The National Association of Home Builders provided data on the cost of the most
ID: 3340924 • Letter: T
Question
The National Association of Home Builders provided data on the cost of the most popular home remodeling projects. Sample data on cost in thousands of dollars for two types of remodeling projects are as follows. Excel File: data10-41.xsx Kitchen Master Bedroom Kitchen Master Bed room 25.2 17.4 22.8 21.9 19.7 18.0 22.9 264 24.8 26.9 23.0 19.7 16.9 21.8 23.6 17.8 24.6 21.0 a. Develop a two point estimate of the difference between the population mean remodeling costs for the two types of projects. Enter negative values as negative numbers. Point estimate S (Report in dollars with no commas in your answer.)Explanation / Answer
TRADITIONAL METHOD
given that,
mean(x)=21.2
standard deviation , s.d1=2.7047
number(n1)=10
y(mean)=22.8
standard deviation, s.d2 =3.5517
number(n2)=8
I.
stanadard error = sqrt(s.d1^2/n1)+(s.d2^2/n2)
where,
sd1, sd2 = standard deviation of both
n1, n2 = sample size
stanadard error = sqrt((7.315/10)+(12.615/8))
= 1.519
II.
margin of error = t a/2 * (stanadard error)
where,
t a/2 = t -table value
level of significance, =
from standard normal table, two tailedand
value of |t | with min (n1-1, n2-1) i.e 7 d.f is 2.365
margin of error = 2.365 * 1.519
= 3.593
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (21.2-22.8) ± 3.593 ]
= [-5.193 , 1.993]
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DIRECT METHOD
given that,
mean(x)=21.2
standard deviation , s.d1=2.7047
sample size, n1=10
y(mean)=22.8
standard deviation, s.d2 =3.5517
sample size,n2 =8
CI = x1 - x2 ± t a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
where,
x1,x2 = mean of populations
sd1,sd2 = standard deviations
n1,n2 = size of both
a = 1 - (confidence Level/100)
ta/2 = t-table value
CI = confidence interval
CI = [( 21.2-22.8) ± t a/2 * sqrt((7.315/10)+(12.615/8)]
= [ (-1.6) ± t a/2 * 1.519]
= [-5.193 , 1.993]
= -5.193$ $1.993
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interpretations:
1. we are 95% sure that the interval [-5.193 , 1.993] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion
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