Question 1084020
Question:
The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of 41 to 47 months.
 
Solution:
Step 1
summarize data:
Life of batteries follow a normal distribution
mean=44 months
standard deviation, SD=3 months
Need percentage of batteries with a life between 41 and 47 months.
 
Step 2: calculate z-values for X=41 and 47 hours
Z(41)=(X-mean)/SD=(41-44)/3=-1.0
Z(47)=(X-mean)/SD=(47-44)/3=+1.0
 
Step 3: look up probabilities (from table) for Z-values
P(Z<=-1)=0.1586
P(Z<=+1)=0.8413
So P(-1<=Z<=1)=.8413-.1586=0.6827
 
Step 4: interpret answer
Since P(-1<=Z<=1)=.8413-.1586=0.6827, we deduce that the expected percentage of batteries having a life between 41 and 47 hours is 68.3%