Mathematics, 15.04.2020 03:01 rjennis002

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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Answer from: maraa013001

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 $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

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:

$\mu = 8000, \sigma = 200, n = 500, s = \frac{200}{\sqrt{500}} = 8.94$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{8250 - 8000}{8.94}$

Z = 27.95

Z = 27.95 has a pvalue of 1.

100% probability that the average lifetime of 500 of these light bulbs is less than 8250 hours

Answer from: Quest

0.32

step-by-step explanation:

2.24/7

Answer from: Quest

1. x> -5

2. x< 8

step-by-step explanation:

1. 2x> -10

divide by 2 from both sides of equation.

2x/2> -10/2

simplify, to find the answer.

-10/2=-5

x> -5 is the correct answer.

2. 3x< 24

divide by 3 from both sides of equation.

3x/3< 24/3

simplify, to find the answer.

24/3=8

x< 8 is the correct answer.

i hope this you, and have a wonderful day!

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