A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.5 years, and standard deviation of 1.7 years.

If you randomly purchase one item, what is the probability it will last longer than 9 years? (Give answer to 4 decimal places.)
Incorrect

To find the probability that the item will last longer than 9 years, we need to calculate the area under the normal distribution curve to the right of 9 years.

To do this, we can use the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We need to standardize our value of 9 years to fit this standard normal distribution.

First, we calculate the z-score for 9 years using the formula:

z = (x - μ) / σ

Where:
- x is the value we want to standardize (9 years in this case)
- μ is the mean of the population (7.5 years in this case)
- σ is the standard deviation of the population (1.7 years in this case)

Plugging in the values:

z = (9 - 7.5) / 1.7
z = 0.8824

Now, we can use a standard normal distribution table or a statistical software to find the cumulative probability of the z-score being less than or equal to -0.8824. We want the probability to the right of this value, so we subtract the cumulative probability from 1.

Using a normal distribution table or software, we find that the cumulative probability for a z-score of -0.8824 is 0.1894.

Therefore, the probability that a randomly purchased item will last longer than 9 years is approximately 1 - 0.1894 = 0.8106, or 81.06% (to 4 decimal places).

9 years is (9-7.5)/1.7 = 0.833σ above the mean.

Now consult your Z table.