Question

Mortgage rates: Following are interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia for one day. It is reasonable to assume that the population is approximately normal. 4.751 4.373 4.177 4.676 4.425 4.228 4.125 4.251 3.951 4.192 4.291 4.414 Send data to Excel Part: 0 / 2 Part 1 of 2 (a) Construct a 98% confidence interval for the mean rate. Round the answer to at least four decimal places. A 98% confidence interval for the mean rate is

Answer 1

98% Confidence interval: (4.144 ,4.498)

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:

`\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}`

where
`x_i` are data points,
`\bar{x}` is the mean and n is the number of observations.

`Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}`

`Mean =\displaystyle(51.854)/(12) = 4.321`

Sum of squares of differences = 0.5602

`s = \sqrt{(0.5602)/(11)} = 0.226`

98% Confidence interval:

`\bar{x} \pm t_(critical)\displaystyle(s)/(√(n))`

Putting the values, we get,

`t_(critical)\text{ at degree of freedom 11 and}~\alpha_(0.02) = \pm 2.718`

`4.321 \pm 2.718((0.226)/(√(12)) ) = 4.321 \pm 0.1773 = (4.144 ,4.498)`

Answer 2

98% confidence interval for the mean rate = [4.1437 , 4.4983]

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 =
`(\sum X)/(n)` =
`(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)` = 4.321

Standard deviation, s =
`\sqrt{(\sum (X-Xbar)^(2) )/(n-1) }` = 0.226

Now, the pivotal quantity for 98% confidence interval for the mean rate is;

P.Q. =
`(Xbar-\mu)/((s)/(√(n) ))` ~
`t_n_-_1`

where, Xbar = sample mean

s = sample standard deviation

n = sample size = 12

So, 98% confidence interval for the mean rate,
`\mu` is ;

P(-2.718 <
`t_1_1` < 2.718) = 0.98

P(-2.718 <
`(Xbar-\mu)/((s)/(√(n) ))` < 2.718) = 0.98

P(Xbar - 2.718 *
`{(s)/(√(n) )}` <
`\mu` < Xbar + 2.718 *
`{(s)/(√(n) )}` ) = 0.98

98% confidence interval for
`\mu` = [Xbar - 2.718 *
`{(s)/(√(n) )}` , Xbar + 2.718 *
`{(s)/(√(n) )}` ]

= [4.321 - 2.718 *
`{(0.226)/(√(12) )}` , 4.321 + 2.718 *
`{(0.226)/(√(12) )}` ]

= [4.1437 , 4.4983]

Therefore, 98% confidence interval for the mean rate = [4.1437 , 4.4983]