Mathematics, 04.03.2021 23:30 mamas12345

The lifetime of battery of a device (in one charge) is normally distributed with mean µ = 18 hours and standard deviation σ = 2 hours. (a) What is the probability that a battery will last more than 20 hours? (b) Find the 10th percentile of the lifetimes. (c) A particular battery lasts 16 hours. What percentile is its lifetime on? (d) What is the probability that the lifetime of a battery is between 17.5 and 18.5? (e) Ten batteries are chosen at random, what is the probability that the mean lifetime is between 17.5 and 18.5?

Answers: 1

Show answers

Answers

Answer from: TPug24

a) 0.1587 = 15.87% probability that a battery will last more than 20 hours.

b) 15.44 hours.

c) Approximately the 16th percentile.

d) 0.1974 = 19.74% probability that the lifetime of a battery is between 17.5 and 18.5.

e) 0.5704 = 57.04% probability that the mean lifetime is between 17.5 and 18.5

Step-by-step explanation:

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 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 $s = \frac{\sigma}{\sqrt{n}}$ .

In this question, we have that:

$\mu = 18, \sigma = 2$

(a) What is the probability that a battery will last more than 20 hours?

This is 1 subtracted by the pvalue of Z when X = 20. So

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

$Z = \frac{20 - 18}{2}$

Z = 1

Z = 1 has a pvalue of 0.8413

1 - 0.8413 = 0.1587

0.1587 = 15.87% probability that a battery will last more than 20 hours.

(b) Find the 10th percentile of the lifetimes.

This is X when Z has a pvalue of 0.1. So X when Z = -1.28

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

$-1.28 = \frac{X - 18}{2}$

X - 18 = -1.28*2

X = 15.44

So 15.44 hours.

We have to find the pvalue of Z when X = 16. So

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

$Z = \frac{16 - 18}{2}$

Z = -1

Z = -1 has a pvalue of 0.1587

Approximately the 16th percentile.

(d) What is the probability that the lifetime of a battery is between 17.5 and 18.5?

This is the pvalue of Z when X = 18.5 subtracted by the pvalue of Z when X = 17.5. So

X = 18.5

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

$Z = \frac{18.5 - 18}{2}$

Z = 0.25

Z = 0.25 has a pvalue of 0.5987

X = 17.5

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

$Z = \frac{17.5 - 18}{2}$

Z = -0.25

Z = -0.25 has a pvalue of 0.4013

0.5987 - 0.4013 = 0.1974

0.1974 = 19.74% probability that the lifetime of a battery is between 17.5 and 18.5.

(e) Ten batteries are chosen at random, what is the probability that the mean lifetime is between 17.5 and 18.5?

Sample of 10 means that, by the Central Limit Theorem, $n = 10, s = \frac{2}{\sqrt{10}} = 0.6325$

X = 18.5

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

$Z = \frac{18.5 - 18}{0.6325}$

Z = 0.79

Z = 0.79 has a pvalue of 0.7852

X = 17.5

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

$Z = \frac{17.5 - 18}{0.6325}$

Z = -0.79

Z = -0.79 has a pvalue of 0.2148

0.7852 - 0.2148 = 0.5704

0.5704 = 57.04% probability that the mean lifetime is between 17.5 and 18.5

Answer from: Quest

step-by-step explanation:

Answer from: Quest

it's equivalent, so we can say that its equal.

let's call the denominator x and the numerador is 18 less then denominator, x - 18.

$\frac{2}{5}=\frac{x-18}{x}$

multiply

2.x = 5.(x - 18)

2x = 5x - 90

90 = 5x - 2x

90 = 3x

x = 90/3

x = 30

so:

$\frac{2}{5}=\frac{30-18}{30}$

$\frac{2}{5}=\frac{12}{30}$

$Answer this question! eric has a riddle "i am thinking of a fraction that is equivalent to 2/5 and$

Another question on Mathematics

Mathematics, 21.06.2019 12:50

Radii ok and nl are perpendicular to om because of the radius-tangent theorem. by definition of perpendicular, angles kom and lnm are right angles. this means that triangles kom and lnm are right triangles. angle lmn is common to both right triangles, so by the triangles kom and lnm are similar.

Answers: 2

Answer

Mathematics, 21.06.2019 17:00

Which measure is equivalent to 110 ml? 0.0110 l 0.110 l 1.10 l 11,000 l

Answers: 2

Answer

Mathematics, 21.06.2019 18:30

Afarm that has 4 x 10^2 acres has 1.2 x 10^5 cattle. what is the average number of cattle per acre? write the answer in scientific notation. answere asap

Answers: 1

Answer

Mathematics, 21.06.2019 20:30

List x1, x2, x3, x4 where xi is the left endpoint of the four equal intervals used to estimate the area under the curve of f(x) between x = 4 and x = 6. a 4, 4.5, 5, 5.5 b 4.5, 5, 5.5, 6 c 4.25, 4.75, 5.25, 5.75 d 4, 4.2, 5.4, 6

Answers: 1

Answer

You know the right answer?

The lifetime of battery of a device (in one charge) is normally distributed with mean µ = 18 hours a...

Questions