Answer:
95% Confidence Interval is; ( -5.344 to 11.344 )
Option b) -5.344 to 11.344 is the correct answer
Step-by-step explanation:
Given the data in the question;
Sample 1 Sample 2
x"₁ = 45 x"₂ = 42
S₁² = 85 S₂² = 90
n₁ = 10 n₂ = 12
df = [ S₁²/n₁ + S₂²/n₂ ]² / [ ((S₁²/n₁)²/n₁-1) + ((S₂²/n₂)²/n₂-1)) ]
we substitute
df = [ 10/10 + 90/12 ]² / [ ((85/10)²/10-1) + ((90/12)²/12-1)) ] = 19.64 ≈ 20
df = 20
with 95% confidence interval
∝ = 1 - 0.95 = 0.05
∝/2 = 0.05/2 = 0.025
now, [tex]t_{\alpha /2,df} = t_{0.025, 20} = 2.086[/tex] { from table }
95% confidence interval for N1 - N2
⇒ (x"₁ - x"₂) ± [tex]t_{\alpha /2,df}[/tex] × √( S₁²/n₁ + S₂²/n₂ )
⇒ (45 - 42) ± 2.086 × √( 85/10 + 90/12 )
⇒ 3 ± 2.086 × 4
⇒ 3 ± 8.344
so;
Lower Limit = 3 - 8.344 = -5.344
Upper Limit = 3 + 8.344 = 11.344
Therefore, 95% Confidence Interval is; ( -5.344 to 11.344 )
Option b) -5.344 to 11.344 is the correct answer
