Question

In 2012, the mean drug offence rate (per 100,000 population) in Canada's Census Metropolitan Areas (CMAs) was 285; the standard deviation was 114. Assume that the distribution of drug offence rates is approximately bell-shaped. (Source: Statistics Canada, Canadian Centre for Justice Statistics, Uniform Crime Reporting Survey)

1. Between what two values would you expect to find about 95% of the rates?

2. Between what two values would you expect to find about 68% of the rates?

3. If a CMA had a drug offence rate of 573 crimes per 100,000 population, would you

   consider this unusual? Explain.

4. If a CMA had a drug offence rate of 214 crimes per 100,000 population, would you

   consider this unusual? Explain.

Answer 1

Suppose X follows normal distribution ( bell-shaped) with mean ( ) = 285,

and standard deviation ( ) = 114

a) Between what two values would you expect to find about 95% of the rates?

Here we want to find x scores such that P( x1 < X < x2) = 0.95

So that P(X < x1 ) =( 1- 0.95)/2 = 0.025

Z score for probability 0.025 is -z = -1.96

so that z = 1.96

Therefore,

x1 = - z* = 285 - 1.96*114 = 61.56

x2 =   + z* = 285 + 1.96*114 = 508.44

So we would expect that about 95% of the rates lies between 61.56 and 508.44, these two values.

Between what two values would you expect to find about 68% of the rates?

Here we want to find x scores such that P( x1 < X < x2) = 0.68

So that P(X < x1 ) =( 1- 0.68)/2 = 0.16

Z score for probability 0.16 is approximately equal to -1

so that -z = -1

and z = 1

Therefore,

x1 = - z* = 285 - 1*114 = 171

x2 =   + z* =285 + 1*114 = 399

So we would expect that about 68% of the rates lies between 171 and 399, these two values.

3) If a CMA had a drug offence rate of 573 crimes per 100,000 population, would you

consider this unusual? Explain.

Yes it is unusual because it is greater than the upper limit of 95% value in part 1)

that is 573 is greater than 508.44 so it is unusual.

4)

If a CMA had a drug offence rate of 214 crimes per 100,000 population, would you

consider this unusual? Explain.

No because it is lies within 95% value in part 1)  

Also note that 214 is lies within the middle of 68% area. So it is usual.