Question

a manufacturer knows that their items have a normally distributed lifespan, with a mean of 13 years, and standard deviation of 4 years.
the 3% of items with the shortest lifespan will last less than how many years?
give your answer to one decimal place.
question help: video

Explanation:

Step1: Find the z - score

We want to find the value \(x\) such that \(P(X\lt x)=0.03\). Using a standard normal table (or a calculator with a normal - distribution function), the z - score \(z\) corresponding to a left - tail probability of \(0.03\) is approximately \(z=-1.88\).

Step2: Use the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 13\) (mean) and \(\sigma = 4\) (standard deviation).
We know \(z=-1.88\), \(\mu = 13\), and \(\sigma = 4\). Rearranging the formula for \(x\) gives \(x=\mu+z\sigma\).
Substitute the values: \(x = 13+(-1.88)\times4\).

Step3: Calculate the value of \(x\)

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Answer:

\(5.5\)