QUESTION 1.

(a). A biologist assumes that there is a linear relationship between the amount of fertilizer supplied to tomato plant and the subsequent yield of tomatoes obtained. Eight(8) tomato plants of the same variety, were selected at random and treated, weekly, with a solution in which x grams(g) of fertilizer was dissolved in a fixed quantity of water. The yield y kilograms (kg) of tomatoes was recorded as follows.

(i). Plot a scatterplot of yield, y against amount of fertilizer, x. ( 3 points)

(ii). Calculate the equation of the least square regression line of y on x. ( 8 points)

(iii). Estimate the yield of a plant treated, weekly, with a 3.2 grams of fertilizer.

( 2 points)

(iv). Indicate why it may not be appropriate to use your equation to predict the yield of a plant treated weekly with 20 grams of fertilizer. ( 2 points)

(b). The table below shows a Verbal Reasoning test scores, x and an English test scores, y for each of a random sample of eight (8) children who took both tests

(i). Calculate the value of the product moment correlation coefficient between the scores in Verbal Reasoning and English. ( 8 points)

(ii).Comment briefly, in context on the result obtained in part(i). ( 2 points)

QUESTION 2.

(a).(i).Suppose that the variables X and Y satisfy the four assumptions for regression inferences such that for samples of size n, each with the same values X₁, X₂, X₃…………………… X_n, for the predictor variable, List the three properties that must hold for the slope b1 of the sample regression line.

(ii).What distribution is used for the inferences for β₁   (2 point).

(iii).In a hypothesis test for the slope β₁ such that Ho:β₁=0 vs. Ha:β₁≠0, write the formula for the test statistic and state its degrees of freedom. (2 point).

(b).When you are asked to use the critical value approach to conduct a Regression t-test state the following.

(i).Purpose     (2 point).

(ii).Assumptions    (2 point).

(iii).All the six steps (7 points) required to enable you to complete such a test.

(c). Consider the following data on the age and price for a sample of eleven (11) Honda cars as displayed below.

At a 5% significance level, do the data provide sufficient evidence to conclude that age is useful as a linear predictor of price for the cars? .Use the critical value approach for your test.

[Hint: Regression equation is y_hat=195.47-20.26x, and Standard error of the estimate Se=12.58]  (10 points).

QUESTION 3.

The following are the age and price (thousands of dollars) data for Corvettes from a car dealership in Sterling, Virginia. Find attached the DDXL output of the data from regression analysis.

Dependent variable is y and independent variable is x.

No Selector, 10 total cases.

R²=93.7%    R² (adjusted)=92.9%

S=14.25 with 10-2=8 degrees of freedom

Source                        Sum of Squares      df      Mean Square         F-ratio

Regression                 24057.9                    1        24057.9                   119

Residual                     1623.71                    8         202.964              

Variable                     Coefficient          s.e of Coeff    t-ratio            prob

Constant                   456.602                  11.43             39.9               <0.0001

X                               -27.9029                  2.56               -10.9              <0.0001

(a).(i). From the DDXL output write the regression equation. (2 points)

(ii).From the DDXL output, figure out the value of S_e. (1 point)

(iii). Calculate the value of S_xx from the databy using the values of x. (5 points).

(b).Obtain a 90% confidence interval for the mean price of all 3-year old cars by using the conditional mean t-interval procedure on page#17 & page#18 of section E class lecture notes.[By using the values of S_e from (a)-(ii) and S_xx from(a)-(iii), you can find (b)]  (7 points)

 (c).Obtain a 90% prediction interval for the price of a 3-year old car by using the predicted value t-interval procedure on page#20 & page#21 of section E class lecture notes. .[By using the values of S_e from (a)-(ii) and S_xx from(a)-(iii), you can find (c)]  (7 points)

(d). By comparing your results from (b) and (c) which interval is wider? and give two reasons why such results is expected.  (3 points)