Some neighborhoods in a particular city are known for high r

Some neighborhoods in a particular city are known for high rate of car theft. The monthly averages of the stolen cars in a given year are shown below. Determine the 99% confidence interval on the mean monthly stolen cars. (Assume Normal population) (6.74, 9.91) (6.08, 9.68) (6.23,8.82) (7.88, 11 99) Refer to question 3. Determine a 95% confidence interval for the population standard deviation of stolen cars in the city. Assuming all measurements are normally distributed. (1.51, 3.52) (2.28, 12.39) (1.50, 3.12) (2.26, 9.71)

Solution

3.

Note that              
              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    7.875          
t(alpha/2) = critical t for the confidence interval =    3.105806516          
s = sample standard deviation =    2.012291774          
n = sample size =    12          
df = n - 1 =    11          
Thus,              
              
Lower bound =    6.070841348          
Upper bound =    9.679158652          
              
Thus, the confidence interval is              
              
(   6.08   ,   9.68   ) [ANSWER, B]

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4.

As              
              
df = n - 1 =    11          
alpha = (1 - confidence level)/2 =    0.05          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    19.67513757          
chi^2(alpha/2) =    4.574813079          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    2.263897767          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    9.736463381          
              
Thus, the confidence interval for the variance is              
              
(   2.263897767   ,   9.736463381   )
              
Also, for the standard deviation, getting the square root of the bounds,              
              
(   1.50   ,   3.12   ) [ANSWER, C]