the life of electric bulbs has a normal distribution with a mean of 35 months and a standard deviation of 4 months. What should the warranty period be if the company that manufactures these bulbs does not want to replace more than 2% of the bulbs?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.02) and its Z score. Insert the value in the above equation and solve for the score. Remember that the Z score will be negative.
Here mean = 35
s.d.= 4
we have asked that
P(z)=0.98
see normal table
It is z=2.05
remember it is standard normal table value but we have to convert it for x variable for which mean is 35 and s.d. is 4.
do the inverse operation for which we convert x into z.
i.e.
x=mean+z*s.d.
x=35+2.05*4
x=47.3
rounding off it will be 48.
That is answer is 48.
To find the warranty period, we need to determine the threshold value below which the company does not want to replace more than 2% of the bulbs.
Step 1: Standardize the threshold value
Since we have a normal distribution, we can use the Z-score formula to standardize the threshold value. The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Threshold value
μ = Mean of the distribution
σ = Standard deviation of the distribution
Step 2: Find the Z-score corresponding to the desired percentage
We want to find the threshold value below which the company does not want to replace more than 2% of the bulbs. This means we need to find the Z-score that corresponds to the 2nd percentile, which is 2%. Using a Z-score table or calculator, we can find that the Z-score corresponding to the 2nd percentile is approximately -2.05.
Step 3: Solve for the threshold value
Using the Z-score formula, we substitute the calculated Z-score and the given mean and standard deviation:
-2.05 = (X - 35) / 4
Now we can solve for X by rearranging the equation:
X - 35 = -2.05 * 4
X - 35 = -8.2
X = 35 - 8.2
X ≈ 26.8
Therefore, the company should provide a warranty period of approximately 26.8 months to ensure that they do not replace more than 2% of the bulbs.