A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.3 years, and standard deviation of 3.2 years.
If you randomly purchase one item, what is the probability it will last longer than 7 years?
To find the probability that the item will last longer than 7 years, we need to find the area under the normal distribution curve to the right of the value 7. We can do this by calculating the z-score and using a standard normal distribution table or calculator.
First, let's calculate the z-score using the formula:
z = (x - μ) / σ
Where:
x = the value we are interested in (7 years)
μ = the mean (11.3 years)
σ = the standard deviation (3.2 years)
z = (7 - 11.3) / 3.2
z = -4.3 / 3.2
z = -1.34 (rounded to two decimal places)
Next, we need to find the area to the right of z = -1.34 in the standard normal distribution table or calculator. Alternatively, we can find the area to the left of z = 1.34 and subtract it from 1.
Using a standard normal distribution table or calculator, the area to the left of z = 1.34 is approximately 0.0901. Therefore, the area to the right of z = -1.34 is approximately 1 - 0.0901 = 0.9099.
Thus, the probability that the item will last longer than 7 years is approximately 0.9099, or 90.99%.