1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Reil [10]
3 years ago
6

2. The time between engine failures for a 2-1/2-ton truck used by the military is

Mathematics
1 answer:
OLEGan [10]3 years ago
7 0

Answer:

A truck "<em>will be able to travel a total distance of over 5000 miles without an engine failure</em>" with a probability of 0.89435 or about 89.435%.

For a sample of 12 trucks, its average time-between-failures of 5000 miles or more is 0.9999925 or practically 1.

Step-by-step explanation:

We have here a <em>random variable</em> <em>normally distributed</em> (the time between engine failures). According to this, most values are around the mean of the distribution and less are far from it considering both extremes of the distribution.

The <em>normal distribution</em> is defined by two parameters: the population mean and the population standard deviation, and we have each of them:

\\ \mu = 6000 miles.

\\ \sigma = 800 miles.

To find the probabilities asked in the question, we need to follow the next concepts and steps:

  1. We will use the concept of the <em>standard normal distribution</em>, which has a mean = 0, and a standard deviation = 1. Why? With this distribution, we can easily find the probabilities of any normally distributed data, after obtaining the corresponding <em>z-score</em>.
  2. A z-score is a kind of <em>standardized value</em> which tells us the <em>distance of a raw score from the mean in standard deviation units</em>. The formula for it is: \\ z = \frac{x - \mu}{\sigma}. Where <em>x</em> is the value for the raw score (in this case x = 5000 miles).
  3. The values for probabilities for the standard normal distribution are tabulated in the <em>standard normal table</em> (available in Statistics books and on the Internet). We will use the <em>cumulative standard normal table</em> (see below).

With this information, we can solve the first part of the question.

The chance that a truck will be able to travel a total distance of over 5000 miles without an engine failure

We can "translate" the former mathematically as:

\\ P(x>5000) miles.

The z-score for x = 5000 miles is:

\\ z = \frac{5000 - 6000}{800}

\\ z = \frac{-1000}{800}

\\ z = -1.25

This value of z is negative, and it tells us that the raw score is 1.25 standard deviations <em>below</em> the population mean. Most standard normal tables are made using positive values for z. However, since the normal distribution is symmetrical, we can use the following formula to overcome this:

\\ P(z

So

\\ P(z

Consulting a standard normal table available on the Internet, we have

\\ P(z

Then

\\ P(z1.25)

\\ P(z1.25)

However, this value is for P(z<-1.25), and we need to find the probability P(z>-1.25) = P(x>5000) (Remember that we standardized x to z, but the probabilities are the same).

In this way, we have

\\ P(z>-1.25) = 1 - P(z

That is, the complement of P(z<-1.25) is P(z>-1.25) = P(x>5000). Thus:

\\ P(z>-1.25) = 1 - 0.10565

\\ P(z>-1.25) = 0.89435  

In words, a truck "<em>will be able to travel a total distance of over 5000 miles without an engine failure</em>" with a probability of 0.89435 or about 89.435%.

We can see the former probability in the graph below.  

The chance that a fleet of a dozen trucks will have an average time-between-failures of 5000 miles or more

We are asked here for a sample of <em>12 trucks</em>, and this is a problem of <em>the sampling distribution of the means</em>.

In this case, we have samples from a <em>normally distributed data</em>, then, the sample means are also normally distributed. Mathematically:

\\ \overline{x} \sim N(\mu, \frac{\sigma}{\sqrt{n}})

In words, the samples means are normally distributed with the same mean of the population mean \\ \mu, but with a standard deviation \\ \frac{\sigma}{\sqrt{n}}.

We have also a standardized variable that follows a standard normal distribution (mean = 0, standard deviation = 1), and we use it to find the probability in question. That is

\\ z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}

\\ z \sim N(0, 1)

Then

The "average time-between-failures of 5000" is \\ \overline{x} = 5000. In other words, this is the mean of the sample of the 12 trucks.

Thus

\\ z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}

\\ z = \frac{5000 - 6000}{\frac{800}{\sqrt{12}}}

\\ z = \frac{-1000}{\frac{800}{\sqrt{12}}}

\\ z = \frac{-1000}{230.940148}

\\ z = -4.330126

This value is so low for z, that it tells us that P(z>-4.33) is almost 1, in other words it is almost certain that for a sample of 12 trucks, its average time-between-failures of 5000 miles or more is almost 1.

\\ P(z

\\ P(z

\\ P(z

The complement of P(z<-4.33) is:

\\ P(z>-4.33) = 1 - P(z or practically 1.

In conclusion, for a sample of 12 trucks, its average time-between-failures of 5000 miles or more is 0.9999925 or practically 1.

You might be interested in
Cameron buys 2.45 pounds of apples and 1.65 pounds of pears. Apples and pears each cost c dollars per pound. If the total cost a
Leto [7]

Answer:

2.45c + 1.65c = 4.12 + 0.75

Step-by-step explanation:

To write an equation to find the value for c, we need to declare what c is first.

c = price of fruit

2.45c + 1.65c = 4.12 + 0.75

Now we multiplied c to 2.45 and 1.65 and added them together, because whatever the value of c is will give us the equivalence of the sum of 4.12 + 0.75.

Now to check if the equation is right, let's solve for c.

2.45c + 1.65c = 4.12 + 0.75

4.1c = 4.87

Now to get the value of c, we divide both sides of the equation by 4.1.

\dfrac{4.1c}{4.1}=\dfrac{4.87}{4.1}

c = 1.19

Now let's substitute the value of c in the equation to see if we got it right.

2.45(1.19) + 1.65(1.19) = 4.12 + 0.75

2.92 + 1.96 = 4.87

4.87 = 4.87

Therefore concluding that the value of c is 1.19.

7 0
3 years ago
The figure consists of a quarter circle and a parallelogram. What is the area of the composite figure? Use 3.14 for n 76, Round
asambeis [7]
A is the answer i think
3 0
2 years ago
What is the value of x if <br> 4x - 9 = 11?<br> A. 1/2<br> B.2<br> C.5<br> D.20
Maru [420]

Answer:

C

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Help anyone can help me do this question 3,I will mark brainlest.​
malfutka [58]

Answer:

see explanation

Step-by-step explanation:

Calculate the slopes between pairs of the 3 points using the slope formula

m = \frac{y_{2}-y_{1}  }{x_{2}-x_{1}  }

with (x₁, y₁ ) = (- 9, 3) and (x₂, y₂ ) = (- 3, 9)

m = \frac{9-3}{-3-(-9)} = \frac{6}{-3+9} = \frac{6}{6} = 1

Repeat with (x₁, y₁ ) = (5, 1) and (x₂, y₂ ) = (- 3, 9)

m = \frac{9-1}{-3-5} = \frac{8}{-8} = - 1

If lines are perpendicular then the product of their slopes = - 1 , then

1 × - 1 = -1

Thus there is a right angle between the 2 lines

Then triangle is right- angled

8 0
3 years ago
One positive number is the square of another positive number. their sum is 132. find the numbers.
Jet001 [13]
X +x²= 132
x +x²-132= 132-132
x²+x -132 = 0
(x-11) (x+12) = 0
x-11 = 0 X+12=0
x-11+11 = 0+11 X+12-12=0-12
x =11 x= -12

Check
x +x²= 132 x +x²= 132
-12 +-12²= 132 11 +11²= 132
-12 + 144 = 132 11+ 121=132
132= 132 132=132

Since the problem says positive the answer is (11,121)
8 0
3 years ago
Other questions:
  • Find an equation of the curve that passes through the point (0, 4) and whose slope at (x, y) is x/y.
    10·1 answer
  • <img src="https://tex.z-dn.net/?f=5%20%5Ctimes%206%20%2B%20%20%5Cfrac%7B1%7D%7B2%7D%20" id="TexFormula1" title="5 \times 6 + \f
    8·1 answer
  • Vertical angles are congruent <br><br> a. never<br> b. sometimes <br> c. always
    14·1 answer
  • I need help class almost over
    10·1 answer
  • Simplify the following surds 45 &lt;------ the 45 is a surd
    12·1 answer
  • find the points of D(-3,-1), E(-1,2), F(4,-2) after a rotation 180° about the origin ASAP PLEASE ILL GIVE BRAINLIEST
    14·1 answer
  • I have an 88 in my math class, according to my teacher if I get a 75% on my final exam my grade will be 86. my final exam is wha
    8·2 answers
  • Help- this literally makes no sense
    5·1 answer
  • Iconic memory is a type of memory that holds visual information for about half a second (0.5 seconds). To demonstrate this type
    5·1 answer
  • The point (5 5/8, 2 1/4) lies on a line that represents a proportional relationship between two units of measure.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!