The owner of a private fishing pond says that the average le

The owner of a private fishing pond says that the average length of fish in the pond is 30cm. Samples of 30 fish are caught from various locations in the pond. The mean length of these fish is 27cm with standard deviation 12cm. i. By using P-value approach, does this data support the claim that the average fish length is less than 30cm? ii. Construct a 95% two-sided CI for this mean. What is your comment on this CI?

Solution

i)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   30  
Ha:    u   <   30  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha =    0.05          
zcrit =    -   1.644853627      
              
Getting the test statistic, as              
              
X = sample mean =    27          
uo = hypothesized mean =    30          
n = sample size =    30          
s = standard deviation =    12          
              
Thus, z = (X - uo) * sqrt(n) / s =    -1.369306394          
              
Also, the p value is              
              
p =    0.08545176          
              
As |z| < 1.645 or P > 0.05, we   FAIL TO REJECT THE NULL HYPOTHESIS.          
              
Thus, there is no significant evidence that the average fish length is less than 30 cm.

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ii)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    27          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    12          
n = sample size =    30          
              
Thus,              
Margin of Error E =    4.294065945          
Lower bound =    22.70593406          
Upper bound =    31.29406594          
              
Thus, the confidence interval is              
              
(   22.70593406   ,   31.29406594   )

Comment: 30 cm is inside this interval. Thus, there is no significant evidence that the average fish length is less than 30 cm.