Deck 12: Simple Linear Regression and Correlation

Question 1

The estimated regression line or least squares line for the simple linear regression model is the line whose equation is given by __________.

Answer 1

Question 2

In the simple linear regression model Y = the quantity E is a random variable, assumed to be normally distributed with E( ) = 0, and V( ) = . The estimated standard error of (the least squares estimated of ), denoted by , is __________ divided by __________, where .

Answer 2

s,

Question 3

In the simple linear regression model the quantity E is a random variable, assumed to be normally distributed with E( ) = 0 and V( ) = . The estimator has a __________ distribution, because it is a linear function of independent __________ random variables.

Answer 3

normal, normal

Question 4

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

Question 5

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

Question 6

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

Question 7

If then the least squares estimate of the slope coefficient of the true regression line = __________.

Question 8

The vertical deviations from the estimated regression line are referred to as the __________.

Question 9

If SSE = 36 and SST = 500, then the proportion of total variation that can be explained by the simple linear regression model is_ _________.

Question 10

If then the least squares estimate of the slope coefficient of the true regression line = __________.

Question 11

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.

Question 12

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.

Question 13

In a simple linear regression problem, the following statistics are given: Then, the error sum of squares is __________.

Question 14

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 is given by

Question 15

Since the mean of is an __________ estimator of .

Question 16

If then the least squares estimate of the intercept of the true regression line = __________.

Question 17

The simple linear regression model is is a random variable assumed to be __________ distributed, with

Question 18

When the estimated regression line is obtained via the principle of least squares, the sum of the residuals (i = 1, 3, …….., n) should in theory be __________.

Question 19

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.

Question 20

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

Question 21

In a simple linear regression, the most commonly encountered pair of hypotheses about is A test of these two hypotheses is often referred to as the __________.

Question 22

Given that , and n = 15, the 95% confidence interval for the slope of the true regression line (__________,__________).

Question 23

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

Question 24

The validity of joint or simultaneous confidence intervals for the expected value of Y when rests on a probability result called the __________ inequality, so the joint confidence intervals are referred to as __________ intervals.

Question 25

In testing the test statistic value is the t - ratio t = __________ divided by __________.

Question 26

If the confidence interval . For the expected value of Y when is computed both for x = a and for x = b to obtain joint confidence intervals for then the joint confidence coefficient on the resulting pair of intervals is at least __________ %.

Question 27

The sample correlation coefficient r equals -1 if and only if all pairs lie on a straight line with __________ slope.

Question 28

A 100(1 - ) % confidence interval for the slope of the true regression line is __________ .

Question 29

In testing 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 = __________.

Question 30

In testing using a sample of 18 observations, the rejection region for .025 level test is __________.

Question 31

Given n pairs of observations 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.

Question 32

The sample correlation coefficient r equals 1 if and only if all pairs lie on a straight line with __________ slope.

Question 33

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

Question 34

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.

Question 35

The null hypothesis can be tested against by constructing an ANOVA table, and rejecting level of significance if the test statistic value f __________, where n is the sample size.

Question 36

The t critical value for a confidence level of 90% for the slope of the regression line, based on a sample of size 20, is t = __________.

Question 37

Let where is some fixed value of x. Then, the mean value of is __________.

Question 38

In testing using a sample of 15 observations, the rejection region for .05 level test is either __________ or __________.

Question 39

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

Question 40

Both the confidence interval for , the expected value of Y when and prediction interval for a future Y observation to be made when are __________ for an near than for an far from .

Question 41

Which of the following statements are not true?

A) The predicted value $\hat { y } _ { i }$
Is the value of y that we would predict or expect when using the estimated regression line with $x = x _ { i }$
B) The predicted value $\hat { y } _ {i}$
Is the height of the estimated regression line above the value $x _ {i }$
For which the $i ^ { t h }$
Observation was made.
C) The residual $y _ { i } - \hat { y } _ { i }$
Is the difference between the observed $y _ { i }$
And the predicted $\hat { y } _ { i }$
D) If the residuals are all large in magnitude, then much of the variability in observed y values appears to be due to the linear relationship between x and y, whereas many small residuals suggest quite a bit of inherent variability in y relative to the amount due to the linear relation.
E) All of the above statements are true.

Question 42

Which of the following statements are true?

A) Before the least squares estimates $\hat { \beta } _ { 1 } \text { and } \hat { \beta } _ { 2 }$
Are computed, a scatter plot should be examined to see whether a linear probabilistic model is plausible.
B) For a fixed $x \text { value } x ^ { + } , \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
(the height of the estimated regression line above $x ^ { + }$
) gives either a point estimate of the expected value of Y when $x = x ^ { + }$
Or a point prediction of the Y value that will result from a single new observation made at $x = x ^ { + }$
)
C) The least squares regression line should not be used to make a prediction for an x value much beyond the range of the data x values.
D) The residuals are 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.
E) All of the above statements are true.

Question 43

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 $\varepsilon$
?

A) The expected value of $\varepsilon$

Is zero.
B) The variance of $\varepsilon$

Is the same for all values of the independent variable x.
C) The error term is normally distributed.
D) The values of the error term are independent of one another.
E) All of the above are required assumptions about $\varepsilon$

)

Question 44

Which of the following statements are true?

A) The simplest deterministic mathematical relationship between two variables x and y is a linear relationship $y = \beta _ { 0 } + \beta _ { 1 } x$
B) The set of pairs (x, y) for which $y = \beta _ { 0 } + \beta _ { 1 } x$
Determines a straight line with slope $\beta _ { 1 }$
And y-intercept $\beta _ { 0 }$
)
C) The slope of a line $y = \beta _ { 0 } + \beta _ { 1 } x$
Is the change in y per a 1-unit increase in x.
D) The y-intercept of a line $y = \beta _ { 0 } + \beta _ { 1 } x$
Is the height at which the line crosses the vertical axis and is obtained by setting x = 0 in the equation.
E) All of the above statements are true.

Question 45

If then the sample correlation coefficient r equals __________.

Question 46

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

A) 18.75
B) 28.42
C) 9.15
D) 9.76
E) 10.50

Question 47

Which of the following statements are not true?

A) In regression analysis, the independent variable is also referred to as the predictor or explanatory variable.
B) In regression analysis, the dependent variable is also referred to as the response variable.
C) A first step in a regression analysis involving two variables is to construct a scatter plot.
D) The simple linear regression model is $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$
Where the quantity $\varepsilon$

Is a random variable, assumed to be normally distributed with $E ( \varepsilon\ ) = 0 \text { and } V ( \varepsilon\ ) = 1$


E) All of the above statements are true.

Question 48

In testing 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 __________ when

Question 49

Which of the following statements are true?

A) The true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$
Is the line of mean values.
B) The height of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$
Above any particular x value is the expected value of Y for that value of x.
C) The slope $\beta _ { 1 }$
Of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$
Is interpreted as the expected change in Y associated with a 1-unit increase in the value of x.
D) The equation
States that the amount of variability in the distribution of Y values is the same at each different value of x (homogeneity of variance).
E) All of the above statements are true.

Question 50

The principle of least squares results in values of $\hat { \beta } _ { 0 } \text { and } \hat { \beta } _ { 1 }$ that minimizes the sum of squared deviations between

A) the observed values of the explanatory variable x and the estimated values $\hat { x }$
B) the observed values of the response variable y and the estimated values $\hat { y }$
C) the observed values of the explanatory variable x and the response variable y
D) the observed values of the explanatory variable x and the response values $\hat { y }$
E) the estimated values of the explanatory variable x and the observed values of the response variable y

Question 51

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

A) The y-intercept is 7
B) y decreases by 3 when x increases by 4
C) y decreases by 3 when x increases by 1
D) The slope of the line is -3
E) All of the above statements are not true.

Question 52

A procedure used to estimate the regression parameters $\beta _ { 1 } \text { and } \beta _ { 2 }$ and to find the least squares line which provides the best approximation for the relationship between the explanatory variable x and the response variable Y is known as the

A) least squares method
B) best squares method
C) regression analysis method
D) coefficient of determination method
E) prediction analysis method

Question 53

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

Question 54

When is true, the test statistic has a t distribution with __________ degrees of Freedom, where n is the sample size.

Question 55

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

A) The normal equations are linear in the unknowns $b _ { 0 } \text { and } b _ { 1 }$
)
B) The least squares estimates are always the unique solution to the system of normal equations.
C) Provided that at least two of the $x _ {i }$
Values are different, the least squares estimates are the unique solution to the system of normal equations.
D) The quantity $\sum y _ { i } ^ { 2 }$
Is not needed to solve the system of normal equations.
E) All of the above statements are true.

Question 56

If $\sum \left( x _ { i} - \bar { x } \right) \left( y _ { i } - \bar { y } \right) = 128 \text { and } \sum \left( x _ { i } - \bar { x } \right) ^ { 2 } = 80$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ is

A) 11.314
B) 8.944
C) 1.600
D) 0.625
E) cannot be determined from the given information

Question 57

Which of the following statements are not true?

A) The objective of regression analysis is the exploit the relationship between two (or more) variables so that we can gain information about one of them through knowing values of the other(s).
B) Saying that variables x and y are deterministically related means that once we are told the value of x, the value of y is completely specified.
C) Regression analysis is the part of statistics that deals with investigation of the relationship between two or more variables related in a deterministic fashion.
D) All of the above statements are true.
E) None of the above statements are true.

Question 58

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

A) 3.60
B) 0.75
C) 1.33
D) 4.80
E) 1.68

Question 59

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 $x = x ^ { + }$ ?

A)
B)
C)
D)
E)

Question 60

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 $x = x ^ { + }$ ?

A)
B)
C)
D)
E)

Question 61

Which of the following statements are true?

A) The confidence interval for
; the expected value of Y when $x = x ^ { + }$
Is centered at the point estimate for
And extends out to each side by an amount that depends on the confidence level and on the extent of variability in the estimator on which the point estimated is based.
B) In some situations, a confidence interval is desired not just for a single x value but for two or more x values.
C) The joint or simultaneous confidence level for a set of K Bonferroni intervals is guaranteed to be at least 100(1 - K $\alpha$
)%)
D) We refer to an interval of plausible values for a future Y as a prediction interval rather than a confidence interval, since a future value of Y is a random variable.
E) All of the above statements are true.

Question 62

Which of the following statements are not true?

A) $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
, where $x ^ { + }$
Is a specified value of the independent variable x, can be regarded either as a point estimate of
(the expected or true average value of Y when $x = x ^ {+}$
) or as a prediction of the Y value that will result from a single observation made when $x = x ^ { + }$
)
B) Before we obtain sample data, both $\hat { \beta } _ { 0 }$
And $\hat { \beta } _ { 1 }$
Are subject to sampling variability - that is, they are both statistics whose values will vary from sample to sample.
C) A confidence interval for a mean y value in regression is based on properties of the sampling distribution of the statistic $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
)
D) All of the above statements are true.
E) None of the above statements are true.

Question 63

In testing $H _ { 0 } : \beta _ { 1 } = 0$ versus $H _ { \pm } : \beta _ { 1 } \neq 0$ using a sample of 20 observations, the rejection region for .01 level of significance test is

A) t $\geq$
-2)878
B) t $\leq$
2)878
C) -2.878 $\leq$
T $\leq$
2)878
D) either t $\geq$
2)878 or t $\leq$
-2)878
E) t = 0

Question 64

Which of the above statements are not true?

A) A t variable obtained by standardizing $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
Leads to a confidence interval and test procedure concerning
(the expected value of Y when $x = x ^ { + }$
))
B) The variable T = $\left[ \hat { Y } - \left( \beta _ { 0 } + \beta _ { 1 } x ^ { +} \right) \right] / S _ { \dot { y } }$
Has a t distribution with n - 1 degrees of Freedom, where n is the sample size and $x ^ { + }$
Is a specified value of the independent variable x.
C) A 100 $( 1 - \alpha ) \%$
Confidence interval for
; the expected value of Y when $x = x ^ { + }$
, is given by
Where n is the sample size.
D) All of the above statements are true.
E) None of the above statements are true.

Question 65

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

A) 95%
B) 90%
C) 85%
D) 80%
E) 75%

Question 66

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

A) SSE / (n - 2)
B) $\sqrt { S S E / ( n - 2 ) }$
C) $[ S S E / ( n - 2 ) ] ^ { 2 }$
D) $\operatorname { SSE } / \sqrt { n - 2 }$
E) $\sqrt { \operatorname { SSE } } / ( n - 2 )$

Question 67

Which of the following statements are not true?

A) The slope $\beta _ { 1 }$
Of the population regression line is the true average change in the independent variable x associated with a 1 - unit increase in the dependent variable y.
B) The slope of the least squares line, $\beta _ { 1 }$
Of the population regression line.
C) Inferences about the slope $\beta _ { 1 }$
Of the population regression line are based on thinking of the slope $\hat { \beta } _ { 1 }$
Of the least squares line as a statistic and investigating its sampling distribution.
D) All of the above statements are true
E) Non of the above statements are true.

Question 68

Which of the following statements are not correct?

A) The coefficient of determination, denoted by $Y^ { 2 }$ ,
Is interpreted as the proportion of observed y variation that cannot be explained by the simple linear regression model.
B) The higher the value of the coefficient of determination, the more successful is the simple linear regression model in explaining y variation.
C) If the coefficient of determination is small, an analyst will usually want to search for an alternative model (either a nonlinear model or a multiple regression model that involves more than a single independent variable).
D) The coefficient of determination can be calculated as the ratio of the regression sum of squares (SSR) to the total sum of squares.
E) All of the above statements are correct.

Question 69

Which of the following statements are true?

A) The denominator of the slope $\hat { \beta } _ { 1 }$
Of the least squares line is $S _ { x x } = \Sigma \left( x _ { i } - \bar { x } \right) ^ { 2 }$
, which is a constant since it depends only on the $x _ { i } ^ { \prime } s$
And not on the $Y _ { i} ^ { \prime } s$
B) The slope $\hat { \beta } _ { 1 }$
Of the least squares line is a linear function of the "independent" random variables $Y _ { 1 } Y _ { 2 } \ldots \ldots , Y _ { n }$
Each of which is normally distributed.
C) The distribution of the slope $\hat { \beta } _ { 1 }$
Of the least squares line is always centered at the value of the slope $\beta _ { 1 }$
Of the population regression line.
D) All of the above statements are true.
E) None of the above statements are true.

Question 70

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

A) 0.030
B) 0.173
C) 0.970
D) 0.985
E) None of the above answers are correct.

Question 71

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

A) .01
B) .02
C) .025
D) .05
E) .99

Question 72

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

A) -1
B) 0
C) between -1 and zero
D) 1
E) between -1 and 1

Question 73

Which of the following statements are true?

A) The assumptions of the simple linear regression model imply that the standardized variable
Has a t distribution with n - 2 degrees of freedom.
B) The estimated standard error of $\hat { \beta } _ { 1 }$
; namely
, will tend to be small when there is little variability in the distribution of $\hat { \beta } _ { 1 }$
And large otherwise.
C) There is an estimated standard error for the statistic $\hat { \beta } _ { 0 }$
From which a confidence interval for the intercept $\beta _ { 0 }$
Of the population regression line can be calculated.
D) The most commonly encountered pair of hypotheses about the slope $\beta _ { 1 }$
Of the population regression line is $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \pm 0$
E) All of the above statements are true.

Question 74

Which of the following statements are not true?

A) The model utility test is the test of $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { \pm } : \beta _ { 1 } \pm 0$
In which case the test statistic value is the t ratio t =
)
B) The null hypothesis $H _ { 0 } : \beta _ { 1 } = 0$
Can be tested against the alternative hypothesis $H _ { \pm } : \beta _ { 1 } \neq 0$
By constructing an ANOVA table and rejecting $H _ { 0 }$
If the test statistic value $f \geq F _ { α , { 1 } , n- 2 }$
, when n is the sample size.
C) The simple linear regression model should not be used for further inferences (estimates of mean value or predictions of future values) unless the model utility test results in acceptance of $H _ { 0 } : \beta _ { 1 } = 0$
For a suitably small significance level $\alpha$
)
D) All of the above statements are true.
E) None of the above statements are true.

Question 75

Which of the following statements are not true?

A) The slope $\hat { \beta } _ { 1 }$
Of the least squares line is an unbiased estimator of the slope coefficient $\beta _ { 1 }$
Of the true regression line.
B) The variance $\hat { \beta } _ { 1 }$
Of the least squares line equals the variance $\sigma ^ { 2 }$
Of the random error $\varepsilon$

Divided by $\sqrt { S _ { xx } }$
, where $S _ {xx } = \sum \left( x _ { i } - \bar { x } \right) ^ { 2 }$
C) Values of $x _ { i}$
All close to one another imply a highly variable estimator $\hat { \beta } _ { 1 }$
Of the slope $\beta _ { 1 }$
Of the true regression line.
D) Values of $x _ { i }$
That are quite spread out results in a more precise estimator $\hat { \beta } _ { 1 }$
Of the slope $\beta _ { 1 }$
Of the true regression line
E) All of the above statements are true

Question 76

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

A) 2.500
B) 0.400
C) 0.667
D) 0.444
E) None of the above answers are correct.

Question 77

Which of the following statements are not true?

A) The total sum of squares is the sum of squared deviations about the sample mean of the observed y values.
B) The error sum of squares is the sum of squared deviations about the least squares line $y = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x .$
C) The ratio of the error sum of squares to the total sum of squares is the proportion of total variation that cannot be explained by the simple linear regression model.
D) The sum of squared deviations about the least squares regression line is always smaller than the sum of squared deviations about any other line.
E) All of the above statements are true.

Question 78

Which of the following statements are not true?

A) 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( $\hat { Y }$
) = $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
)
B) $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { +}$
Is an unbiased estimator for $\beta _ { 0 } + \beta _ { 1 } x ^ { + }$
(i)e., for
)
C) The estimation $\hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x ^ { + }$
For
Is more precise when $x ^ { + }$
Is near the center of the $x _ { i}$
's then when it is far from the x values at which observations have been made.
D) All of the above statements are true.
E) None of the above statements are true.

Question 79

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

A) z $\geq$
1)645
B) z $\leq$
-1)645
C) -1.645 $\leq$
Z $\leq$
1)645
D) either z $\geq$
1)645 or z $\leq$
-1)645
E) z = 1.96

Question 80

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

A) .9357
B) .0643
C) .1286
D) .4357
E) .3714