Question 1: In 2007, the property crime rate (per 100,000 people) for the 24 sta

Question

Question 1: In 2007, the property crime rate (per 100,000 people) for the 24 states west of the Mississppi River was 3331; the standard deviation was 729. Assume the distribution of crime rates is bell-shaped. Using the Empirical Rule:

(a) What percentage of western states would you except to have property crime rates between 2602 and 4060?

(b) What percetage of western states would you except to have property crime rates between 1873 and 4789?

Question 2: Quantitative SAT scores have a mean of 500 and a standard deviation of 100,while ACT scores have a mean of 21 and a standard deviation of 5. Assume both type of scores have distributions that are bell-shaped. Using the Empirical Rule, which is more unusual? A quantitative SAT score of 750 or an ACT score of 28? Show work please.

Explanation / Answer

QUESTION 1.

a)
We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    2602      
x2 = upper bound =    4060      
u = mean =    3331      
          
s = standard deviation =    729      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1      
z2 = upper z score = (x2 - u) / s =    1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.158655254      
P(z < z2) =    0.841344746      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.682689492 = 68.27% [ANSWER]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    1873      
x2 = upper bound =    4789      
u = mean =    3331      
          
s = standard deviation =    729      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -2      
z2 = upper z score = (x2 - u) / s =    2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.022750132      
P(z < z2) =    0.977249868      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.954499736 = 95.45% [ANSWER]

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