Exam 16: A: Simple Linear Regression and Correlation

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Melanoma per 100,000 7 11 9 12 15 10 7 5 -{Sunshine and Melanoma Narrative} Determine the least squares regression line.

Wayne Newton Concert At a recent Wayne Newton concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number af Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Wayne Newton Concert Narrative} Estimate with 95% confidence the average number of concerts attended by all 45 year-old individuals.

Allman Brothers Concert At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number af Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Allman Brothers Concert Narrative} Interpret the value of the slope of the regression line.

Speed vs Gas Mileage An economist wanted to analyze the relationship between the speed of a car (x)and its gas mileage (y).As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured.These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 -{Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.

In order to estimate with 95% confidence the expected value of y for a given value of x in a simple linear regression problem,a random sample of 10 observations is taken.Which of the following t-table values listed below would be used?

Given the least squares regression line $\hat { y } = 5 - 2 x$ :

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,

In simple linear regression,the coefficient of correlation r and the least squares estimate b₁ of the population slope $\beta$ ₁:

Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Analysis of Variance Source DF SS MS F p Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 -{Oil Quality and Price Narrative} For what values of API gravity degrees do we feel comfortable making predictions for oil price?

The residual is defined as the difference between:

Allman Brothers Concert At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number af Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Allman Brothers Concert Narrative} Estimate the number of Allman Brothers concerts attended by a 64 year old person.

The unbiased estimator of the variance of the error variable is found by taking ____________________ divided by n - 2.

In the first-order linear regression model,the population parameters of the y-intercept and the slope are,respectively,

The symbol for the sample coefficient of correlation is:

Accidents and Rain A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below. Day Rain Number of Accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 -{Accidents and Rain Narrative} Estimate the number of accidents in a day with 0.25 inches of rain.

Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Analysis of Variance Source DF SS MS F p Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 -When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same,the standard error of the estimate will be 1.0.

Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Melanoma per 100,000 7 11 9 12 15 10 7 5 -{Sunshine and Melanoma Narrative} Calculate the residual corresponding to the pair (x,y)= (8,15).

U V's and Skin Cancer A medical statistician wanted to examine the relationship between the amount of UV's (x)and incidence of skin cancer (y).As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight states around the country.These data are shown below. Average Daily UV's 5 7 6 7 8 6 4 3 Skin Cancer per 100,000 7 11 9 12 15 10 7 5 -{UV's and Skin Cancer Narrative} What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

The graph of a confidence interval for the expected value of y is represented by two parallel lines,one on either side of the regression line.