Question 1

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. $$\begin{array} { | c | c | c | } 
\hline \text { Day } & \text { Precipitation } & \text { Number of accidents } \
\hline 1 & 0.05 & 5 \
\hline 2 & 0.12 & 6 \
\hline 3 & 0.05 & 2 \
\hline 4 & 0.08 & 4 \
\hline 5 & 0.10 & 8 \
\hline 6 & 0.35 & 14 \
\hline 7 & 0.15 & 7 \
\hline 8 & 0.30 & 13 \
\hline 9 & 0.10 & 7 \
\hline 10 & 0.20 & 10 \
\hline
\end{array}$$ 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.

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Question 2

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between age and number of concerts attended.

Accepted Answer

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Question 3

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. $$\begin{array} { | c | c | } 
\hline \text { Oil degrees API } & \text { Price per barrel (in } \$ \text { ) } \
\hline 27.0 & 12.02 \
\hline 28.5 & 12.04 \
\hline 30.8 & 12.32 \
\hline 31.3 & 12.27 \
\hline 31.9 & 12.49 \
\hline 34.5 & 12.70 \
\hline 34.0 & 12.80 \
\hline 34.7 & 13.00 \
\hline 37.0 & 13.00 \
\hline 41.0 & 13.17 \
\hline 41.0 & 13.19 \
\hline 38.8 & 13.22 \
\hline 39.3 & 13.27 \
\hline
\end{array}$$ A partial Minitab output follows.
Descriptive Statistics $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \
\hline \text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \
\hline \text { Frice } & 13 & 12.730 & 0.457 & 0.127 \
\hline
\end{array}$$ Covariances $$\begin{array} { | l | r | r | } 
\hline & \text { Degrees } & \text { Price } \
\hline \text { Degrees } & 21.281667 & \
\hline \text { Price } & 2.026750 & 0.208833 \
\hline
\end{array}$$ Regression Analysis $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Fredictor } & \text { Coef } & \text { StDev } & \mathrm { T } & \mathrm { P } \
\hline \text { Constant } & 9.4349 & 0.2867 & 32.91 & 0.000 \
\hline \text { Degrees } & 0.095235 & 0.008220 & 11.59 & 0.000 \
\hline
\end{array}$$ S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7%
Analysis of Variance $$\begin{array} { | l | r | r | r | r | r | } 
\hline \text { Source } & \text { DF } & \text { SS } & \text { MS } & \mathrm { F } & \mathrm { P } \
\hline \text { Regression } & 1 & 2.3162 & 2.3162 & 134.24 & 0.000 \
\hline \text { Residual Error } & 11 & 0.1898 & 0.0173 & & \
\hline \text { Total } & 12 & 2.5060 & & & \
\hline
\end{array}$$ a. Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the quality of oil and price per barrel.
b. Determine the least squares regression line.
c. Redraw the scatter diagram and plot the least squares regression line on it.
d. Interpret the value of the slope of the regression line.

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Question 4

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  a. Predict with 95% confidence the number of concerts attended by a 45-year-old individual.
b. Predict with 95% confidence the average number of concerts attended by all 45-year-old individuals.

Accepted Answer

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Question 5

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  a. Determine the standard error of estimate and describe what this statistic tells you about the model's fit.
b. Interpret the coefficient of correlation.
c. Interpret the coefficient of determination, and discuss what its value tells you about the two variables.

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Question 6

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  a. Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the age and number of concerts attended by the respondents.
b. Determine the least squares regression line.
c. Plot the least squares regression line.
d. Interpret the value of the slope of the regression line.

Accepted Answer

The answer of At a recent music concert, a survey...

Question 7

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. $$\begin{array} { | c | c | } 
\hline \text { Oil degrees API } & \text { Price per barrel (in } \$ \text { ) } \
\hline 27.0 & 12.02 \
\hline 28.5 & 12.04 \
\hline 30.8 & 12.32 \
\hline 31.3 & 12.27 \
\hline 31.9 & 12.49 \
\hline 34.5 & 12.70 \
\hline 34.0 & 12.80 \
\hline 34.7 & 13.00 \
\hline 37.0 & 13.00 \
\hline 41.0 & 13.17 \
\hline 41.0 & 13.19 \
\hline 38.8 & 13.22 \
\hline 39.3 & 13.27 \
\hline
\end{array}$$ A partial computer output follows.
Descriptive Statistics $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \
\hline \text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \
\hline \text { Frice } & 13 & 12.730 & 0.457 & 0.127 \
\hline
\end{array}$$ Covariances $$\begin{array} { | l | r | r | } 
\hline & \text { Degrees } & \text { Price } \
\hline \text { Degrees } & 21.281667 & \
\hline \text { Price } & 2.026750 & 0.208833 \
\hline
\end{array}$$ Regression Analysis $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Fredictor } & \text { Coef } & \text { StDev } & \mathrm { T } & \mathrm { P } \
\hline \text { Constant } & 9.4349 & 0.2867 & 32.91 & 0.000 \
\hline \text { Degrees } & 0.095235 & 0.008220 & 11.59 & 0.000 \
\hline
\end{array}$$ S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7%
Analysis of Variance $$\begin{array} { | l | r | r | r | r | r | } 
\hline \text { Source } & \text { DF } & \text { SS } & \text { MS } & \mathrm { F } & \mathrm { P } \
\hline \text { Regression } & 1 & 2.3162 & 2.3162 & 134.24 & 0.000 \
\hline \text { Residual Error } & 11 & 0.1898 & 0.0173 & & \
\hline \text { Total } & 12 & 2.5060 & & & \
\hline
\end{array}$$ Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between the quality of oil and price per barrel.

Accepted Answer

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Question 8

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  a. Use the regression equation $\hat { y }$ = -3.0115 + 0.1257x to determine the predicted values of y.
b. Use the predicted values and the actual values of y to calculate the residuals.
c. Plot the residuals against the predicted values $\hat { y }$ .
d. Does it appear that heteroscedasticity is a problem? Explain.
e. Draw a histogram of the residuals.
f. Does it appear that the errors are normally distributed? Explain.
g. Use the residuals to compute the standardised residuals.
h. Identify possible outliers.

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Question 9

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. $$\begin{array} { | c | c | c | } 
\hline \text { Day } & \text { Precipitation } & \text { Number of accidents } \
\hline 1 & 0.05 & 5 \
\hline 2 & 0.12 & 6 \
\hline 3 & 0.05 & 2 \
\hline 4 & 0.08 & 4 \
\hline 5 & 0.10 & 8 \
\hline 6 & 0.35 & 14 \
\hline 7 & 0.15 & 7 \
\hline 8 & 0.30 & 13 \
\hline 9 & 0.10 & 7 \
\hline 10 & 0.20 & 10 \
\hline
\end{array}$$ Estimate with 95% confidence the mean daily number of accidents when the daily precipitation is 0.25 inches.

Accepted Answer

The answer of A statistician investigating the relationship between the...

Question 10

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. $$\begin{array} { | c | c | } 
\hline \text { Oil degrees API } & \text { Price per barrel (in } \$ \text { ) } \
\hline 27.0 & 12.02 \
\hline 28.5 & 12.04 \
\hline 30.8 & 12.32 \
\hline 31.3 & 12.27 \
\hline 31.9 & 12.49 \
\hline 34.5 & 12.70 \
\hline 34.0 & 12.80 \
\hline 34.7 & 13.00 \
\hline 37.0 & 13.00 \
\hline 41.0 & 13.17 \
\hline 41.0 & 13.19 \
\hline 38.8 & 13.22 \
\hline 39.3 & 13.27 \
\hline
\end{array}$$ A partial computer output follows.
Descriptive Statistics $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \
\hline \text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \
\hline \text { Frice } & 13 & 12.730 & 0.457 & 0.127 \
\hline
\end{array}$$ Covariances $$\begin{array} { | l | r | r | } 
\hline & \text { Degrees } & \text { Price } \
\hline \text { Degrees } & 21.281667 & \
\hline \text { Price } & 2.026750 & 0.208833 \
\hline
\end{array}$$ Regression Analysis $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Fredictor } & \text { Coef } & \text { StDev } & \mathrm { T } & \mathrm { P } \
\hline \text { Constant } & 9.4349 & 0.2867 & 32.91 & 0.000 \
\hline \text { Degrees } & 0.095235 & 0.008220 & 11.59 & 0.000 \
\hline
\end{array}$$ se = 0.1314 R2 = 92.46% R2(adjusted) = 91.7%
Analysis of Variance $$\begin{array} { | l | r | r | r | r | r | } 
\hline \text { Source } & \text { DF } & \text { SS } & \text { MS } & \mathrm { F } & \mathrm { P } \
\hline \text { Regression } & 1 & 2.3162 & 2.3162 & 134.24 & 0.000 \
\hline \text { Residual Error } & 11 & 0.1898 & 0.0173 & & \
\hline \text { Total } & 12 & 2.5060 & & & \
\hline
\end{array}$$ a. Determine the standard error of estimate and describe what this statistic tells you.
b. Determine the coefficient of determination and discuss what its value tells you about the two variables.
c. Calculate the Pearson correlation coefficient. What sign does it have? Why?

Accepted Answer

The answer of The quality of oil is measured in...

Question 11

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. $$\begin{array} { | c | c | c | } 
\hline \text { Day } & \text { Precipitation } & \text { Number of accidents } \
\hline 1 & 0.05 & 5 \
\hline 2 & 0.12 & 6 \
\hline 3 & 0.05 & 2 \
\hline 4 & 0.08 & 4 \
\hline 5 & 0.10 & 8 \
\hline 6 & 0.35 & 14 \
\hline 7 & 0.15 & 7 \
\hline 8 & 0.30 & 13 \
\hline 9 & 0.10 & 7 \
\hline 10 & 0.20 & 10 \
\hline
\end{array}$$ Predict with 95% confidence the number of accidents that occur when there is 0.40 inches of rain.

Accepted Answer

The answer of A statistician investigating the relationship between the...

Question 12

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. $$\begin{array} { | c | c | } 
\hline \text { Oil degrees API } & \text { Price per barrel (in } \$ \text { ) } \
\hline 27.0 & 12.02 \
\hline 28.5 & 12.04 \
\hline 30.8 & 12.32 \
\hline 31.3 & 12.27 \
\hline 31.9 & 12.49 \
\hline 34.5 & 12.70 \
\hline 34.0 & 12.80 \
\hline 34.7 & 13.00 \
\hline 37.0 & 13.00 \
\hline 41.0 & 13.17 \
\hline 41.0 & 13.19 \
\hline 38.8 & 13.22 \
\hline 39.3 & 13.27 \
\hline
\end{array}$$ A partial computer output follows.
Descriptive Statistics $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \
\hline \text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \
\hline \text { Frice } & 13 & 12.730 & 0.457 & 0.127 \
\hline
\end{array}$$ Covariances $$\begin{array} { | l | r | r | } 
\hline & \text { Degrees } & \text { Price } \
\hline \text { Degrees } & 21.281667 & \
\hline \text { Price } & 2.026750 & 0.208833 \
\hline
\end{array}$$ Regression Analysis $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Fredictor } & \text { Coef } & \text { StDev } & \mathrm { T } & \mathrm { P } \
\hline \text { Constant } & 9.4349 & 0.2867 & 32.91 & 0.000 \
\hline \text { Degrees } & 0.095235 & 0.008220 & 11.59 & 0.000 \
\hline
\end{array}$$ S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7%
Analysis of Variance $$\begin{array} { | l | r | r | r | r | r | } 
\hline \text { Source } & \text { DF } & \text { SS } & \text { MS } & \mathrm { F } & \mathrm { P } \
\hline \text { Regression } & 1 & 2.3162 & 2.3162 & 134.24 & 0.000 \
\hline \text { Residual Error } & 11 & 0.1898 & 0.0173 & & \
\hline \text { Total } & 12 & 2.5060 & & & \
\hline
\end{array}$$ Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between the quality of oil and price per barrel.

Accepted Answer

The answer of The quality of oil is measured in...

Question 13

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \
\hline \text { Number of concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \
\hline
\end{array}$$ $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \
\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \
\hline
\end{array}$$  a. Calculate the Pearson correlation coefficient. What sign does it have? Why?
b. Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between age and number of concerts attended.

Accepted Answer

The answer of At a recent music concert, a survey...

Quiz 15: Simple Linear Regression and Correlation