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?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
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 to the left of 2500 hours.
Here's how you can do it step by step:
Step 1: Standardize the value of 2500 hours using the standard score formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
In this case, x = 2500 hours, μ = 2500 hours, and σ = 500 hours.
So, z = (2500 - 2500) / 500 = 0.
Step 2: Look up the area to the left of z = 0 in the standard normal distribution table.
Since z = 0, the area to the left of z = 0 is 0.5000 (or 50%).
Therefore, approximately 50% of light bulbs have a life less than 2500 hours.
Note: In a standard normal distribution, the mean is 0 and the standard deviation is 1. By standardizing the value of interest, we can use the standard normal distribution table to determine the corresponding area. In this case, since the mean and standard deviation of the light bulb lives were given, we needed to standardize the value before using the table.
To find the percentage of light bulbs that have a life less than 2500 hours, we need to compute the area under the normal distribution curve up to 2500 hours.
First, let's standardize the value by subtracting the mean and dividing by the standard deviation:
Z = (2500 - 2500) / 500 = 0
Since the mean is 2500 and we are interested in values less than 2500, we are looking for the area to the left of Z = 0.
Using the standard normal distribution table or a calculator, we can find that the area to the left of 0 is 0.5000, or 50%.
Therefore, 50% of light bulbs have a life less than 2500 hours.