1 In order to estimate the difference between the yearly inc

1. In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the United States, the following information was gathered. East West n1 = 40 n2 = 45 = 72 (in $1,000) = 78 (in $1,000) s1 = 6 (in $1,000) s2 = 8 (in $1,000) df=80

a. What is the point estimate of the difference between the two population means?

b. At 90 % confidence, what is the margin of error?

c. What is the 90 % confidence interval for the difference between the two population means?

d. Use critical value and p-value approaches and test to determine if the average yearly income of marketing managers in the East is significantly different from the West. Use =0.10

Solution

a)
point estimate of the difference = Y(Mean) - X(Mean) = 78-72 - 6

b)
Margin of Error = Z a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
   x1 = Mean of Sample 1, x2 = Mean of sample2
sd1 = SD of Sample 1, sd2 = SD of sample2
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
ME = [ Z a/2 * Sqrt( 36/40+64/45)]
= [ Z a/2 * Sqrt( 2.3222) ]
= [ 1.645 * Sqrt( 2.3222) ]
= [2.5068]

c)
CI = x1 - x2 ± Z a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
Where,
x1 = Mean of Sample 1, x2 = Mean of sample2
sd1 = SD of Sample 1, sd2 = SD of sample2
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x1)=72
Standard deviation( sd1 )=6
Sample Size(n1)=40
Mean(x2)=78
Standard deviation( sd2 )=8
Sample Size(n1) =45
CI = [ ( 72-78) ±Z a/2 * Sqrt( 36/40+64/45)]
= [ (-6) ± Z a/2 * Sqrt( 2.3222) ]
= [ (-6) ± 1.645 * Sqrt( 2.3222) ]
= [-8.5068 , -3.4932]

d)
Set Up Hypothesis
Null, average yearly income of marketing managers in the East is same Ho: u1 = u2
Alternate,average yearly income of marketing managers in the East is significantly different H1: u1 != u2
we use Test Statistic (Z) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
Zo=72-78/Sqrt((36/40)+(64/45))
Zo =-3.94
| Zo | =3.94
Critical Value
The Value of |Z | at LOS 0.1% is 1.645
We got |Zo | =3.937 & | Z | =1.645
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != -3.94 ) = 0.00008
Hence Value of P0.1 > 0.00008,Here we Reject Ho

We conclude that average yearly income of marketing managers in the East is significantly different