Mathematics, 29.06.2021 23:20 granniejo

A quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a variance of 2916 with a mean life of 518 minutes. If the claim is true, in a sample of 81 batteries, what is the probability that the mean battery life would be greater than 526.4 minutes

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Answer from: sksksjs

0.0808 = 8.08% probability that the mean battery life would be greater than 526.4 minutes

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 normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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.

The design engineer claims they have a variance of 2916 with a mean life of 518 minutes.

This means that $\sigma = \sqrt{2916} = 54, \mu = 518$

Sample of 81 batteries

This means that $n = 81, s = \frac{54}{\sqrt{81}} = 6$

What is the probability that the mean battery life would be greater than 526.4 minutes?

1 subtracted by the p-value of Z when X = 526.4. So

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

By the Central Limit Theorem

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

$Z = \frac{526.4 - 518}{6}$

Z = 1.4

Z = 1.4 has a p-value of 0.9192

1 - 0.9192 = 0.0808

0.0808 = 8.08% probability that the mean battery life would be greater than 526.4 minutes

Answer from: Quest

9 sandwiches per hour

step-by-step explanation:

when we have a rate, we want a 1 in the denominator

27/3 = 9

27 sandwiches/ 3 hours = 9 sandwiches/hour

Answer from: Quest

a, b, c, d are correct.

step-by-step explanation:

we have g(x) = 3 $\sqrt{x}$ and are required to find which option is a translation of g(x).

now, 'translation' is a transformation which means to shift the function in any direction and the general form of translation is g(x) + k, where k is any real number.

so, according to the options we see that all are correct as they satisfy the general form of translation.

also, it can be seen from the graph below that the function g(x) = 3 $\sqrt{x}$ is shifted from its position in all the four options.