Exam 12: Simple Linear Regression and Correlation

If $\sum x _ { i } = 15 , \sum y _ { 2 } = 36 , \sum x _ { i } y _ { i } = 210 , \sum x _ { i } ^ { 2 } = 30 , \text { and } n = 20$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ is

In simple linear regression model $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ which of the following statements are not required assumptions about the random error term $\varepsilon$ ?

Which of the following statements are not true?

A data set consists of 15 pairs of observations $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \ldots \ldots \left( x _ { 15 } , y _ { 15 } \right)$ If each $x _ { i }$ is replaced by $3 x _ { i }$ and if each $y _ { 1 }$ is replaced by $4 y _ { i } ,$ then the sample correlation coefficient r

In a simple linear regression problem, the following statistics are given: $\sum x _ {i } = 10 , \sum x _ { i } ^ { 2 } = 55 , \sum x _ {i } y _ { i } = 130 , \sum y _ { 2 } = 40 , \sum y _ { i } ^ { 2 } = 330 , \hat { \beta } _ { 0 } = 2.50 \text { and } \hat { \beta } _ { 1 } = 1.75$ Then, the error sum of squares is __________.

In testing $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \neq 0$ the t test statistic value is found to be t = 2.15. Should the null hypothesis be tested by constructing an ANOVA table, the F test would result in a test statistic value f = __________.

The null hypothesis $H _ { 0 } : \beta _ { 1 } = 0$ can be tested against $H _ { \pm } : \beta _ { 1 } \neq 0$ by constructing an ANOVA table, and rejecting $H _ { 0 } \text { at } \alpha$ level of significance if the test statistic value f $\geq$ __________, where n is the sample size.

In testing $H _ { 0 } : \rho = 0 \text { versus } H _ { \pm } : > 0$ using a sample of size 25, the test statistic value is found to be t = 2.50. The corresponding P-value for the test is __________, and we __________ $H _ { 0 }$ when $\alpha = .005$

Which of the following statements are not true regarding the normal equations $n b _ { 0 } + \left( \sum x _ { i } \right) b _ { 1 } = \sum y _ { i } \text { and } \left( \sum x _ { i } \right) b _ { 0 } + \left( \sum x _ { i } ^ { 2 } \right) b _ { 1 } = \sum x _ { i } y _ { i }$ ?

The simple linear regression model is $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ where $\varepsilon$ is a random variable assumed to be normally distributed with $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 3 }$ Let $x ^ { + }$ denote a particular value of the independent variable x. Which of the following identities are true regarding the expected or mean value of Y when $x = x ^ { + }$ ?

Which of the following statements are not true?

When the estimated regression line is obtained via the principle of least squares, the sum of the residuals $y _ { i } - \hat { y } _ { i }$ (i = 1, 3, …….., n) should in theory be __________.

A study contains a plot of the following data pairs, where x = pressure of extracted gas (microns) and y = extraction time (min): x 40 130 155 160 260 275 325 370 420 480 y 2.5 3.0 3.1 3.3 3.7 4.1 4.3 4.8 5.0 5.4 a. Estimate $\sigma$ and the standard deviation of $\hat { \beta } _ { 1 }$ b. Suppose the investigators had believed prior to the experiment that on average there would be an increase of .006 min. in extraction time associated with an increase of 1 micron in pressure. Use the P-value approach with a significance level of .10 to decide whether the data contradicts this prior belief.

Which of the following statements are not correct?

If $\sum x _ { i } = 28 , \sum y _ { i } = 54 , \sum x _ { i } y _ { 2 } = 156 , \sum x _ { i } ^ { 2 } = 82 , \text { and } n = 10$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ is

If the sample correlation coefficient r equals -.80, then the value of the coefficient of determinations is __________.

The accompanying observations on x = hydrogen concentration (ppm) using a gas chromatography method and y = concentration using a new sensor method were obtained in a recent study x 47 62 65 70 70 78 95 100 114 118 y 38 62 53 67 84 79 93 106 117 116 x 124 127 140 140 140 150 152 164 198 221 y 127 114 134 139 142 156 149 154 200 215 Construct a scatter plot. Does there appear to be a very strong relationship between the two types of concentration measurements? Do the two methods appear to be measuring roughly the same quantity? Explain your reasoning.

In simple linear regression analysis, the __________, denoted by __________, can be interpreted as a measure of how much variability in y left unexplained by the model - that is, how much cannot be attributed to a linear relationship.

In simple linear regression analysis, a quantitative measure of the total amount of variation in observed y values is given by the __________, denoted by __________.

The simple linear regression model is $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon \text {, where } \varepsilon$ is a random variable assumed to be normally distributed with $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 3 } \text {. Let } x ^ { + }$ denote a particular value of the independent variable x. Which of the following identities are true regarding the variance of Y when $x = x ^ { + }$ ?