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?
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
To find the number of years that the 1% of items with the shortest lifespan will last, we need to calculate the z-score associated with this percentile.
The z-score formula is given by:
z = (X - μ) / σ
Where:
X = the value we want to find
μ = the mean
σ = the standard deviation
In this case, we are looking for the X value (number of years) such that only 1% of the items have a shorter lifespan. We want to find the z-score associated with the 1st percentile, which is also known as the critical value.
Using a standard normal table or a calculator, we can find that the z-score corresponding to the 1st percentile is approximately -2.33.
Now we can rearrange the z-score formula to solve for X:
X = μ + (z * σ)
Plugging in the values:
X = 9.5 + (-2.33 * 2.2)
X = 9.5 - 5.126
X = 4.374
Therefore, the 1% of items with the shortest lifespan will last less than approximately 4.374 years.
To determine the number of years that the 1% of items with the shortest lifespan will last, we need to find the corresponding z-score. The z-score represents the number of standard deviations an observation is from the mean in a normal distribution.
First, let's calculate the z-score corresponding to the 1st percentile. The 1st percentile is the value below which 1% of the observations fall. Since the normal distribution is symmetric, we can find this value by subtracting the area above the 1st percentile (which is 0.01) from 0.5, which represents the area to the left of the mean in a standard normal distribution.
Using a standard normal distribution table or a statistical calculator, the z-score corresponding to an area of 0.5 - 0.01 = 0.49 is approximately -2.33.
Next, we need to convert this z-score back into the original measurement units (years). We can do this using the formula:
z = (x - mean) / standard deviation
Rearranging the formula to solve for x, we get:
x = z * standard deviation + mean
Substituting the values we have:
x = -2.33 * 2.2 + 9.5
Calculating, we find:
x ≈ 4.85
Therefore, the 1% of items with the shortest lifespan will last less than approximately 4.85 years.