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?
Normally distributed data are symmetric about the mean, so 50% goes above and the other 50% goes below the mean.
To find the percentage of light bulbs that have a life less than 2500 hours, we need to calculate the area under the normal distribution curve up to the value of 2500 hours. We can use the standard normal distribution table or z-table to find this.
Let's start by finding the z-score, which represents the number of standard deviations a given value is from the mean. We can use the formula:
z = (x - μ) / σ
where:
- x is the value we want to calculate the area up to (2500 hours in this case),
- μ is the mean of the distribution (2500 hours),
- σ is the standard deviation of the distribution (500 hours).
Plugging in the values, we get:
z = (2500 - 2500) / 500
z = 0
Since the z-score is 0, it means the value is equal to the mean.
Now, we need to find the area under the normal distribution curve up to the z-score of 0. Looking up the z-table, we find that the area to the left of z = 0 is 0.5000. This means that 50% of the light bulbs have a life less than 2500 hours.
So, the answer is 50%.