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Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks

Gnesinka [82]

Answer:

a) 3.75

b) 23.61% probability that fewer than 3 tanks will be found to be leaking

c) 0% the probability that at least 600 of these tanks are leaking

Step-by-step explanation:

For each tank there are only two possible outcomes. EIther they leak, or they do not. The probability of a tank leaking is independent of other tanks. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

To solve question c), i am going to approximate the binomial distribution to the normal.

Normal probability distribution

Problems of normally distributed samples 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.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

It is estimated that 25% of these tanks leak.

This means that $p = 0.25$

15 tanks chosen at random

This means that $n = 15$

a.What is the expected number of leaking tanks in such samples of 15?

$E(X) = np = 15*0.25 = 3.75$

b.What is the probability that fewer than 3 tanks will be found to be leaking?

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{15,0}.(0.25)^{0}.(0.75)^{15} = 0.0134$

$P(X = 1) = C_{15,1}.(0.25)^{1}.(0.75)^{14} = 0.0668$

$P(X = 2) = C_{15,2}.(0.25)^{2}.(0.75)^{13} = 0.1559$

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0134 + 0.0668 + 0.1559 = 0.2361$

23.61% probability that fewer than 3 tanks will be found to be leaking

c.Now you do a larger study, examining a random sample of 2000 tanks nationally. What is the probability that at least 600 of these tanks are leaking?

Now we have n = 2000. So

$\mu = E(X) = np = 2000*0.25 = 500$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{2000*0.25*0.75} = 19.36$

This probability is 1 subtracted by the pvalue of Z when X = 600. So

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

$Z = \frac{600 - 500}{19.36}$

$Z = 5.16$

$Z = 5.16$ has a pvalue of 0.

0% the probability that at least 600 of these tanks are leaking

4 0

4 years ago

ANSWER ASAP PLEASE. How many lines of symmetry does this figure have?

elena-s [515]

Answer:

4

Step-by-step explanation:

7 0

3 years ago

The height of one solid limestone square pyramid is 21 m. A similar solid limestone square pyramid has a height of 30 m. The vol

Elina [12.6K]

Answer:

Part a) The scale factor of the smaller pyramid to the larger pyramid in simplest form is $\frac{7}{10}$

Part b) The ratio of the volume of the smaller pyramid to the larger pyramid is $\frac{343}{1,000}$

Part c) The volume of the smaller pyramid is $4,116\ m^{3}$

Step-by-step explanation:

Part a) The scale factor of the smaller pyramid to the larger pyramid in simplest form

we know that

If two figures are similar, then the ratio of its corresponding sides is equal and this ratio is called the scale factor

so

Let

z----> the scale factor

x----> the height of the smaller pyramid

y----> the height of the larger pyramid

$z=\frac{x}{y}$

substitute the values

$z=\frac{21}{30}$

Simplify

$z=\frac{7}{10}$ -----> scale factor in simplest form

Part b) Ratio of the volume of the smaller pyramid to the larger pyramid

we know that

If two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cube

so

Let

z----> the scale factor

x----> the volume of the smaller pyramid

y----> the volume of the larger pyramid

$z^{3}=\frac{x}{y}$

we have

$z=\frac{7}{10}$

substitute

$\frac{7}{10}^{3}=\frac{x}{y}$

Rewrite

$\frac{x}{y}=\frac{343}{1,000}$ -----> ratio of the volume of the smaller pyramid to the larger pyramid

Part c) The volume of the smaller pyramid

we know that

If two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cube

so

Let

z----> the scale factor

x----> the volume of the smaller pyramid

y----> the volume of the larger pyramid

$z^{3}=\frac{x}{y}$

we have

$z=\frac{7}{10}$

$y=12,000\ m^{3}$

substitute the values and solve for x

$(\frac{7}{10})^{3}=\frac{x}{12,000}$

$x=\frac{343}{1,000}(12,000)=4,116\ m^{3}$

8 0

3 years ago

10 Y What is the vertical change from Point A to Point B? 9 8 What is the horizontal change from Point A to Point B? 7 6 What is

Alina [70]

Answer:

0.5

Step-by-step explanation:

From the graph given ;

The vertical distance ; y - axis :

(y2 - y1) ; y2 = 2 ; y1 = 1

y2 - y1 = 2 - 1 = 1

The horizontal distance ; the x - axis :

(x2 - x1) ; x2 = 4 ; x1 = 2

x2 - x1 = 4 - 2 = 2

The slope ; the rate of change :

Change in y / change in x

The rate of change ;

(y2 - y1) / (x2 - x1) = 1 / 2

7 0

3 years ago

Refer to the figure below to complete the following problem.

Romashka-Z-Leto [24]

Answer:

C. 30°

Step-by-step explanation:

8 0

4 years ago