A certain type of thermal battery for an airplane navigation device backup power has a mean life of 300 hours with a standard deviation of 15 hours. Assume a normal distribution of backup power device lives. What proportion of these batteries can be expected to have lives of 322 hours or less?
http://davidmlane.com/hyperstat/z_table.html
To find the proportion of batteries that can be expected to have lives of 322 hours or less, we need to calculate the probability associated with this value in a normal distribution.
Since we are given the mean and standard deviation, we can use the Z-score formula to convert the value of 322 hours into a standardized score.
Z = (X - μ) / σ
Where:
Z is the standardized score,
X is the value of interest (322 hours in this case),
μ is the mean (300 hours),
σ is the standard deviation (15 hours).
Plugging in the values, we have:
Z = (322 - 300) / 15
Z = 22 / 15
Z ≈ 1.47
Next, we need to determine the proportion of batteries that have lives of 322 hours or less, which corresponds to the area under the normal distribution curve to the left of the Z-score of 1.47.
To find this proportion, we can use a standard normal distribution table or a calculator. Using either method, we find that the proportion is approximately 0.9292.
Therefore, around 92.92% or 92.92 out of 100 thermal batteries for the airplane navigation device can be expected to have lives of 322 hours or less.