Sure! Let's break this down step-by-step using the empirical rule ([tex]$68\%-95\%-99.7\%$[/tex] rule) which applies to normal distributions.
### Step 1: Understand the Problem
- Mean ([tex]\(\mu\)[/tex]): 750 hours
- Standard Deviation ([tex]\(\sigma\)[/tex]): 75 hours
- Lower Bound: 525 hours
- Upper Bound: 750 hours
### Step 2: Convert Hours to Z-Scores
First, we'll convert the given hours into z-scores. The z-score for a value [tex]\(X\)[/tex] is calculated using the formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
#### Calculate the z-score for the lower bound (525 hours):
[tex]\[ z_{\text{lower}} = \frac{525 - 750}{75} = -3.0 \][/tex]
#### Calculate the z-score for the upper bound (750 hours):
[tex]\[ z_{\text{upper}} = \frac{750 - 750}{75} = 0.0 \][/tex]
### Step 3: Determine the Probability from Z-Scores
Using the empirical rule:
- The empirical rule states that approximately 68% of the data lies within one standard deviation from the mean.
- Specifically, 34.13% of the data lies between the mean and one standard deviation below the mean ([tex]\( \mu - \sigma \)[/tex]).
- The total area under the normal curve is 1 (or 100%).
#### Find the probability corresponding to [tex]\( z = -3.0 \)[/tex]:
- By the empirical rule, the probability of a z-score of [tex]\(-1\)[/tex] to the left of the mean (which corresponds to 525 hours) is the remaining area in the left tail, which is:
[tex]\[ \text{Probability below z = -1} = 0.1587 \][/tex]
#### Find the probability corresponding to [tex]\( z = 0 \)[/tex]:
- A [tex]\( z \)[/tex]-score of 0 corresponds to the mean, and 50% of the data is below the mean because the mean splits the distribution exactly in half.
[tex]\[ \text{Probability below z = 0} = 0.5 \][/tex]
### Step 4: Calculate the Desired Probability
The probability that a light bulb lasts between 525 and 750 hours is the area under the normal curve between [tex]\( z = -3.0 \)[/tex] and [tex]\( z = 0 \)[/tex].
[tex]\[ \text{Probability (525 < b < 750)} = \text{Probability (b < 750)} - \text{Probability (b < 525)} \][/tex]
Substituting the corresponding probabilities:
[tex]\[ \text{Probability (525 < b < 750)} = 0.5 - 0.1587 = 0.3413 \][/tex]
### Conclusion
The probability that a given light bulb lasts between 525 and 750 hours is 0.3413, or 34.13%.
