Question

Suppose that a lifetime of lightbulbs is approximately normally distributed with a mean of 36 hours and a standard deviation of 3.2 hours. What proportion of lightbulbs will last more than 61 hours?

a) Approximately 2.5%
b) Approximately 0.4%
c) Approximately 15.9%
d) Approximately 1.5%

Answer 1

To find the proportion of lightbulbs that will last more than 61 hours, calculate the z-score for 61 hours and find the corresponding area under the standard normal curve. Subtract this area from 1 to get the final answer.

Step-by-step explanation:

To find the proportion of lightbulbs that will last more than 61 hours, we need to calculate the area under the normal distribution curve to the right of 61 hours. This can be done by calculating the z-score for 61 hours and then finding the corresponding area under the standard normal curve using a z-table or a calculator.

First, we calculate the z-score using the formula: z = (x - μ) / σ where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 61, μ = 36, and σ = 3.2. Plugging in these values, we get: z = (61 - 36) / 3.2 = 7.8125.

Next, we find the area to the right of z = 7.8125 under the standard normal curve. Using a z-table or a calculator, we can find that the area to the right of z = 7.8125 is approximately 0. Since we need to find the proportion of lightbulbs that will last more than 61 hours, we subtract this area from 1 to get the final answer.

Therefore, the proportion of lightbulbs that will last more than 61 hours is approximately 1 - 0 = 1, or 100%.