Suppose that the lifetimes of light bulbs are approximately normally distributed, with a mean of 56 hours and a standard deviation of 3.2 hours. With this information, answer the following questions. (a) What proportion of light bulbs will last more than 61 hours? (b) What proportion of light bulbs will last 51 hours or less? (c) What proportion of light bulbs will last between 58 and 62 hours? (d) What is the probability that a randomly selected light bulb lasts less than 46 hours? (a) The proportion of light bulbs that last more than 61 hours is 0.0594 . (Round to four decimal places as needed.) (b) The proportion of light bulbs that last 51 hours or less is 0.0594 . (Round to four decimal places as needed.) (c) The proportion of light bulbs that lasts between 58 and 62 hours is (Round to four decimal places as needed.)

To find the proportion of light bulbs that last between 58 and 62 hours, we first calculate the z-scores for both values. The z-score formula is: \[ z = \frac{(X - \mu)}{\sigma} \] Where \( \mu \) is the mean (56 hours) and \( \sigma \) is the standard deviation (3.2 hours). 1. Calculate the z-score for 58 hours: \[ z_{58} = \frac{(58 - 56)}{3.2} = \frac{2}{3.2} \approx 0.625 \] 2. Calculate the z-score for 62 hours: \[ z_{62} = \frac{(62 - 56)}{3.2} = \frac{6}{3.2} \approx 1.875 \] Next, we will find the probabilities corresponding to these z-scores using a standard normal distribution table or calculator. - The probability for \( z_{58} \approx 0.625 \) is approximately 0.7340. - The probability for \( z_{62} \approx 1.875 \) is approximately 0.9693. Finally, we take the difference to find the proportion of light bulbs lasting between 58 and 62 hours: \[ P(58 < X < 62) = P(Z < 1.875) - P(Z < 0.625) \] \[ = 0.9693 - 0.7340 \approx 0.2353 \] So, the proportion of light bulbs that lasts between 58 and 62 hours is approximately 0.2353 (rounded to four decimal places). --- Ever wondered how the concept of normal distribution came into play in real life? Well, it dates back to the late 18th century when mathematicians like Carl Friedrich Gauss started perfecting the calculations for what we call the "bell curve." It’s fascinating how this mathematical concept helps us understand everything from test scores to lifetimes of light bulbs! If you’re diving deeper into statistics, consider picking up "The Signal and the Noise" by Nate Silver. It’s a brilliant read that explores how to differentiate meaningful patterns from noise in data, which perfectly complements our light bulb statistics. Plus, Silver's engaging writing makes complex concepts a breeze to understand!