Assume the lifespan of light bulbs manufactured by Bright Inc. can be modeled with a normal distribution with a mean of 300 days and a standard deviation of 40 days. 70% of light bulbs produced by Bright Inc. survive longer than how many days?
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To find how many days 70% of light bulbs produced by Bright Inc. survive longer than, we need to use the concept of the cumulative distribution function (CDF) of a normal distribution.
Here's how we can calculate it step by step:
Step 1: Calculate the z-score
The z-score represents the number of standard deviations the value is away from the mean of the distribution. It can be calculated using the formula:
z = (x - μ) / σ
Where:
- x: The value you're interested in (number of days)
- μ: The mean of the distribution (300 days)
- σ: The standard deviation of the distribution (40 days)
In this case, since we want to find the number of days that 70% of the light bulbs survive longer than, we need to find the z-score for the corresponding percentile.
Step 2: Find the z-score for the percentile
To find the z-score for the desired percentile, we can use a standard normal distribution table or a calculator. Since we're looking for the percentage greater than the value, we want the z-score that corresponds to the complement of the desired percentile.
In this case, we want to find the z-score for the complement of 70%, which is 30%.
Using a standard normal distribution table, a z-score of approximately 0.52 corresponds to a percentile of 30%.
Step 3: Solve for x
Now that we have the z-score, we can solve for x by rearranging the z-score formula:
z = (x - μ) / σ
Rearranging, we get:
x = z * σ + μ
Plugging in the values:
x = 0.52 * 40 + 300
x ≈ 80 + 300
x ≈ 380
Therefore, 70% of light bulbs produced by Bright Inc. survive longer than approximately 380 days.