Exam 12: Simple Linear Regression and Correlation

If the error sum of squares is 12 and the total sum of squares is 400, then the proportion of observed y variation explained by the simple linear regression model is

Given that , and n = 15, the 95% confidence interval for the slope $\beta _ { 1 }$ of the true regression line (__________,__________).

Let $\hat { Y } = \hat { \beta } _ { 0 } + \hat { \beta } x ^ { + }$ where $x ^ { + }$ is some fixed value of x. Then, the mean value of $\hat { Y }$ is $E ( \hat { Y } ) =$ __________.

If y = 2x + 5, then y__________ by __________when x increases by 1.

If $S _ {xx } = 2 , S _ { yy } = .5 ,\text {, and } S _ { xy } = - .75$ then the sample correlation coefficient r equals __________.

Suppose that in a certain chemical process the reaction time y (hour) is related to the temperature $(\left.{ \degree}{\mathrm{F}}\right)$ in the chamber in which the reaction takes place according to the simple linear regression model with equation y = 5.00 - .01x and $\sigma$ = .075. a. What is the expected change in reaction time for a $1 ^ { \circ }$ F increase in temperature? For a 10 ${ \circ }$ F increase in temperature? b. What is the expected reaction time when temperature is 200 ${ \circ }$ F? When temperature is 250 ${ \circ }$ F? c. Suppose five observations are made independently on reaction time, each one for a temperature of 250 ${ \circ }$ F. What is the probability that all five times are between 2.4 and 2.6 hours? d. What is the probability that two independently observed reaction times for temperatures $1 ^ { \circ }$ apart are such that the time at the higher temperature exceeds the time at the lower temperature?

You are told that a 95% CI for expected lead content when traffic flow is 15, based on a sample of n = 10 observations, is (462.1, 597.7). Calculate a CI with confidence level 99% for expected lead content when traffic flow is 15.

The assumptions of the simple of the simple linear regression model imply that the standardized variable has a t distribution with __________ degrees of freedom.

A first step in a regression analysis involving two variables is to construct a __________. In such a plot, each (x,y) is represented as a point plotted on a two-dimensional coordinate system.

If $\hat { \beta } _ { 1 } = - 1.0 , \sum x _ { i } = 10 , \sum y _ { i } = 20 , \text { and } n = 15$ then the least squares estimate of the intercept $\beta _ { 0 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x \text { is } \hat { \beta } _ { 0 }$ = __________.

A confidence interval refers to a parameter, or population characteristic, whose value is fixed but unknown to us. In contrast, a future value of Y is not a parameter but instead a random variable; for this reason we refer to an interval of plausible values for a future Y as a __________ rather than a confidence interval.

Since the mean of $\hat { \beta } _ { 1 } \text { is } E ( \hat { \beta }_{1} ) = \beta _ { 1 } \text {, then } \hat { \beta }_{1}$ is an __________ estimator of $\beta _ { 1 }$ .

A data set consists of 20 pairs of observations $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \ldots \ldots \left( x _ { 20 } , y _ { 20 } \right)$ If each $x _ { i }$ is replaced by $x _ { i } - 1$ and if each $y _ { i }$ is replaced by $y _ { i } - 2$ then the sample correlation coefficient r

The __________ is a measure of how strongly related two variables x and y are in a sample.

An experiment is conducted to investigate how the behavior of mozzarella cheese varied with temperature. Consider the accompanying data on x = temperature and y = elongation (%) at failure of the cheese. x 59 63 67 72 74 78 83 y 118 182 247 208 197 160 132 a. Construct a scatter plot in which the axes intersect at (0,0). Mark 0, 20, 40, 60, 80, and 100 on the horizontal axis and 0, 50, 100, 150, 200, and 250 on the vertical axis. b. Construct a scatter plot in which the axes intersect at (55,100). Does this plot seem preferable to the one in part (a)? Explain your reasoning. c. What do the plots of parts (a) and (b) suggest about the nature of the relationship between the two variables?

In testing $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ {\pm } : \beta _ { 1 } > 0$ using a sample of 18 observations, the rejection region for .025 level test is $t \geq$ __________.

An investigation of the relationship between traffic flow x (1000's of cars per 24 hours) and lead content y bark on trees near the highway ( $\mu g / g$ dry wt) yielded the data in the accompanying table. x 8.3 8.3 12.1 12.1 17.0 17.0 17.0 24.3 24.3 24.3 33.6 y 227 312 362 521 640 539 728 945 738 759 1263 The summary statistics are: $n=11, \sum x_{i}=198.3, \sum y_{i}=7034$ , $\sum x _ { i } ^ { 2 } = 4198.03 , \sum y _ {i } ^ { 2 } = 5,390,382 , \sum x _ { i } y _ { i } = 149,354.4$ In addition, the least squares estimates are given by: $\hat { \beta } _ { 0 } = - 12.84159 \text {, and } \hat { \beta } _ { 1 } = 36.18385$ Carry out the model utility test using the ANOVA approach for the traffic flow/lead-content data of Example 12.6. Verify that it gives a result equivalent to that of the t test.

Which of the following statements are not true if $y = - 3 x + 7$ ?

In testing $H _ { 0 } : \rho = .80 \text { versus } H _ { \pm } : \rho < .80$ the rejection region for .05 level of significance test is

The test statistic value for testing $H _ { 0 } : \rho = .5 \text { ver sus } \rho \neq .5$ is found to be z = 1.52. The corresponding P-value for the test is