Applied Linear Statistical Models 5th ed

Introduction to Linear Regression Models

Book Link: Applied Linear Statistical Models 5th ed

Chapter 1: Linear Regression with One Predictor Variable

1.1.

Refer to the sales volume example on page 3. Suppose that the number of units sold is measured accurately, but clerical errors are frequently made in determining the dollar sales. Would the relation between the number of units sold and dollar sales still be a functional one? Discuss.

Solution: It depends.

  • Could be a functional relationship if clerical errors are consistently adding/substracting a value in dertermining the dollar sales.

    Example: the clerical error is to add 0.01 to the dollar sales.

    $Y - 0.01 = 2X$ => $Y = 2X + 0.01$ is still a functional relationship

  • Otherwise, the relationship is a statistical relation.

1.2.

The members of a health spa pay annual membership dues of $\$$300 plus a charge of $\$$2 for each visit to the spa. Let Y denote the dollar cost for the year for a member and X the number of visits by the member during the year. Express the relation between X and Y mathematically. Is it a functional relation or a statistical relation?

Solution: It is a functional relation.

$Y = 300 + 2X$

1.3.

Experience with a certain type of plastic indicates that a relation exists between the hardness (measured in Brinell units) of items molded from the plastic ($Y$) and the elapsed time since termination of the molding process ($X$). It is proposed to study this relation by means of regression analysis. A participant in the discussion objects, pointing out that the hardening of the plastic “is the result of a natural chemical process that doesn’t leave anything to chance, so the relation must be mathematical and regression analysis is not appropriate.” Evaluate this objection.

Solution:

The true relation is functional relation, i.e. mathematical relation without chance.

In the practice, the measurement random error is introduced becuase of accuracy of equiments and reading errors generated by human meansing. Hence, the observed $Y$ and $X$ have statistical relation.

1.4.

In Table 1.1, the lot size $X$ is the same in production runs 1and 24 but the work hours $Y$ differ. What feature of regression model (1.1) is illustrated by this?

idea

solution


1.29.

Refer to regression model (1.1) $Y_i = \beta_0+\beta_1X_i+\varepsilon_i$. Assume that $X = 0$ is within the scope of the model. What is the implication for the regression function if $\beta_0 = 0$ so that the model is $Y_i = \beta_0+\beta_1X_i+\varepsilon_i$? How would the regression function plot on a graph?

Solution:

$Y_i = \beta_0+\beta_1X_i+\varepsilon_i \stackrel{\beta_0=0}{\Rightarrow} Y_i = \beta_1X_i+\varepsilon_i$ on the $i$th trial.

The regression function is $E(Y) = \beta_1X$. The intercept is 0.

The regression function is a linear line and passes the origin dot (0,0).

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