Question

In 2011, the mean rate of violent crime (per 100,000 people) for the 24 states west of the Mississippi River was 406. The standard deviation was 177. Assume that the distribution of violent crime rates is approximately unimodal and symmetric. Please show work.

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

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

c) If a western state had a violent crime rate of 584 crimes per 100,000 people, would you consider this unusual? Explain.

d) If a western state had a violent crime rate of 30 crimes per 100,000 people, would you consider this unusual? Explain.

Answer 1

Let X be the violent crime rate in a state west of the Mississippi River. X has a mean and standard deviation

Since the distribution of violent crime rates is approximately unimodal and symmetric, we can say that X is approximately normally distributed.

We will use the following empirical rule about the normal distribution

• Approximately 68% of the values in a normal distribution lie within +- 1 standard deviation from the mean
• Approximately 95% of the values in a normal distribution lie within +- 2 standard deviation from the mean
• Approximately 99.7% of the values in a normal distribution lie within +- 3 standard deviation from the mean

a) Between what two values would you expect to find about 95% of the rates?
Using the empirical rule, we can say that 95% of the rates are between +- 2 standard deviations from the mean. That is 95% of the crime rates are between

ans: you expect to find about 95% of the rates between 52 and 760

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

Using the empirical rule, we can say that 68% of the rates are between +- 1 standard deviations from the mean. That is 68% of the crime rates are between

ans: you expect to find about 68% of the violent crime rates between 229 and 583

c) If a western state had a violent crime rate of 584 crimes per 100,000 people, would you consider this unusual? Explain.

First we will find the z-score of the value 584.

This means that 584 is approximately just above 1 standard deviation from the mean crime rate.

Approximately 68% of the values in a normal distribution lie within +- 1 standard deviation from the mean. Since the distribution is symmetric, we can say that  (100-68)/2=16% of the crime rates would be above 1 standard deviation from the mean.

In other words there is a 16% chance that a western state has a crime rate which is 583 and above, which is not unusual.

ans: We would not consider a western state having a violent crime rate of 584 crimes per 100,000 people as unusual as there is a 16% chance that a western state will have a crime rate which is 583 or above.

d) If a western state had a violent crime rate of 30 crimes per 100,000 people, would you consider this unusual? Explain.

First we will find the z-score of 30.

This means that 30 is below 2 standard deviations from the mean crime rate.

Approximately 95% of the values in a normal distribution lie within +- 2 standard deviation from the mean. Since the distribution is symmetric, we can say that  (100-95)/2=2.5% of the crime rates would be below 2 standard deviations from mean. From part a) we know that 52 is 2 standard deviations below the mean and 30 is well below 52.

In other words there is less than 2.5% chance that a western state has a crime rate which is 30 and below , which is very unusual.

ans: We would consider a western state having a violent crime rate of 30 crimes per 100,000 people as unusual as there is only a 2.5% chance that a western state will have a crime rate which is 52 or below.