Exam 12: Simple Linear Regression and Correlation

If $S _ { x, y } = 562.3 , V ( x ) = 53.4 , \text { and } n = 20$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x \text { is } \hat { \beta } _ { 1 }$ = __________.

In testing $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \pm 0$ using a sample of 15 observations, the rejection region for .05 level test is either $t \geq$ __________ or $t \leq$ __________.

Hydrogen content is conjectured to be an important factor in porosity of aluminum alloy castings. The accompanying data on x = content and y = gas porosity for one particular measurement technique have been reported: x .18 .20 .21 .21 .21 .22 .23 y .46 .70 .41 .45 .55 .44 .24 x .23 .24 .24 .25 .28 .30 .37 y .47 .22 .80 .88 .70 .72 .75 MINITAB gives the following output in response to a CORRELATION command: Correlation of Hydrogen and Porosity = 0.449 a. Test at level .05 to see whether the population correlation coefficient differs from 0. b. If a simple linear regression analysis had been carried out, what percentage of observed variation in porosity could be attributed to the model relationship?

Both the confidence interval for , the expected value of Y when $x = x ^ { + }$ , and prediction interval for a future Y observation to be made when $x = x ^ { + }$ , are __________ for an $x ^ { + }$ near $\bar { x }$ than for an $x ^ { + }$ far from $\bar { x }$ .

In testing $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \neq 0$ using a sample of 22 observations, the test statistic value is found to be t = -2.528. the approximated P-value of the test is

Given n pairs of observations $\left( x _ { 1 } y _ { 1 } \right) , \left( x _ { 2 } y _ { 2 } \right) , \ldots \ldots \ldots , \left( x _ { n } , y _ { n } \right)$ if large x's are paired with large y's and small x's are paired with small y's, then a __________ relationship between the variables is implied. Similarly, it is natural to speak of x and y having a __________ relationship if large x's are paired with small y's and small x's are paired with large y's.

When $H _ { 0 } : \rho = 0$ is true, the test statistic $T = R \sqrt { n - 2 } / \sqrt { 1 - R ^ { 2 } }$ has a t distribution with __________ degrees of Freedom, where n is the sample size.

In the simple linear regression model Y = $\beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ the quantity E is a random variable, assumed to be normally distributed with E( $\varepsilon$ ) = 0, and V( $\varepsilon$ ) = $\sigma ^ { 2 }$ . The estimated standard error of $\hat { \beta } _ { 1 }$ (the least squares estimated of $\beta _ { 1 }$ ), denoted by , is __________ divided by __________, where $s = \sqrt { S S E ( n - 2 ) } \text { and } S _ { xx } = \sum \left( x _ { i } - \bar {x} \right) ^ { 2 }$ .

A 100(1 - $\alpha$ ) % confidence interval for the slope $\beta _ { 1 }$ of the true regression line is $\hat { \beta } _ { 1 } \pm$ __________ $\cdot$ .

If y = -2x - 8, then the y-intercept is __________.

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 $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 2 }$ The estimated standard deviation $\hat { \sigma }$ is given by

The validity of joint or simultaneous confidence intervals for the expected value of Y when $x = x _ { 1 } ^ { + } , x _ { 2 } ^ { + } , \ldots . , x _ { k } ^ { +}$ rests on a probability result called the __________ inequality, so the joint confidence intervals are referred to as __________ intervals.

The vertical deviations $y _ { 1 } - \hat { y } _ { 1 } , y _ { 2 } - \hat { y } _ { 2 } , \ldots \ldots , y _ { n } - \hat { y } _ { n }$ from the estimated regression line are referred to as the __________.

Which of the following statements are not true?

The value of the sample correlation coefficient r is always between __________ and __________.

In simple linear regression analysis, if the residual sum of squares is zero, then the coefficient of determination $r ^ { 2 }$ must be

Suppose the expected cost of a production run is related to the size of the run by the equation y = 4000 + 10x. Let Y denote an observation on the cost of a run. If the variable size and cost are related according to the simple linear regression model, could it be the case that $P ( Y > 5500 \text { when } x = 100 ) = .05 \text { and } P ( Y > 6500 \text { when } x = 200 ) = .10 \text { ? }$ Explain.

A scatter plot, along with the least squares line, of x = rainfall volume $\left( \mathrm { m } ^ { 3 } \right)$ and y = runoff volume $\left( \mathrm { m } ^ { 3 } \right)$ for a particular location were given. The accompanying values were read from the plot. x 5 12 14 17 23 30 40 47 y 4 10 13 15 15 25 27 46 x 55 67 72 81 96 112 127 y 38 46 53 70 82 99 100 a. Does a scatter plot of the data support the use of the simple linear regression model? b. Calculate point estimates of the slope and intercept of the population regression line. c. Calculate a point estimate of the true average runoff volume when rainfall volume is 50. d. Calculate a point estimate of the standard deviation $\sigma$ e. What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall?

A study reports on an investigation of methods for age determination based on tooth characteristics. With x = percentage of root with transparent dentine and y = age (years), consider the following representative data for anterior teeth: x 15 19 31 39 41 44 47 48 55 64 y 23 52 65 55 32 60 78 59 61 60 a. Calculate a 95% CI for the expected change in age associated with a 1% increase in transparent dentine content. What does the interval suggest about usefulness of the model? b. Carry out a test of model utility based on the P-