Respuesta :
Answer:
a) The 95% confidence interval for the true mean reduction of intrusion attemps per day is (-2.898, 11.698).
b) As negative values are also included in the confidence interval, we can not claim that there is a significant reduction in the rate of intrusion attempts.
Step-by-step explanation:
The mean and standard deviation for the blocked intrusion attempts on each day during the first two weeks of the month are:
[tex]M_1=\dfrac{1}{14}\sum_{i=1}^{14}(56+47+49+37+38+60+50+43+43+59+50+56+54+58)\\\\\\ M_1=\dfrac{700}{14}=50[/tex]
[tex]s_1=\sqrt{\dfrac{1}{(n-1)}\sum_{i=1}^{14}(x_i-M)^2}\\\\\\s_1=\sqrt{\dfrac{1}{13}\cdot [(56-(50))^2+(47-(50))^2+(49-(50))^2+...+(54-(50))^2+(58-(50))^2]}\\\\\\[/tex]
[tex]s_1=\sqrt{\dfrac{1}{13}\cdot [(36)+(9)+...+(64)]}\\\\\\s_1=\sqrt{\dfrac{754}{13}}=\sqrt{58}\\\\\\s_1=7.616[/tex]
The mean and standard deviation for the blocked intrusion attempts on each day during the next 15 days of the month are:
[tex]M_2=\dfrac{1}{15}\sum_{i=1}^{15}(53+21+32+49+45+38+44+33+32+43+53+46+36+48+39+35+37+36+39+45)\\\\\\ M_2=\dfrac{684}{15}=45.6[/tex]
[tex]s_2=\sqrt{\dfrac{1}{(n-1)}\sum_{i=1}^{15}(x_i-M)^2}\\\\\\s_2=\sqrt{\dfrac{1}{14}\cdot [(53-(45.6))^2+(21-(45.6))^2+...+(45-(45.6))^2]}\\\\\\[/tex]
[tex]s_2=\sqrt{\dfrac{1}{14}\cdot [(54.76)+(605.16)+(184.96)+...+(0.36)]}\\\\\\s_2=\sqrt{\dfrac{1786.4}{14}}=\sqrt{127.6}\\\\\\s_2=11.296[/tex]
a) We have to calculate a 95% confidence interval for the difference between means.
The sample 1 (first two weeks), of size n1=14 has a mean of 50 and a standard deviation of 7.616.
The sample 2 (last two weeks), of size n2=15 has a mean of 45.6 and a standard deviation of 11.296.
The difference between sample means is Md=4.4.
[tex]M_d=M_1-M_2=50-45.6=4.4[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{7.616^2}{14}+\dfrac{11.296^2}{15}}\\\\\\s_{M_d}=\sqrt{4.143+8.507}=\sqrt{12.65}=3.557[/tex]
The critical t-value for a 95% confidence interval is t=2.052.
The margin of error (MOE) can be calculated as:
[tex]MOE=t\cdot s_{M_d}=2.052 \cdot 3.557=7.298[/tex]
Then, the lower and upper bounds of the confidence interval are:
[tex]LL=M_d-t \cdot s_{M_d} = 4.4-7.298=-2.898\\\\UL=M_d+t \cdot s_{M_d} = 4.4+7.298=11.698[/tex]
The 95% confidence interval for the true mean reduction of intrusion attemps per day is (-2.898, 11.698).
b) We could claim a significant reduction in the rate of intrusion if the confidence interval bounds were both positive but, in this case, the confidence interval for the true mean reduction also includes negative values, so we can not be sure that there is significant improvement.
