Answer:
The confidence interval is  
Step-by-step explanation:
From  the question we are given a data set 
      25.5, 26.1, 26.8, 23.2, 24.2, 28.4, 25.0, 27.8, 27.3, and 25.7.
The mean of the this sample data is  
        
where is the sample size with values  n =  10
          
           
The standard deviation is evaluated as 
              
substituting values 
      
                   
      
The  confidence level is given as 90% hence the level of significance is calculated as 
     
      %
%
    
Now the critical values of  is obtained from the normal distribution table as
 is obtained from the normal distribution table as 
       
The reason  we are obtaining  the critical values of  instead of
 instead of   is because  we are considering two tails of the area under the normal curve
 is because  we are considering two tails of the area under the normal curve 
   The margin of error is evaluated as 
             
substituting values 
            
            
The  90%, two sided confidence interval is mathematically evaluated as 
            
            
            
Given that the lower and the upper limit is greater than  25 then we can assure the manufactures  that the battery life exceeds 25 hours