 Regression 24 Residual 25 Total 26 27 28 Intercept 29 Age, x 1 1830.655719 1830.655719 202.462392 7 63.29368091 9.041954415 2.012E-06 1893.9494 Coefficients Standard Error Stat 0.8386661442.150022966 0.39007311 0.7080799344.245330302 5.92266259-4.245330302 5.92266259 0.489120491 0.034375077 14.228928 2.012E-06 0.407836351 0.570404631 0.407836351 0.570404631 P-value Lower 95% Upper 95% Lower 95.0% r95.0% U

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