A manufacturer knows that their items have a normally distributed lifespan, with a mean of 8.4 years, and standard deviation of 1.2 years. If you randomly purchase one item, what is the probability it will last longer than 6 years? Round answer to four decimal places Check Answer

To find the probability that an item lasts longer than 6 years, we first calculate the z-score using the formula: \( z = \frac{(X - \mu)}{\sigma} \) Where \( X \) is the lifespan we're interested in (6 years), \( \mu \) is the mean (8.4 years), and \( \sigma \) is the standard deviation (1.2 years). So, \( z = \frac{(6 - 8.4)}{1.2} = \frac{-2.4}{1.2} = -2.0 \) Now, we can find the probability associated with a z-score of -2.0 using the standard normal distribution table or a calculator, which gives us approximately 0.0228. This value represents the probability of an item lasting less than 6 years. Therefore, to find the probability of lasting longer than 6 years, we subtract this from 1: \( P(X > 6) = 1 - P(X < 6) \) \( P(X > 6) = 1 - 0.0228 = 0.9772 \) So, the probability that a randomly purchased item will last longer than 6 years is approximately **0.9772** (rounded to four decimal places). --- Did you know that the idea of a normal distribution dates back to the early 19th century? It was first introduced by mathematicians Carl Friedrich Gauss and Pierre-Simon Laplace, who used it to explain errors in measurements and create what we know today as the "bell curve." This curve nicely illustrates how values are distributed around the mean, with most items clustering close to the average lifespan. In everyday life, the principles of normal distribution have a wide array of applications! For instance, it helps businesses forecast product lifespans, optimize inventory, and ensure customer satisfaction. Imagine if a car manufacturer could predict how long their vehicles would last based on sales data; they can design warranties and services around those insights, ultimately enhancing customer experience while managing costs effectively!