Question

The city council of Pine Bluffs is considering increasing the number of police in an effort to reduce crime. Before making a final decision, the council asked the chief of police to survey other cities of similar size to determine the relationship between the number of police and the number of crimes reported. The chief gathered the following sample information.
City Police Number of Crimes City Police Number of Crimes
Oxford 15 17 Holgate 17 7
Starksville17 13 Carey 12 21
Danville 25 5 Whistler 11 19
Athens 27 7 Woodville 22 6
a) Determine the regression equation. (Round your answers to 4 decimal places. Negative amounts should be indicated by a minus sign.)
b) Estimate the number of crimes for a city with 20 police officers. (Round your answer to the nearest whole number.)

Answer 1

Answer:

Part a

The regression equation is, $\hat y = 29.3877 - 0.9596x$

Part b

The number of crimes for a city at 20 police officers was appointed is approximately 10.

Step-by-step explanation:

x          y         xy         x²         y²

15      17        255      225         289

17      7            119      289          49

17      13         221       289        169

12      21          252      144          441

25      5         125       625          25

11       19          209        121           361

27     7           189        729           49

22    6         132       484          36

146    95      502      2906      1419

Therefore,

$\hat B_1=\frac{n \sum xy - (\sum x) (\sum y)}{n (\sum x^2) - (\sum x)^2}$

$= \frac{8(1502) - (146)(95)}{8(2906)-(146)^2} \\\\= \frac{-1854}{1932} \\\\= - 0.9596$

The calculation of the intercept coefficient is as follows:

$\hat B_0= \frac{\sum y }{n} - \hat B_1 (\frac{\sum x}{n} )$

$= \frac{95}{8} - (-0.9596)(\frac{146}{8} )\\\\= 11.875 + 17.5127\\\\= 29.3877$

The regression equation is, $\hat y = 29.3877 - 0.9596x$

b)

The calculation is as follows:

$\hat y= 29.3877 - 0.9596x\\\\= 29.3877-0.9596(20)\\\\=29.3877-19.1920\\\\=10.1957$

≅ 10

The number of crimes for a city at 20 police officers was appointed is approximately 10.

The number of crimes for a city at 20 police officers was appointed is approximately 10.

That is, the number of polices increases then the number of crimes will be decreases.

Answer 2

Using technology such as Excel or linear regression calculator, the regression equation and predicted number of crimes for a city with 20 police officers are :

• Ŷ = - 0.5169X + 20.6631
• Predicted number of crimes = 10

From the data given:

City police (X) : 15, 17, 25, 27, 7, 12, 11, 22

Number of crimes (Y) : 17, 13, 5, 7, 7, 21, 19, 6

Using a regression calculator, the linear regression equation which models the data is ;

• Ŷ = - 0.5169X + 20.6631

B.)

The number of crimes for a city with 20 police officers can be calculated thus :

• X = 20

Substitute the value of X into the equation

Ŷ = -0.5169(20) + 20.6631

Ŷ = 10.3251

Hence, the predicted number of crimes for a city with 20 police officers is 10.

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