Question

Duracell released an advanced long lasting AAA battery, which it claims can last on average for at least 200 days with the standard deviation of 16 days in any remote control devices. A random sample of 100 batteries was selected and the results showed that new batteries lasted on average for 198 days. Construct 90% confidence interval for the population mean and interpret it. Also conduct the five-step hypothesis test at α = 0.05 to determine whether the battery manufacturer’s claim is true or not and afterwards, compute the p-value for it.

Answer 1

a)

Level of Significance ,    α =    0.1
population std dev ,    σ =    16
Sample Size ,   n =    100
Sample Mean,    x̅ =   198
z value=   z α/2=   1.6449 [excel formula =normsinv(0.10/2) ]
      
Standard Error , SE =   σ/√n =   1.6000
margin of error ,   E=Z*SE =   2.6318
confidence interval is       
Interval Lower Limit=   x̅ - E =    195.3682
Interval Upper Limit=   x̅ + E =    200.6318

we are 90% confident that population mean days for battery will lie within confidence interval.
b)

Ho :   µ ≥ 200 (claim)
Ha :   µ <   200  
          
Level of Significance ,    α =    0.05  
population std dev ,    σ =    16  
Sample Size ,   n =    100  
Sample Mean,    x̅ =   198  
          
          
Standard Error , SE =   σ/√n =   1.6000  
          
Z-test statistic=   (x̅ - µ )/SE =    -1.2500  
          
  
p-Value   =   0.1056 [excel formula , =normsdist(-1.25)
Conclusion:     p-value>α, Do not reject null hypothesis

so, there is not enough evidence to reject the  battery manufacturer’s claim