Respuesta :
Answer:
Step-by-step explanation:
The formula for determining the confidence interval for the difference of two population means is expressed as
Confidence interval = (x1 - x2) ± z√(s²/n1 + s2²/n2)
Where
x1 = sample mean of group 1
x2 = sample mean of group 2
s1 = sample standard deviation for data 1
s2 = sample standard deviation for data 2
For data 1(graduate)
number of samples, n1 = 21
Mean, x1 = (13 + 7 + 15 + 10 + 5 + 5 + 2 + 3 + 12 + 16 + 15 + 39 + 8 + 14 + 10 + 17 + 3 + 27 + 15 + 5 + 5)/21 = 11.7
Standard deviation = √(summation(x - mean)/n
Summation(x - mean) = (13 - 11.7)^2 + (7 - 11.7)^2 + (15 - 11.7)^2 + (10 - 11.7)^2 + (5 - 11.7)^2 + (5 - 11.7)^2 + (2 - 11.7)^2 + (3 - 11.7)^2 + (12 - 11.7)^2 + (16 - 11.7)^2 + (15 - 11.7)^2 + (39 - 11.7)^2 + (8 - 11.7)^2 + (14 - 11.7)^2 + (10 - 11.7)^2+ (17 - 11.7)^2 + (3 - 11.7)^2 + (27 - 11.7)^2 + (15 - 11.7)^2 + (5 - 11.7)^2 + (5 - 11.7)^2 = 1533.25
s = √1533.25/21 = 8.54
For data 2,
n2 = 10
Mean, x2 = (6 + 8 + 14 + 6 + 5 + 13 + 10 + 10 + 13 + 5)/10 = 9
Summation(x - mean) = (6 - 9)^2 + (8 - 9)^2 + (14 - 9)^2 + (6 - 9)^2 + (5 - 9)^2 + (13 - 9)^2 + (10 - 9)^2 + (10 - 9)^2 + (13 - 9)^2 + (5- 9)^2 = 110
s2 = √110/10 = 3.32
For a 95% confidence interval, we would determine the z score from the t distribution table because the number of samples are small
Degree of freedom =
(n1 - 1) + (n2 - 1) = (21 - 1) + (10 - 1) = 29
z = 2.045
x1 - x2 = 11.7 - 9 = 2.7
√(s1²/n1 + s2²/n2) = √(8.54²/21 + 3.32²/10) = √(3.473 + 1.10224)
= 2.14
Margin of error = 2.045 × 2.14 = 4.38
The upper boundary for the confidence interval is
2.7 + 4.38 = 7.08
The lower boundary for the confidence interval is
2.7 - 4.38 = - 1.68
We are confident that the difference in population means between students in the class who planned to go to graduate school and those who did not is between - 1.68 and 7.08
