A manufacturer knows that their items have a normally distributed lifespan, with a mean of 9.5 years, and standard deviation of 1.6 years.

Question

The 9% of items with the shortest lifespan will last less than how many years?

Give your answer to one decimal place.

Answer 1

Hi Bailey,

Final answer is 7.4 years. See explanation and all work shown below.

As mentioned previously, I'm happy to offer a free session as I noticed you have several questions posted :). Let me know!

To find the lifespan at which the 9% of items with the shortest lifespan will last, we can use the z-score formula for a normal distribution. We'll find the z-score corresponding to the 9th percentile and then convert it back to the actual number of years.

First, find the z-score corresponding to the 9th percentile using a standard normal distribution table or calculator. The z-score represents how many standard deviations a value is from the mean.

For the 9th percentile, we're interested in the lower tail of the distribution, so it's a negative z-score. You can use a standard normal distribution table to find that the z-score is approximately -1.34 (rounded to two decimal places).

Now, we can use the z-score formula to find the corresponding value in years:

z = (X - μ) / σ

Where:

1. z is the z-score (-1.34).
2. X is the value we want to find.
3. μ is the mean (9.5 years).
4. σ is the standard deviation (1.6 years).

Now, plug in the values:

-1.34 = (X - 9.5) / 1.6

Next, isolate X (the number of years) by multiplying both sides by 1.6 and adding 9.5:

-1.34 * 1.6 = X - 9.5

-2.144 = X - 9.5

X = -2.144 + 9.5

X ≈ 7.36 years

So, the 9% of items with the shortest lifespan will last less than approximately 7.4 years (rounded to one decimal place).

Thank you,

Benjamin M.

Answer 2

I have solved the question above in this video. Please let me know if you have any additional questions. Take care!

-Dr. C