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Exam 11: Correlation Coefficient and Simple Linear Regression Analysis

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Question 51

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Calculate the correlation coefficient.

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Question 52

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The following results were obtained from a simple regression analysis: $\hat { y }$ = 37.2895 - 1.2024x, r2 = 0.6744, $s _ { b _ { 1 } }$ = 0.2934 -For each unit change in x,the estimated change in the mean of y is equal to:

A) 36.0871
B) 0.6774
C) 37.2895
D) 0.2934
E) -1.2024

F) A) and B)
G) A) and C)

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Question 53

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When using simple regression analysis,if there is a strong positive correlation between the independent and dependent variable,then we can conclude that an increase in the value of the independent variable causes an increase in the value of the dependent variable.

In simple regression analysis,the quantity $\sum ( y - \bar { y } ) ^ { 2 }$ is called the total variation.

In a simple linear regression analysis,when the constant variance assumption for the error term holds,a plot of the residual versus x:

Consider the following partial computer output from a simple linear regression analysis: $\begin{array} { l l c c c } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm { T } & \mathrm { P } \\\text { Constant } & 5566.1 & 254.0 & 21.91 & 0.000 \\\text { Independent Var } & - 210.35 & 24.19 & - &\end{array}$ S = _________ R-Sq = Analysis of Variance $\begin{array}{lccrcc}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & 3963719 & 3963719 & 75.59 & 0.000 \\\text { Residual Error } & 14 &\_\_\_ & 52439 & &\\\text { Total }&15&\_\_\_\end{array}$ -Calculate the standard error of the model.

An experiment was performed on a certain metal to determine if the strength is a function of heating time.Results based on 10 metal sheets are given below.Use the simple linear regression model. $\sum X$ = 30 $\sum X ^ { 2 }$ = 104 $\sum Y$ = 40 $\sum Y ^ { 2 }$ = 178 $\sum X Y$ = 134 -Find the estimated y-intercept.

A data set with 7 observations yielded the following.Use the simple linear regression model where y is the dependent variable and x is the independent variable. $\sum x$ = 21.57 $\sum X ^ { 2 }$ = 68.31 $\sum Y ^ { 2 }$ = 188.9 $\sum Y ^ { 2 }$ = 5,140.23 $\sum X Y$ = 590.83 SSE = 1.06 -Use the least squares regression equation,

A local grocery store wants to predict the daily sales in dollars. The manager believes that the amount of newspaper advertising significantly affects the store sales. The manager randomly selects 7 days of data consisting of daily grocery store sales (in thousands of dollars) and advertising expenditures (in thousands of dollars) . The Excel/Mega-Stat output given below summarizes the results of fitting a simple linear regression model using this data. Regression Analysis $\begin{array}{rc}\mathrm{r}^{2} 0.762 & \mathrm{n} 7 \\\mathrm{R} 0.873 & \mathrm{k} \text { 1 } \\\text { Std. Error 11.547 } & \text { Dep. Var. Sales }\end{array}$ ANOVA table $\begin{array}{rrrrrr}\hline \text { Source } & S S & d f & M S & F & p \text {-value } \\\hline \text { Regression } & 2,133.3333 & 1 & 2,133.3333 & 16.00 & .0103 \\\text { Residual } & 666.6667 & 5 & 133.3333 & & \\\hline \text { Total } & 2,800.0000 & 6 & & &\\\hline\end{array}$ $\text { Regression output }$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Confidence interval }$ $\begin{array}{rrrrrrrr}\hline\text { Variables} & \text { Coefficients } & \text { std. error } & t(d f=5) & \text { p-value } & 95 \% &95 \% \text { upper } \\& & & & & \text { lower } & \\\hline \text { Intercep } & 63.3333 & 7.9682 & 7.948 & .0005 & 42.8505 & 83.8162 \\\text { Advertising } & 6.6667 & 1.6667 & 4.000 & .0103 & &\end{array}$ -In testing the population slope for significance at a significance level of .05,what is the rejection point condition for the two-sided t test?

If there is significant autocorrelation present in a data set,the error terms are not ________________.

The following results were obtained as a part of simple linear regression analysis: R²= 0.9162 F test statistic = 81.87 At $\alpha$ = 0) 05,the null hypothesis of no linear relationship between the dependent variable and the independent variable _____.

The point estimate of the error variance in a regression model is:

Determine the 95% prediction interval for the strength of a metal sheet when the average heating time is 2.5 minutes.

_____ is a statistical technique in which we use observed data to relate a dependent variable to one or more predictor (independent)variables.

What is the coefficient of determination?

The simple linear regression model assumes there is a _____ relationship between the dependent variable and the independent variable.

Consider the following partial computer output from a simple linear regression analysis. $\begin{array} { l c c c c } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\text { Constant } & 67.05 & 20.90 & 3.21 & 0.012 \\\text { Independent Var } & 5.8167 & 0.7085 & \_\_\_& 0.000\end{array}$ S = _________ R-Sq = _______ Analysis of Variance $\begin{array} { l c c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & - & 34920 & 67.39 & 0.000 \\\text { Residual Error } & 8 & - & 518 & & \\\text { Total } & 9 & 39065 & & &\end{array}$ -Calculate the SSE.

A local tire dealer wants to predict the number of tires sold each month. The dealer believes that the number of tires sold is a linear function of the amount of money invested in advertising. The dealer randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars) . Based on the data set with 6 observations, the simple linear regression model yielded the following results. $\sum x$ = 24 $\sum X ^ { 2 }$ =124 $\sum Y$ =42 $\sum Y ^ { 2 }$ =338 $\sum X Y$ =196 -What is the value of SSE?

The dependent variable is the variable that is being described,predicted,or controlled.

Consider the following partial computer output from a simple linear regression analysis. $\begin{array}{l}\begin{array} { l l l l l } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\text { Constant } & 4.8615 & 0.5201 & 9.35 & 0.000 \\\text { Independent Var } - & 0.34655 & 0.05866 &\_\_\_ &\end{array}\\\mathrm { S } = 0.4862 \quad \mathrm { R } - \mathrm { Sq } =\_\_\_\_\end{array}$ Analysis of Variance $\begin{array}{lccccc}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & \_\_& & 34.90 & 0.000\\\text { Residual Error }&13& \_\_& \_\_\\\text { Total }&14&11.3240\end{array}$ -Write the equation of the least squares line.