To solve the problem of finding the probability that a light bulb lasts between 675 and 825 hours, given that the lifetimes are normally distributed with a mean (μ) of 750 hours and a standard deviation (σ) of 75 hours, we can use the empirical rule, also known as the 68%-95%-99.7% rule.
Firstly, let's identify the given parameters:
- Mean (μ) = 750 hours
- Standard deviation (σ) = 75 hours
- Lower bound = 675 hours
- Upper bound = 825 hours
According to the empirical rule:
- 68% of the data lies within 1 standard deviation (σ) of the mean (μ).
- 95% of the data lies within 2 standard deviations (2σ) of the mean (μ).
- 99.7% of the data lies within 3 standard deviations (3σ) of the mean (μ).
### Step-by-Step Solution:
1. Calculate the Z-scores for the lower and upper bounds:
- The Z-score formula is: [tex]\( Z = \frac{(X - \mu)}{\sigma} \)[/tex]
For the lower bound (675 hours):
[tex]\[
Z_{\text{lower}} = \frac{675 - 750}{75} = \frac{-75}{75} = -1
\][/tex]
For the upper bound (825 hours):
[tex]\[
Z_{\text{upper}} = \frac{825 - 750}{75} = \frac{75}{75} = 1
\][/tex]
2. Interpret using the 68%-95%-99.7% rule:
- A Z-score of -1 corresponds to 1 standard deviation below the mean.
- A Z-score of 1 corresponds to 1 standard deviation above the mean.
Given that the empirical rule states that approximately 68% of the data in a normal distribution lies within 1 standard deviation from the mean:
[tex]\[
P(\mu - \sigma \leq X \leq \mu + \sigma) = 68\%
\][/tex]
3. Result:
- The desired range, 675 to 825 hours, is within 1 standard deviation of the mean (750 hours ± 75 hours).
- Therefore, the probability that a given light bulb lasts between 675 and 825 hours is 68%.
So, the probability that a given light bulb lasts between 675 and 825 hours is 68%.
