Respuesta :
Answer:
98% Confidence interval: (4.144 ,4.498)
Step-by-step explanation:
We are given the following in the question:
4.751, 4.373, 4.177, 4.676, 4.425, 4.228, 4.125, 4.251, 3.951, 4.192, 4.291, 4.414
Formula:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{51.854}{12} = 4.321[/tex]
Sum of squares of differences = 0.5602
[tex]s = \sqrt{\dfrac{0.5602}{11}} = 0.226[/tex]
98% Confidence interval:
[tex]\bar{x} \pm t_{critical}\displaystyle\frac{s}{\sqrt{n}}[/tex]
Putting the values, we get,
[tex]t_{critical}\text{ at degree of freedom 11 and}~\alpha_{0.02} = \pm 2.718[/tex]
[tex]4.321 \pm 2.718(\frac{0.226}{\sqrt{12}} ) = 4.321 \pm 0.1773 = (4.144 ,4.498)[/tex]
Answer:
98% confidence interval for the mean rate = [4.1437 , 4.4983]
Step-by-step explanation:
We are given the interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia for one day ;
4.751, 4.373, 4.177, 4.676, 4.425, 4.228, 4.125, 4.251, 3.951, 4.192, 4.291, 4.414
Now, Firstly we will find Mean of above data, Xbar ;
Mean, Xbar = [tex]\frac{\sum X}{n}[/tex] = [tex]\frac{4.751 +4.373+ 4.177+ 4.676+ 4.425+ 4.228+ 4.125 +4.251 +3.951 +4.192+ 4.291 +4.414}{12}[/tex] = 4.321
Standard deviation, s = [tex]\sqrt{\frac{\sum (X-Xbar)^{2} }{n-1} }[/tex] = 0.226
Now, the pivotal quantity for 98% confidence interval for the mean rate is;
P.Q. = [tex]\frac{Xbar-\mu}{\frac{s}{\sqrt{n} }}[/tex] ~ [tex]t_n_-_1[/tex]
where, Xbar = sample mean
s = sample standard deviation
n = sample size = 12
So, 98% confidence interval for the mean rate, [tex]\mu[/tex] is ;
P(-2.718 < [tex]t_1_1[/tex] < 2.718) = 0.98
P(-2.718 < [tex]\frac{Xbar-\mu}{\frac{s}{\sqrt{n} }}[/tex] < 2.718) = 0.98
P(Xbar - 2.718 * [tex]{\frac{s}{\sqrt{n} }}[/tex] < [tex]\mu[/tex] < Xbar + 2.718 * [tex]{\frac{s}{\sqrt{n} }}[/tex] ) = 0.98
98% confidence interval for [tex]\mu[/tex] = [Xbar - 2.718 * [tex]{\frac{s}{\sqrt{n} }}[/tex] , Xbar + 2.718 * [tex]{\frac{s}{\sqrt{n} }}[/tex] ]
= [4.321 - 2.718 * [tex]{\frac{0.226}{\sqrt{12} }}[/tex] , 4.321 + 2.718 * [tex]{\frac{0.226}{\sqrt{12} }}[/tex] ]
= [4.1437 , 4.4983]
Therefore, 98% confidence interval for the mean rate = [4.1437 , 4.4983]
