Exam 18: Simple Linear Regression and Correlation

In testing the hypotheses: $H _ { 0 } : \rho _ { s } = 0$ $H _ { 1 } : D _ { s } \neq 0$ , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

The Spearman rank correlation coefficient must be used to determine whether a relationship exists between two variables when:

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Book } & \text { Number of pages } & \text { Selling price (\$) } \\\hline 1 & 844 & 55 \\\hline 2 & 727 & 50 \\\hline 3 & 360 & 35 \\\hline 4 & 915 & 60 \\\hline 5 & 295 & 30 \\\hline 6 & 706 & 50 \\\hline 7 & 410 & 40 \\\hline 8 & 905 & 53 \\\hline 9 & 1058 & 65 \\\hline 10 & 865 & 54 \\\hline 11 & 677 & 42 \\\hline 12 & 912 & 58 \\\hline\end{array}$ Interpret the value of the slope of the regression line.

One method of diagnosing autocorrelation is to plot the residuals against the time periods to see whether some pattern emerges.

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: $\sum ( x - \bar { x } ) ( y - \bar { y } )$ = 2250, $S _ { x }$ = 10, $\sum x$ = 50, $\sum y$ = 75 The least squares estimates of the slope and y-intercept are respectively:

Which of the following is not a required condition for the error variable $\varepsilon$ in the simple linear regression model?

When the sample size n is greater than 30, the Spearman rank correlation coefficient $r _ { s }$ is approximately normally distributed with:

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Determine the least squares regression line.

When the actual values y of a dependent variable and the corresponding predicted values are the same, the standard error of estimate, $S _ { \varepsilon }$ , will be -1.0.

In simple linear regression, the coefficient of correlation r and the least squares estimate $b _ { 1 }$ of the population slope $\beta _ { 1 }$ :

The manager of a fast food restaurant wants to determine how sales in a given week are related to the number of discount vouchers (#) printed in the local newspaper during the week. The number of vouchers and sales ($000s) from 10 randomly selected weeks is given below with Excel regression output. Determine the standard error of the estimate and describe what this statistic sells you about the regression line.

The manager of a fast food restaurant wants to determine how sales in a given week are related to the number of discount vouchers (#) printed in the local newspaper during the week. The number of vouchers and sales ($000s) from 10 randomly selected weeks is given below with Excel regression output.

The coefficient of determination is the coefficient of correlation squared.

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Predict with 95% confidence the income of an individual with 10 years of education.

The value of the sum of squares for regression, SSR, can never be smaller than 1.

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) impacts daily hot coffee sales revenue (in $00's). A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature of that day noted. Excel output given below. Coffee sales revenue Temperature 6.50 25 10.00 17 5.50 30 4.50 35 3.50 40 28.00 9 Test the hypothesis that the population intercept is positive, at the 5% level of significance.

A statistician investigating the relationship between the amount of precipitation (in inches) and the number of car accidents gathered data for 10 randomly selected days. The results are presented below. Day Precipitation Number of accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 8 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 Calculate the Spearman rank correlation coefficient, and test to determine at the 5% significance level whether we can infer that a linear relationship exists between the number of accidents and the amount of precipitation.

Which value of the coefficient of correlation r indicates a stronger correlation than − 0.85?

The standardised residual is defined as:

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be: