Answers
1)
Hₒ : σ₁ = σ₂
H₁ : σ₁ ≠ σ₂
Test statistic:
F = s₁² / s₂² = 2² / 2.5² = 0.64
Degree of freedom:
df₁ = n₁-1 = 12
df₂ = n₂-1 = 9
Critical value(s):
Lower tailed critical value, FL = F.INV(0.05/2, 12, 9) = 0.2910
Upper tailed critical value, FU = F.INV(1-0.05/2, 12, 9) = 3.8682
P-value :
P-value = 2*F.DIST.RT(0.64, 12, 9) = 1.5366
Conclusion:
As p-value > α, we fail to reject the null hypothesis.
Conclusion: Both are equal
2)
Ho : µ1 - µ2 = 0
Ha : µ1-µ2 ╪ 0
Level of Significance , α = 0.05
Sample #1 ----> 1
mean of sample 1, x̅1= 4.700
standard deviation of sample 1, s1 = 2.000
size of sample 1, n1= 13
Sample #2 ----> 2
mean of sample 2, x̅2= 6.800
standard deviation of sample 2, s2 = 2.500
size of sample 2, n2= 10
difference in sample means = x̅1-x̅2 = 4.7000 - 6.8 = -2.10
pooled std dev , Sp= √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) = 2.2281
std error , SE = Sp*√(1/n1+1/n2) = 0.9372
t-statistic = ((x̅1-x̅2)-µd)/SE = ( -2.1000 - 0 ) / 0.94 = -2.2408
Degree of freedom, DF= n1+n2-2 = 21
t-critical value , t* = 2.080 (excel formula =t.inv(α/2,df)
Decision: | t-stat | > | critical value |, so, Reject Ho
p-value = 0.035979 (excel function: =T.DIST.2T(t stat,df) )
Conclusion: p-value <α , Reject null hypothesis
There is enough evidence that both mean are different
c)
Degree of freedom, DF= n1+n2-2 = 21
t-critical value = t α/2 = 2.0796 (excel formula =t.inv(α/2,df)
pooled std dev , Sp= √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) = 2.2281
std error , SE = Sp*√(1/n1+1/n2) = 0.9372
margin of error, E = t*SE = 2.0796 * 0.94 = 1.95
difference of means = x̅1-x̅2 = 4.7000 - 6.800 = -2.1000
confidence interval is
Interval Lower Limit= (x̅1-x̅2) - E = -2.1000 - 1.9490 = -4.049
Interval Upper Limit= (x̅1-x̅2) + E = -2.1000 + 1.9490 = -0.151
Please let me know in case of any doubt.
Thanks in advance!
Please upvote!