A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.4 years, and standard deviation of 1.6 years.
The 10% of items with the shortest lifespan will last less than how many years?
Give your answer to one decimal place.
5.2 years
To find the number of years that represents the 10% of items with the shortest lifespan, we need to use the concept of z-scores.
First, we need to find the z-score that corresponds to the 10th percentile. The z-score represents the number of standard deviations away from the mean a value is. We can use a standard normal distribution table or a calculator to find the z-score.
Using the standard normal distribution table, we find that the z-score corresponding to the 10th percentile is approximately -1.28 (rounded to two decimal places).
Next, we need to use the formula for converting a z-score to an actual value, which is:
Value = Mean + (z-score * Standard Deviation)
Plugging in the values, we have:
Value = 7.4 + (-1.28 * 1.6)
Calculating this, we get:
Value ≈ 7.4 - 2.048
So, the 10% of items with the shortest lifespan will last less than approximately 5.4 years (rounded to one decimal place).
To find the time in years that the 10% of items with the shortest lifespan will last, we need to determine the z-score associated with the 10th percentile.
The z-score formula for a normally distributed variable is given by:
z = (x - mean) / standard deviation
To find the z-score corresponding to the 10th percentile, we need to find the z-value that has 10% of the area under the curve to the left of it. This can be found using a standard normal distribution table or calculator.
Using the standard normal distribution table, we find that the z-score corresponding to the 10th percentile is approximately -1.28.
We can use the z-score formula to find the corresponding value in years:
-1.28 = (x - 7.4) / 1.6
Rearranging the equation, we have:
(x - 7.4) = -1.28 * 1.6
(x - 7.4) = -2.048
x = 7.4 - 2.048
x ≈ 5.352
Therefore, the 10% of items with the shortest lifespan will last less than approximately 5.4 years.