Year one heightscm 54 40 49 64 54 79 72 46 66 47 57 23 48 57

Year one heights(cm): 54, 40, 49, 64, 54, 79, 72, 46, 66, 47, 57, 23, 48, 57, 73.
In the following year a group of seedlings were randomly selected and had the following heights(cm): 44, 20, 28, 41, 45, 15, 31, 38, 37, 32.

Determine whether mean seedling height changed significantly from the first year to the second. Indicate null and alternatice hyptheses, along with conclusions for each.

Solution

As the significance level is not given, we assume it is 0.05.

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   =   0  
Ha:   u1 - u2   =/   0  

At level of significance =    0.05          

As we can see, this is a    two   tailed test.      
Calculating the means of each group,              
              
X1 =    55.26666667          
X2 =    33.1          
              
Calculating the standard deviations of each group,              
              
s1 =    14.340087          
s2 =    9.960477454          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    15          
n2 = sample size of group 2 =    10          
Thus, df = n1 + n2 - 2 =    23          
Also, sD =    4.861102494          
              
Thus, the t statistic will be              
              
t = [X1 - X2 - uD]/sD =    4.560008083          
              
where uD = hypothesized difference =    0          
              
Now, the critical value for t is              
              
tcrit =    +/-   2.06865761      
              
              
Also, using p values,              
              
p =    0.00013937          
              
As t > 2.0687, and P < 0.05, we REJECT THE NULL HYPOTHESIS.          

Hence, there is significant evidence that the mean seedling height changed significantly from the first year to the second. [CONCLUSION]