the battery lifetime is normally distributed for large samples with a standard deviation of 70 days. if 16% of the batteries have a lifetime of 850 or higher, what is the average battery life?
about 780
http://davidmlane.com/hyperstat/z_table.html
open
http://davidmlane.com/normal.html
enter 70 into SD
click on <above> and enter 850
adjust Mean until you get .16 in the area (probability) window
I had 780.39 days as the mean.
To find the average battery life, we need to use the concept of z-scores and the cumulative distribution function (CDF) of the normal distribution.
First, let’s find the z-score corresponding to the value of the battery life of 850 days. The z-score formula is:
z = (x - μ) / σ
Where:
x = value of the battery life (850 days)
μ = mean (average) battery life
σ = standard deviation of the battery life (70 days)
Rearranging the formula, we have:
μ = x - (z * σ)
Now, let's find the z-score using the CDF function. Since we want to find the value that corresponds to the top 16% of the distribution, we subtract 0.16 from 1 to get 0.84.
Using a standard normal distribution table or a statistical software, we find that the z-score corresponding to a cumulative probability of 0.84 is approximately 0.99.
Plugging in the values, we have:
μ = 850 - (0.99 * 70)
μ = 850 - 69.3
μ ≈ 780.7
Therefore, the average battery life is approximately 780.7 days.