Question

Asked Jul 19, 2019

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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 |
14.3 | 31.8 | 24.5 | 14.3 | 7.5 | 27.7 | 16.2 | 20.1 |

y |
24 | 70 | 39 | 23 | 15 | 30 | 15 | 25 |

(a) Make a scatter diagram of the data.

Then visualize the line you think best fits the data.

(b) Use a calculator to verify that Σ*x* = 156.4, Σ*x*^{2} = 3510.46, Σ*y* = 241, Σ*y*^{2} = 9501 and Σ*xy* = 5542.6.

Compute*r*. (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.

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(b) Use a calculator to verify that Σ

Compute

As

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.

Step 1

**(a) Scatter diagram of the data:**

The least-squares regression line is the best fitted to the given data.

The procedure to draw scatterplot with least-squares line using EXCEL is as follows.

- Select the data
*x*and*y*. - Go to
**Insert**. - Select
**Scatter**under**Charts**. - Under
**chart elements**, check**Trendline**.

The scatter diagram for the data that is obtained using EXCEL is as follows

Step 2

**(b) Computation of r:**

Using calculator, the following preliminary calculations verified.

*∑x* = (14.3 + 31.8 + 24.5 + 14.3 + 7.5 + 27.7 + 16.2 + 20.1) = 156.4.

*∑x*^{2} = (14.3^{2} + 31.8^{2} + 24.5^{2} + 14.3^{2} + 7.5^{2} + 27.7^{2} + 16.2^{2} + 20.1^{2}) = 3,510.46.

*∑y* = (24 + 70 + 39 + 23 + 15 + 30 + 15 + 25) = 241.

*∑y*^{2} = (24^{2} + 70^{2} + 39^{2} + 23^{2} + 15^{2} + 30^{2} + 15^{2} + 25^{2}) = 9,501.

*∑xy* = (14.3×24) + (31.8×70) + (24.5×39) + (14.3×23) + (7.5×15) + (27.7×30) + (16.2×15) + (20.1×25) = 5,542.6.

Here, the number of pair of observations, *n* = 8.

The EXCEL formula to find the value of correlation coefficient, *r* is “=CORREL(array1, array2)”.

The correlation coefficient, *r* is calculated as **0.825** using the EXCEL formula “=CORREL(A3:A10,B3:B10)”.

Step 3

**Interpretation of correlation coefficient:**

The correlation coefficient is 0.825 which is a positive value. This represents that as student enrolment on a university campus (*x*) increases, the number of bur...

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