A manufacturer knows that their items have a normally distributed lifespan, with a mean of 9.5 years, and standard deviation of 2.2 years.
The 1% of items with the shortest lifespan will last less than how many years?
Give your answer to one decimal place.
To find the number of years that the 1% of items with the shortest lifespan will last, we need to calculate the value of the z-score that corresponds to the 1st percentile of the normal distribution.
The formula to convert a given percentile to a z-score is:
z = (X - μ) / σ
Where:
- z is the z-score
- X is the value we want to convert to a percentile
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case, we want to find X, so we rearrange the equation to solve for X:
X = μ + (z * σ)
For the 1st percentile, we need to find the z-score corresponding to it. We can use a standard normal distribution table or a statistical calculator to find it. The z-score corresponding to the 1st percentile is -2.326.
Substituting the values into the equation:
X = 9.5 + (-2.326 * 2.2)
X = 9.5 - 5.1052
X ≈ 4.4
Therefore, the 1% of items with the shortest lifespan will last less than approximately 4.4 years.