Questions: Acme Light Bulbs has determined their light bulbs lifetimes are Normally distributed with a mean of 850 hours and a standard deviation of 50 hours. What percent of their bulbs will last longer than 880 hours?

Solution Steps

To determine the percentage of light bulbs that will last longer than 880 hours, we need to calculate the z-score for 880 hours using the given mean and standard deviation. Then, we can use the z-score to find the corresponding probability from the standard normal distribution.

To determine the percentage of light bulbs that will last longer than 880 hours, we first calculate the z-score using the formula: \[ z = \frac{x - \mu}{\sigma} \] where \( x = 880 \), \( \mu = 850 \), and \( \sigma = 50 \).

Substituting the values, we get: \[ z = \frac{880 - 850}{50} = 0.6 \]

Next, we find the probability that a bulb lasts less than 880 hours using the cumulative distribution function (CDF) of the standard normal distribution: \[ P(X < 880) = \Phi(z) = \Phi(0.6) \approx 0.7257 \]

To find the percentage of bulbs that last longer than 880 hours, we subtract the above probability from 1: \[ P(X > 880) = 1 - \Phi(0.6) \approx 1 - 0.7257 = 0.2743 \]

Converting this probability to a percentage: \[ \text{Percentage} = 0.2743 \times 100 \approx 27.43\% \]

Final Answer