Question

A researcher is studying the difference between average survival time between women and men suffering from the same critical disease. Information from two randomly selected samples is as follows:

Women: mean₁ = 16.9 s₁ = 3.58 n₁ = 60

Men: mean₂ = 18.9 s₂ = 4.15 n₂ = 60

Construct a 95% confidence interval for the true mean difference in survival times between women and men.

Select one:

a. -2.91 < μ₁ - μ₂ < -1.09

b. -3.59 < μ₁ - μ₂ < -0.42

c. -3.16 < μ₁ - μ₂ < -0.84

d. Not enough information for answer.

e. -3.39 < μ₁ - μ₂ < -0.61

Answer 1

TRADITIONAL METHOD
given that,
mean(x)=16.9
standard deviation , σ1 =3.58
population size(n1)=60
y(mean)=18.9
standard deviation, σ2 =4.15
population size(n2)=60
I.
standard error = sqrt(s.d1^2/n1)+(s.d2^2/n2)
where,
sd1, sd2 = standard deviation of both
n1, n2 = sample size
standard error = sqrt((12.8164/60)+(17.2225/60))
= 0.71
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 0.71
= 1.39
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (16.9-18.9) ± 1.39 ]
= [-3.39 , -0.61]
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DIRECT METHOD
given that,
mean(x)=16.9
standard deviation , σ1 =3.58
number(n1)=60
y(mean)=18.9
standard deviation, σ2 =4.15
number(n2)=60
CI = x1 - x2 ± Z 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)
Za/2 = Z-table value
CI = confidence interval
CI = [ ( 16.9-18.9) ±Z a/2 * Sqrt( 12.8164/60+17.2225/60)]
= [ (-2) ± Z a/2 * Sqrt( 0.5) ]
= [ (-2) ± 1.96 * Sqrt( 0.5) ]
= [-3.39 , -0.61]
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interpretations:
1. we are 95% sure that the interval [-3.39 , -0.61] contains the difference between
true population mean U1 - U2
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 difference between
true population mean U1 - U2
3. Since this Cl does contain a zero we can conclude at 0.05 true mean
difference is zero

Answer:
95% sure that the interval [-3.39 , -0.61]
option:e