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15.04.2020 •
Mathematics
The lifetime of a certain lightbulb is a random variable with the expectation of 8000 hours and a standard deviation of 200 hours. What is the probability that the average lifetime of 500 of these light bulbs is less than 8250 hours?
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Ответ:
100% probability that the average lifetime of 500 of these light bulbs is less than 8250 hours
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation , the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
What is the probability that the average lifetime of 500 of these light bulbs is less than 8250 hours?
This is the pvalue of Z when X = 8250. So
By the Central Limit Theorem
has a pvalue of 1.
100% probability that the average lifetime of 500 of these light bulbs is less than 8250 hours
Ответ:
the - means negative which also means below 0