To find the probability that a given light bulb lasts between 675 and 900 hours, we start by understanding that the lifespans are normally distributed with a mean (μ) of 750 hours and a standard deviation (σ) of 75 hours.
1. Identify the mean and standard deviation:
- Mean (μ): 750 hours
- Standard deviation (σ): 75 hours
2. Determine the bounds for the problem:
- Lower bound: 675 hours
- Upper bound: 900 hours
3. Convert the original values to z-scores:
The z-score formula is given by:
[tex]\[ z = \frac{(X - \mu)}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value for which we are calculating the z-score.
- For the lower bound (675 hours):
[tex]\[ z_{\text{lower}} = \frac{(675 - 750)}{75} = \frac{-75}{75} = -1.0 \][/tex]
- For the upper bound (900 hours):
[tex]\[ z_{\text{upper}} = \frac{(900 - 750)}{75} = \frac{150}{75} = 2.0 \][/tex]
4. Find the cumulative distribution function (CDF) values:
Use the standard normal distribution table or CDF tables to determine the area under the curve up to these z-scores.
- [tex]\( P(Z \leq -1.0) \)[/tex] corresponds to the cumulative probability up to the z-score of -1.0.
- [tex]\( P(Z \leq 2.0) \)[/tex] corresponds to the cumulative probability up to the z-score of 2.0.
5. Calculate the probability that the light bulb lasts between 675 and 900 hours:
The probability (P) can be found by subtracting the CDF value of the lower bound z-score from the CDF value of the upper bound z-score.
- The CDF value for [tex]\( Z = -1.0 \)[/tex] is approximately 0.1587.
- The CDF value for [tex]\( Z = 2.0 \)[/tex] is approximately 0.9772.
Therefore, the probability (P) is:
[tex]\[ P(-1.0 < Z < 2.0) = P(Z \leq 2.0) - P(Z \leq -1.0) \][/tex]
[tex]\[ P = 0.9772 - 0.1587 = 0.8185 \][/tex]
So, the probability that a light bulb lasts between 675 and 900 hours is approximately 0.8185, or 81.85%.
