The average number of violent crimes per 100000 inhabitants

The average number of violent crimes per 100,000 inhabitants in the U.S. in 2006 was 431.52. Assume we know the population standard deviation for the number of violent crimes per 100,000 inhabitants is 238.38. Suppose we want to see if there has been a significant decrease in the number of violent crimes. We take a sample of 30 states in 2013 and find that the average number of violent crimes per 100,000 inhabitants is 365.46. Use =.10.

               a. What are the null and alternative hypotheses?

               b. What is the critical value?

               c. Calculate the test statistic.

               d. Find the p-value.

               e. What conclusion is made here?

Solution

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   431.52   [ANSWER]
Ha:    u   <   431.52   [ANSWER]
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b)

              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical z, as alpha =    0.1   ,      
alpha =    0.1          
zcrit =    -   1.281551566   [ANSWER]  
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C)              
Getting the test statistic, as              
              
X = sample mean =    365.46          
uo = hypothesized mean =    431.52          
n = sample size =    30          
s = standard deviation =    238.38          
              
Thus, z = (X - uo) * sqrt(n) / s =    -1.517851839 [ANSWER]

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d)          
              
Also, the p value is              
              
p =    0.064525877   [ANSWER]

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As P < 0.10, we reject Ho.

Then, there is significant evidence that there has been a decrease in the number of violent crimes. [ANSWER]