Questions: 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 (a) What proportion of light bulbs will last more than 62 hours? (b) What proportion of light bulbs will last 52 hours or less? (c) What proportion of light bulbs will last between 57 and 62 hours? (d) What is the probability that a randomly selected light bulb lasts less than 45 hours? (a) The proportion of light bulbs that last more than 62 hours is (Round to four decimal places as needed.)

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
(a) What proportion of light bulbs will last more than 62 hours?
(b) What proportion of light bulbs will last 52 hours or less?
(c) What proportion of light bulbs will last between 57 and 62 hours?
(d) What is the probability that a randomly selected light bulb lasts less than 45 hours?
(a) The proportion of light bulbs that last more than 62 hours is (Round to four decimal places as needed.)

Transcript text: w for the Final Exam Part 1 of 4 Points: 0.75 of 1 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 (a) What proportion of light bulbs will last more than 62 hours? (b) What proportion of light bulbs will last 52 hours or less? (c) What proportion of light bulbs will last between 57 and 62 hours? (d) What is the probability that a randomly selected light bulb lasts less than 45 hours? (a) The proportion of light bulbs that last more than 62 hours is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Proportion of Light Bulbs Lasting More Than 62 Hours

To find the proportion of light bulbs that last more than 62 hours, we calculate $ P(X > 62) $. This can be expressed as: \[ P(X > 62) = 1 - P(X \leq 62) \] Using the cumulative distribution function (CDF) for a normal distribution with mean $ \mu = 56 $ and standard deviation $ \sigma = 3.2 $, we find: \[ P(X \leq 62) \approx 0.9696 \] Thus, the proportion of light bulbs that last more than 62 hours is: \[ P(X > 62) \approx 1 - 0.9696 = 0.0304 \]

Step 2: Proportion of Light Bulbs Lasting 52 Hours or Less

Next, we determine the proportion of light bulbs that last 52 hours or less, which is given by $ P(X \leq 52) $. This is directly obtained from the CDF: \[ P(X \leq 52) \approx 0.1056 \]

Step 3: Proportion of Light Bulbs Lasting Between 57 and 62 Hours

To find the proportion of light bulbs that last between 57 and 62 hours, we calculate $ P(57 < X \leq 62) $. This can be expressed as: \[ P(57 < X \leq 62) = P(X \leq 62) - P(X \leq 57) \] From the CDF, we have: \[ P(X \leq 62) \approx 0.9696 \quad \text{and} \quad P(X \leq 57) \approx 0.6227 \] Thus, the proportion of light bulbs that last between 57 and 62 hours is: \[ P(57 < X \leq 62) \approx 0.9696 - 0.6227 = 0.3469 \]