Leaf length is normally distributed for leaves on a large oa

Leaf length is normally distributed for leaves on a large oak tree. A sample of is collected with sizes 3, 3, 5, 6, 6, 6, 7, 8, 8, 9 and 9.

If the leaves of a typical oak tree have a mean of length 5, is the mean leaf length for this oak tree different at a 5% significance level?

Is the mean leaf length for this oak tree different at a 1% significance level?

Is the mean leaf length for this oak tree different at a 10% significance level?

Solution

Set Up Hypothesis
Null, H0: U=5
Alternate, mean leaf length for this oak tree different, H1: U!=5
Test Statistic
Population Mean(U)=5
Sample X(Mean)=6.364
Standard Deviation(S.D)=2.111
Number (n)=11
we use Test Statistic (t) = x-U/(s.d/Sqrt(n))
to =6.364-5/(2.111/Sqrt(10))
to =2.143
| to | =2.143

AT 0.05 LOS, Critical Value
The Value of |t | with n-1 = 10 d.f is 2.228
We got |to| =2.143 & | t | =2.228
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho

AT 0.01 LOS
The Value of |t | with n-1 = 10 d.f is 3.169
We got |to| =2.143 & | t | =3.169
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho

AT 0.1 LOS
The Value of |t | with n-1 = 10 d.f is 1.812
We got |to| =2.143 & | t | =1.812
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho