If light bulbs have lives that are normally distributed with a mean of
2500 hours and a standard deviation of 500 hours, what percentage of
light bulbs have a life less than 2500 hours?
A. About 25%
B. About 50%
C. About 75%
D. About 68%
Answer B
To determine the percentage of light bulbs that have a life less than 2500 hours, we need to calculate the area under the normal distribution curve to the left of 2500 hours.
In this case, we have a normal distribution with a mean of 2500 hours and a standard deviation of 500 hours. The mean represents the center of the distribution, and the standard deviation tells us how spread out the values are from the mean.
To calculate the percentage, we need to find the z-score (standard score) for 2500 hours. The z-score is calculated using the formula:
z = (X - μ) / σ
Where X is the given value (2500 hours), μ is the mean (2500 hours), and σ is the standard deviation (500 hours).
Plug in the values:
z = (2500 - 2500) / 500
Simplify:
z = 0 / 500
z = 0
The z-score is 0. A z-score of 0 represents the mean of the distribution.
Now, we can use a z-table or calculator to find the area to the left of the z-score of 0. The area represents the percentage of values less than 2500 hours.
Since the z-score of 0 is at the mean, the area to the left of it is 50%. Therefore, approximately 50% of light bulbs have a life less than 2500 hours.
So the correct answer is option B: About 50%.
To find the percentage of light bulbs that have a life less than 2500 hours, we can use the concept of the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1.
To convert the given normal distribution with a mean of 2500 hours and a standard deviation of 500 hours to a standard normal distribution, we can use the formula:
z = (x - μ) / σ
where z is the z-score, x is the value we want to convert, μ is the mean, and σ is the standard deviation.
In this case, if we want to find the percentage of light bulbs that have a life less than 2500 hours, we need to convert 2500 hours to a z-score using the given mean and standard deviation.
z = (2500 - 2500) / 500 = 0
The z-score of 0 represents the mean in the standard normal distribution. Since we want to find the percentage of light bulbs with a life less than 2500 hours, we need to find the area to the left of the z-score 0, which is 50%.
Therefore, the correct answer is B. About 50% of light bulbs have a life less than 2500 hours.