# Mathematics

**MTH 242 FALL 2014**

*Instructions:* Answer all questions. All questions carry equal points. I expect you to show as detailed work as possible to enable you earn full credit. Also remember to begin a new question on a fresh page as you have been doing for your homework problems. Present neat and eligible work.

**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.

Plant | A | B | C | D | E | F | G | H |

x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 |

y | 3.9 | 4.4 | 5.8 | 6.6 | 7.0 | 7.1 | 7.3 | 7.7 |

(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

Child | A | B | C | D | E | F | G | H |

x | 112 | 113 | 110 | 113 | 112 | 114 | 109 | 113 |

y | 69 | 65 | 75 | 70 | 70 | 75 | 68 | 76 |

(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…………………… Xn, 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.

Age(yr) x | 5 | 4 | 6 | 5 | 5 | 5 | 6 | 6 | 2 | 7 | 7 |

Price($100) y | 85 | 103 | 70 | 82 | 89 | 98 | 66 | 95 | 169 | 70 | 48 |

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 yhat=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.

x(Age) | 6 6 6 2 2 5 4 5 1 4 |

y(Price) | 290 280 295 425 384 315 355 328 425 325 |

*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 Se. **(1 point)**

(iii). Calculate the value of Sxx from the data by 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 Se from (a)-(ii) and Sxx 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 Se from (a)-(ii) and Sxx 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)**