Given two dependent random samples with the following results: Use this data to

Question

Given two dependent random samples with the following results:

Use this data to find the 90% confidence interval for the true difference between the population means. Assume that both populations are normally distributed.

Step 1 of 4:

Find the point estimate for the population mean of the paired differences. Let x1 be the value from Population 1 and x2 be the value from Population 2 and use the formula d=x2x1 to calculate the paired differences. Round your answer to one decimal place.

Step 2 of 4:

Calculate the sample standard deviation of the paired differences. Round your answer to six decimal places.

Step 3 of 4:

Calculate the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

Step 4 of 4:

Construct the 90% confidence interval. Round your answers to one decimal place

Explanation / Answer

step.1
Given that,
mean(x)=36
standard deviation , s.d1=8.3267
number(n1)=7
y(mean)=37.4286
standard deviation, s.d2 =7.3452
number(n2)=7
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, = 0.1
from standard normal table, two tailed t /2 =1.943
since our test is two-tailed
reject Ho, if to < -1.943 OR if to > 1.943
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =36-37.4286/sqrt((69.33393/7)+(53.95196/7))
to =-0.3
| to | =0.3
critical value
the value of |t | with min (n1-1, n2-1) i.e 6 d.f is 1.943
we got |to| = 0.34041 & | t | = 1.943
make decision
hence value of |to | < | t | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -0.3404 ) = 0.745
hence value of p0.1 < 0.745,here we do not reject Ho
ANSWERS
---------------
null, Ho: u1 = u2
alternate, H1: u1 != u2
test statistic: -0.3
critical value: -1.943 , 1.943
decision: do not reject Ho
p-value: 0.745

step.2

standard deviation , s.d1=8.326700
standard deviation, s.d2 =7.345200
sample standard deviation of the paired differences = X1-X2 =8.326700-7.345200 =0.981500

step.3

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 1 d.f is 6.313752
margin of error = 6.314 * 6.072959
= 38.344662

step.4

TRADITIONAL METHOD
given that,
mean(x)=36
standard deviation , s.d1=8.3267
number(n1)=7
y(mean)=37.4286
standard deviation, s.d2 =7.3452
number(n2)=2
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((69.3/7)+(54/2))
= 6.1
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 1 d.f is 6.3
margin of error = 6.314 * 6.1
= 38.3
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (36-37.4286) ± 38.3 ]
= [-39.8 , 36.9]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
mean(x)=36
standard deviation , s.d1=8.3267
sample size, n1=7
y(mean)=37.4286
standard deviation, s.d2 =7.3452
sample size,n2 =2
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 = [( 36-37.4286) ± t a/2 * sqrt((69.3/7)+(54/2)]
= [ (-1.4) ± t a/2 * 6.1]
= [-39.8 , 36.9]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 90% sure that the interval [-39.8 , 36.9] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population proportion

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.