Question

Problem 10-28

Some citizens complained to city council members that there should be equal protection under the law against the occurrence of crimes. The citizens argued that this equal protection should be interpreted as indicating that high-crime areas should have more police protection than low-crime areas. Therefore, police patrols and other methods for preventing crime (such as street lighting or cleaning up abandoned areas and buildings) should be used proportionately to crime occurrence.

         The city has been broken down into 20 geographic areas, each containing 4,950 residences. The police recognize that not all crimes and offenses are reported: people do not want to become involved, consider the offenses too small to report, are too embarrassed to make a police report, or do not take the time, among other reasons. Every month, because of this, the police are contacting by phone a random sample of 990 of the 4,950 residences for data on crime. (Respondents are guaranteed anonymity.) The 990 sampled from each area showed the following incidence of crime during the past month:

AREA	NUMBER OF CRIMES	SAMPLE SIZE	CRIME RATE
1	15	990	0.015
2	5	990	0.005
3	20	990	0.020
4	19	990	0.019
5	15	990	0.015
6	30	990	0.030
7	11	990	0.011
8	18	990	0.018
9	13	990	0.013
10	4	990	0.004
11	22	990	0.022
12	16	990	0.016
13	13	990	0.013
14	15	990	0.015
15	11	990	0.011
16	33	990	0.033
17	5	990	0.005
18	22	990	0.022
19	7	990	0.007
20	33	990	0.033
	327

a. Determine P−P−, S_p, UCL and LCL for a p chart of 95 percent confidence (at Z = 1.96). (Round your answers to 4 decimal places.)

P−P−
Sp
UCL
LCL

b. What can you say about the process?

	Process is in statistical control.
	Process is out of statistical control.

Answer 1

Given data as per the question :

n = 20 Samples.

N = 990 Sample Size

NO. of crimes (p) = 327

z = 1.96

Therefore $\bar{p}= \frac{p}{N * n}$

= 327/ 20* 990 = 0.0165

$\sigma _{p} = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}$

=   $\frac{\sqrt{0.0165(1-0.0165)}}{20}$

$\sigma$_p = 0.028

$UCL = \bar{p} + z * \sigma _{p}$ = 0.0165 + 1.96 (0.028) = 0.07138

$LCL = \bar{p} - z * \sigma _{p}$ = 0.0165 - 1.96 (0.028) = −0.03838 = 0 ( IF required)

(b)

The process is in statistical control since all the crime rates are under limits.

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