In: Statistics and Probability
Do larger universities tend to have more property crime? University crime statistics are affected by a variety of factors. The surrounding community, accessibility given to outside visitors, and many other factors influence crime rate. Let x be a variable that represents student enrollment (in thousands) on a university campus, and let y be a variable that represents the number of burglaries in a year on the university campus. A random sample of n = 8 universities in California gave the following information about enrollments and annual burglary incidents.
x | 14.5 | 30.2 | 24.5 | 14.3 | 7.5 | 27.7 | 16.2 | 20.1 |
---|---|---|---|---|---|---|---|---|
y | 25 | 69 | 39 | 23 | 15 | 30 | 15 | 25 |
(a)
Make a scatter diagram of the data. Then visualize the line you think best fits the data. (Submit a file with a maximum size of 1 MB.)
CrimeonCampus.xlsx
This answer has not been graded yet.
(b)
Use a calculator to verify that Σ(x) = 155.0,
Σ(x2) = 3417.02, Σ(y) = 241,
Σ(y2) = 9411 and Σ(x y) =
5419.7.
Compute r. (Enter a number. Round to 3 decimal
places.)
As x increases, does the value of r imply that
y should tend to increase or decrease? Explain your
answer.
Given our value of r, y should tend to decrease as x increases.Given our value of r, y should tend to remain constant as x increases. Given our value of r, y should tend to increase as x increases.Given our value of r, we can not draw any conclusions for the behavior of y as x increases.
(b)
Verify the given sums Σx, Σy, Σx2, Σy2, Σx y, and the value of the sample correlation coefficient r. (For each answer, enter a number. Round your value for r to three decimal places.)
Σx =
Σy =
Σx2 =
Σy2 =
Σx y =
r =
(c)
Find , and . Then find the equation of the least-squares line = a + b x. (For each answer, enter a number. Round your answers for and to two decimal places. Round your answers for a and b to three decimal places.)
= x bar =
= y bar =
= value of a coefficient + value of b coefficient x
(d)
Graph the least-squares line. Be sure to plot the point (, ) as a point on the line. (Select the correct graph.)
(e)
Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (For each answer, enter a number. Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)
r2 =
explained = %
unexplained = %
(f)
The calves you want to buy are 25 weeks old. What does the
least-squares line predict for a healthy weight (in kg)? (Enter a
number. Round your answer to two decimal places.)
kg
(a) Using excel:
We notice an upward trend from the above plot:
(b) Using excel we may verify Σ(x) = 155.0, Σ(x2) = 3417.02, Σ(y) = 241, Σ(y2) = 9411 and Σ(x y) = 5419.7 as follows:
x | y | x2 | y2 | xy |
14.5 | 25 | 210.25 | 625 | 362.5 |
30.2 | 69 | 912.04 | 4761 | 2083.8 |
24.5 | 39 | 600.25 | 1521 | 955.5 |
14.3 | 23 | 204.49 | 529 | 328.9 |
7.5 | 15 | 56.25 | 225 | 112.5 |
27.7 | 30 | 767.29 | 900 | 831 |
16.2 | 15 | 262.44 | 225 | 243 |
20.1 | 25 | 404.01 | 625 | 502.5 |
155 | 241 | 3417.02 | 9411 | 5419.7 |
The Pearson's correlation r can be computed using the formula:
r = 0.795
The correlation coefficient r measures the strength and direction of the linear relationship between x and y. It ranges from -1 to 1; negative and positive values indicating a negative and positive linear relationship respectively. Values close to unity, depicts a strong linear relationship and those close to zero implies weak or no linear relationship.Here , R =0.795. This may be interpreted as: There is a moderately strong positive linear relationship between the two variables.
This implies that given our value of r, y should tend to increase as x increases
(b) Verifying r: Using CORREL function in excel,
We get r = 0.795
(c) Estimating the intercept (a) and Slope (b) coefficients :
Substituting the values,
= 1.813
a = 30.125 - (1.813)(19.375)
= -4.999
The fitted regression equation can be expressed as:
y = -4.999 + 1.813x
(d) Fitting a least square line:
(e) Also, here, r2 = (0.795)2 = 0.632
About 63.2% of the variation in y can be explained by the corresponding variation in x and the least-squares line. And about 1 - 0.632 = 36.8% percentage of the variation in y is unexplained.
r2 = 0.632
Explained = 63.2%
Unexplained = 36.8%