A company installs 5000 light bulbs, each with an average life of 500 hours, standard deviation of 100 hours, and
distribution approximated by a normal curve. Find the percentage of bulbs that can be expected to last the period of
time.
77) At least 500 hours
To find the percentage of bulbs that can be expected to last at least 500 hours, we need to calculate the area under the normal curve to the right of 500 hours.
1. Calculate the z-score: The z-score measures the number of standard deviations a data point is from the mean. It is calculated using the following formula:
z = (x - μ) / σ
Where:
- x is the value we want to calculate the probability for (500 hours)
- μ is the mean (500 hours)
- σ is the standard deviation (100 hours)
So the z-score in this case is:
z = (500 - 500) / 100 = 0
2. Find the cumulative probability: The cumulative probability is the area under the normal curve to the right of the z-score. We can use a standard normal distribution table or a calculator to find this value.
In this case, since the z-score is 0, the cumulative probability is 0.5. This means that 50% of the bulbs can be expected to last at least 500 hours.
So the percentage of bulbs that can be expected to last at least 500 hours is 50%.
To find the percentage of bulbs that can be expected to last at least 500 hours, we need to use the normal distribution.
First, we need to find the Z-score for the value of 500 hours. The Z-score measures the number of standard deviations an observation or data point is from the mean.
The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
X is the value we want to find the Z-score for (500 hours)
μ is the mean (average life of the bulbs, which is 500 hours)
σ is the standard deviation (100 hours)
Plugging in the values, we get:
Z = (500 - 500) / 100 = 0 / 100 = 0
Now that we have the Z-score, we can find the percentage of bulbs that can be expected to last at least 500 hours using a Z-table.
The Z-table provides the cumulative probability associated with a given Z-score. Since we want to find the percentage of bulbs that can be expected to last at least 500 hours, we need to find the cumulative probability beyond the Z-score of 0.
Looking up the Z-score of 0 in the Z-table, we find that the cumulative probability is 0.5. This means that 50% of bulbs are expected to last at least 500 hours.
Therefore, the percentage of bulbs that can be expected to last at least 500 hours is 50%.
A normal distribution is symmetrical about the mean, which means that 50% of the cases are above the mean, and 50%, below.
So how many % of the lightbulbs will last at least 500 hours?