Question

A certain manufacturer makes 400w light bulbs. Assume that these light bulbs have lifetimes that are normally distributed with a mean lifetime of 430 hours

and a standard deviation of 75 hours.
Use this table or the ALEKS calculator to find the percentage of light bulbs with lifetimes longer than 574 hours. For your intermediate computations, use four or
more decimal places. Give your final answer to two decimal places (for example 98.23%).

Answer 1

Explanation :

To find the percentage of light bulbs with lifetimes longer than 574 hours, we need to calculate the area under the normal distribution curve to the right of 574 hours.

First, we calculate the z-score for 574 hours using the formula: z = (x - μ) / σ

where x is the given value (574 hours), μ is the mean (430 hours), and σ is the standard deviation (75 hours).

z = (574 - 430) / 75

z = 1.92

Using the ALEKS calculator or a standard normal distribution table, we find that the area to the left of z = 1.92 is approximately 0.9732.

To find the percentage of light bulbs with lifetimes longer than 574 hours, we subtract the area to the left from 1:

Percentage = 1 - 0.9732 = 0.0268

Rounded to two decimal places, the percentage of light bulbs with lifetimes longer than 574 hours is 2.68%.

Answer 2

2.87%

Explanation:

To find the percentage of light bulbs with lifetimes longer than 574 hours, we can use the standard normal distribution table or an online calculator.

First, we need to standardize the value of 574 hours using the formula z = (x - μ) / σ, where x is the value we want to standardize, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:

z = (574 - 430) / 75 = 1.92 (rounded to two decimal places)

Next, we need to find the area to the right of the z-score of 1.92 in the standard normal distribution table or calculator. This represents the percentage of light bulbs with lifetimes longer than 574 hours.

Using a standard normal distribution table or calculator, we find that the area to the right of a z-score of 1.92 is approximately 0.0287 (rounded to four decimal places).

To convert this to a percentage, we multiply by 100:

0.0287 * 100 = 2.87% (rounded to two decimal places)

Therefore, the percentage of light bulbs with lifetimes longer than 574 hours is approximately 2.87%.