Exam 12: Simple Linear Regression and Correlation

In simple linear regression analysis, SST is the total sum of squares, SSE is the error sum of squares, and SSR is the regression sum of squares. The coefficient of determination $Y ^ { 2 }$ is given by $Y ^ { 2 } =\underline{\quad\quad}$ $/ S S T$ or $Y ^ { 2 } = 1 -\underline{\quad\quad}$ / $( \operatorname { SST } )$

An experiment to measure the macroscopic magnetic relaxation time in crystals $\text { ( } \mu \mathrm { sec } )$ as a function of the strength of the external biasing magnetic field (KG) yielded the following data x 11.0 12.5 15.2 17.2 19.0 20.8 y 187 225 305 318 367 365 x 22.0 24.2 25.3 27.0 29.0 y 400 435 450 506 558 The summary statistics are $\sum x _ { i } = 223.2 , \sum y _ { i } = 4116 , \sum x _ { i } ^ { 2 } = 4877.50 , \sum x _ {i } y _ { i } = 90,096.1$ and $\sum y _ { i } ^ { 2 } = 1,666,782$ Compute the following: a. A 95% CI for expected relaxation time when field strength equals 18. b. A 95% PI for future relaxation time when field strength equals 18. c. Simultaneous confidence intervals for expected relaxation time when field strength equals 15, 18, and 20; your joint confidence coefficient should be at least 97%.

Which of the following statements are not true?

Toughness and fibrousness of asparagus are major determinants of quality. This was the focus of a study reported in "Post-Harvest Glyphosphate Application Reduces Toughening, Fiber Content, and Lignification of Stored Asparagus Spears" (J. of the Amer. Soc. Of Horticultural Science, 1988: 569-572). The article reported the accompanying data (read from a graph) on x = shear force (kg) and y = percent fiber dry weight. x 46 48 55 57 60 72 81 85 94 y 2.18 2.10 2.13 2.28 2.34 2.53 2.28 2.62 2.63 x 109 121 132 137 148 149 184 185 187 y 2.50 2.66 2.79 2.80 3.01 2.98 3.34 3.49 3.26 $n = 18 \quad \sum x _ { i } = 1950 \quad \sum x _ { i} ^ { 2 } = 251,970$ $\sum y _ {i } = 47.92 \sum y _ { i} ^ {2 } = 130.6074 \quad \sum x _ { i } y _ { i } = 5530.92$ a. Calculate the value of the sample correlation coefficient. Based on this value, how would you describe the nature of the relationship between the two variables? b. If a first specimen has a larger value of shear force than does a second specimen, what tends to be true of percent dry fiber weight for the two specimens. c. If shear force is expressed in pounds, what happens to the value of r? Why? d. If the simple linear regression model were fit to this data, what proportion of observed variation in percent fiber dry weight could be explained by the model relationship? e. Carry out a test at significance level .01 to decide whether there is a positive linear association between the two variables.

In general, the variable whose value is fixed by the experimenter will be denoted by x and will be called the independent, predictor, or __________ variable. For fixed x, the second variable will be random; we denote this random variable and its observed value by Y and y, respectively, and refer to it as the dependent or __________ variable.

The simple linear regression model is $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon , \text { where the quantity } \varepsilon$ is a random variable assumed to be __________ distributed, with $E (\varepsilon ) =$$\underline{\quad\quad}$ and $V ( \varepsilon ) =$$\underline{\quad\quad}$

Which of the following statements are not true?

The quantity $\varepsilon$ in the simple linear regression model $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ is a random variable, assumed to be normally distributed with $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 2 }$ Based on 20 observations, if the residual sum of squares is 8, then the estimated standard deviation $\hat { \sigma }$ is

Infestation of crops by insects has long been of great concern to farmers and agricultural scientists. A study reports data on x = age of a cotton plant (days) and y = % damaged squares. Consider the accompanying n = 12 observations: x 9 12 12 15 18 18 y 11 12 23 30 29 52 x 21 21 27 30 30 33 y 41 65 60 72 84 93 a. Why is the relationship between x and y not deterministic? b. Does a scatter plot suggest that the simple linear regression model will describe the relationship between the two variables? c. The summary statistics are $\sum x _ { i} = 246 , \sum x _ {i } ^ { 2 } = 5742 , \sum y _ { i } = 572 , \sum y _ { i } ^ { 2 } = 35,634 , \text { and } \sum x _ {i } y _ { i } = 14,022$ Determine the equation of the least squares line. d. Predict the percentage of damaged squares when the age is 20 days by giving an interval of plausible values.

A 95% confidence interval for the expected value of Y is constructed first for x = 2, then for x = 3, then for x = 4, and finally for x = 5. This yields a set of four confidence intervals for which the joint or simultaneous confidence level is guaranteed to be at least

Which of the following statements are not true about the sample correlation coefficient r?

The accompanying data on x = current density (mA/cm $2$ ) and y = rate of deposition $\text { ( } \mu \mathrm { m } / \mathrm { min } )$ appeared in a recent study. Do you agree with the claim by the article's author that "a linear relationship was obtained from the tin-lead rate of deposition as a function of current density"? Explain your reasoning. x 20 40 60 80 y .24 1.20 1.71 2.22

In testing $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \neq 0$ the test statistic value is the t - ratio t = __________ divided by __________.

A reasonable rule of thumb is to say that the correlation is weak if __________ $\leq | r | \leq$ __________, strong if __________ $\leq | r | \leq$ __________, and moderate otherwise.

The simple linear regression model provides a very good fit to a data set on rainfall and runoff volume. The equation of the least squares line is $\hat { y } = - 1.128 + .82697 X , Y^ { 2 } = .975 , \text { and } s = 5.24$ a. Use the fact that $s _ { y } = 1.44$ when rainfall volume is 40 m $3$ to predict runoff in a way that conveys information about reliability and precision. Does the resulting interval suggest that precise information about the value of runoff for this future observation is available? Explain your reasoning. b. Calculate a PI for runoff when rainfall is 50 using the same prediction level as in part (a). What can be said about the simultaneous prediction level for the two intervals you have calculated?

Which of the following statements are true?

The sample correlation coefficient r equals 1 if and only if all $\left( x _ { i } , y _ { i } \right)$ pairs lie on a straight line with __________ slope.

Which of the following statements are not true?

Which of the following statements are not true?

Which of the following statements are not true?