To solve the problem in your screenshot, we need to calculate the probability that a randomly purchased item lasts longer than 7 years, given that the lifespan of the items is normally distributed with a mean of 3.9 years and a standard deviation of 1.1 years.

This is a normal distribution problem where we want to find $P(X > 7)$. We can standardize the value using the Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X = 7$ (the value we're interested in)
• $\mu = 3.9$ (mean lifespan)
• $\sigma = 1.1$ (standard deviation)

Let's compute the Z-score and then find the probability using the standard normal distribution table (or a calculator).

I'll calculate it for you.The Z-score is approximately $2.818$.

Next, we will use this Z-score to find the probability that a randomly selected item will last longer than 7 years. To do this, we need to find the area to the right of this Z-score under the standard normal curve, which is $P(Z > 2.818)$.

I'll calculate this probability for you.The probability that an item will last longer than 7 years is approximately $0.002$, or 0.2%.

Would you like further details on this calculation or have any questions?

Here are 5 related questions you might find interesting:

1. What is the probability that an item will last less than 2 years?
2. How would the probability change if the mean lifespan increased?
3. What is the Z-score interpretation in normal distribution problems?
4. How would we calculate the probability for a range of lifespans, such as between 4 and 6 years?
5. What does the standard deviation tell us about the variation in lifespan?

Tip: Z-scores help compare individual data points to the overall distribution by showing how far a value is from the mean in standard deviations.