A new extended-life light bulb has an average service life of 750 hours, with a standard deviation of 50 hours. If the service life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?
-3 3
1. A new extended-life light bulb has an average life of 750 hours, with a standard deviation of 50 hours. If the life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?
To find the percentage of the distribution between 600 and 900 hours, we need to use the concept of standard deviation and the properties of the normal distribution.
Step 1: Calculate the z-scores for the lower and upper limits.
First, we calculate the z-score for the lower limit of 600 hours:
z1 = (600 - μ) / σ
where μ is the average service life (750 hours) and σ is the standard deviation (50 hours).
z1 = (600 - 750) / 50 = -3
Next, we calculate the z-score for the upper limit of 900 hours:
z2 = (900 - μ) / σ
z2 = (900 - 750) / 50 = 3
Step 2: Look up the percentage associated with z-scores.
We can now look up the percentage associated with the z-scores -3 and 3 in the standard normal distribution table (also known as the z-table). The standard normal distribution table provides the cumulative percentage of the distribution up to a given z-score.
From the z-table, we find that the percentage associated with z = -3 is approximately 0.0013.
Similarly, the percentage associated with z = 3 is also approximately 0.9987.
Step 3: Calculate the percentage between the limits.
To find the percentage between the two limits, we subtract the lower percentage from the higher percentage:
Percentage between 600 and 900 hours = Percentage(z2) - Percentage(z1)
= 0.9987 - 0.0013
= 0.9974
Step 4: Convert to a percentage.
The value obtained from step 3 is presented as a decimal. To convert it to a percentage, we multiply by 100:
Percentage between 600 and 900 hours = 0.9974 * 100 = 99.74%
Therefore, approximately 99.74% of the distribution will be between 600 and 900 hours.
Use z-scores.
Formula:
z = (x - mean)/sd
Find two z-scores, using 600 for x and also 900 for x. Mean = 750 and sd = 50.
Once you find the two z-scores, use a z-distribution table to determine your probability between the two z-scores.