Question
4.40 CFLBs: A manufacturer of compact fluorescent light bulbs advertises that the distribution of the
lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000
hours.
(a) What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?
(please round to four decimal places) (b) Describe the distribution of the mean
lifespan of 15 light bulbs.
right skewed
approximately normal with
approximately normal with
hours?
What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500
(please round to four decimal places)
lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000
hours.
(a) What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?
(please round to four decimal places) (b) Describe the distribution of the mean
lifespan of 15 light bulbs.
right skewed
approximately normal with
approximately normal with
hours?
What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500
(please round to four decimal places)
Ask by Olson Wood. in the United States
Mar 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a) The probability that a randomly chosen light bulb lasts more than 10,500 hours is approximately 0.0668.
(b) The distribution of the mean lifespan of 15 light bulbs is approximately normal with a mean of 9,000 hours and a standard deviation of approximately 258.19 hours. The probability that the mean lifespan of 15 light bulbs is more than 10,500 hours is approximately 0.0000.
Solution
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Bonus Knowledge
To find the probability that a randomly chosen light bulb lasts more than 10,500 hours, we first convert this lifespan into a z-score using the formula:
Here,
Using the z-table or a standard normal distribution calculator, the probability of a z-score greater than 1.5 is approximately 0.0668. Therefore, the probability that a randomly chosen light bulb lasts more than 10,500 hours is roughly 0.0668 or 0.0668 when rounded to four decimal places.
Now, for the mean lifespan of 15 light bulbs, due to the Central Limit Theorem, the distribution of the sample mean will follow an approximately normal distribution with:
To find the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours:
First, calculate the z-score:
Using the z-table, the probability of a z-score greater than 5.8 is effectively 0, which implies it’s extremely unlikely for the mean lifespan of those 15 light bulbs to exceed 10,500 hours. Thus, it’s roughly 0.0000 when rounded to four decimal places.
