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
svetlana [45]
3 years ago
15

An automobile manufacturer finds that 1 in every 2500 automobiles produced has a particular manufacturing defect. ​(a) Use a bin

omial distribution to find the probability of finding 4 cars with the defect in a random sample of 7000 cars. ​(b) The Poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. Repeat​ (a) using a Poisson distribution and compare the results.
Mathematics
1 answer:
Advocard [28]3 years ago
6 0

Answer:

a) 0.1558 = 15.58% probability of finding 4 cars with the defect in a random sample of 7000 cars.

b) 0.1557 = 15.57% probability of finding 4 cars with the defect in a random sample of 7000 cars. These probabilities are very close, which means that the approximation works.

Step-by-step explanation:

Binomial 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.

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

In which

x is the number of sucesses

e = 2.71828 is the Euler number

\mu is the mean in the given interval.

To use the Poisson approximation for the binomial, we have that:

\mu = np

1 in every 2500 automobiles produced has a particular manufacturing defect.

This means that p = \frac{1}{2500} = 0.0004

a) Use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 7000 cars.

This is P(X = 4) when n = 7000. So

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

P(X = 4) = C_{7000,4}.(0.0004)^{4}.(0.9996)^{6996} = 0.1558

0.1558 = 15.58% probability of finding 4 cars with the defect in a random sample of 7000 cars.

(b) The Poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p.

Using the approximation:

\mu = np = 7000*0.0004 = 2.8. So

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 4) = \frac{e^{-2.8}*(2.8)^{4}}{(4)!} = 0.1557

0.1557 = 15.57% probability of finding 4 cars with the defect in a random sample of 7000 cars. These probabilities are very close, which means that the approximation works.

You might be interested in
A circular swimming pool has a radius of 28 ft. There is a path all the way around the pool that is 4 ft wide. A fence is going
rewona [7]

Answer:

Step-by-step explanation:

4 0
3 years ago
(FASTEST AND BEST ANSWER GETS BRAINLIEST)
Lorico [155]

Answer:

Step-by-step explanation:

5 0
2 years ago
A car is traveling at 45mph. What is the equation that models the total distance d traveled after h. What is the graph of the eq
Ostrovityanka [42]
D=45h

For example: 45mph times 2 hours equals 90 miles traveled
4 0
3 years ago
What is 34-5x6? Help pls thx
azamat

Answer:

4

Step-by-step explanation:

Order of operations multiplication then subtraction so 5*6=30

34-30=4

8 0
3 years ago
Triangle ABC is similar to triangle DEF.
valina [46]

Step-by-step explanation:

angle B = angle E

angle C = angle F

AC= DF

BC = EF

7 0
3 years ago
Other questions:
  • Help help help help​
    12·1 answer
  • What is the major difference between Fourier Transform and Z-Transform??.
    10·1 answer
  • 5. A number cube has faces numbered 1 through 6, and a coin has two sides, heads and tails. The number cube will be rolled once,
    11·1 answer
  • HELP<br> If 45 cookies will server 15 students. How many cookies are needed 30 students?
    11·2 answers
  • 11. The perimeter of a rectangle is 60 cm. The length is four times the width. What
    15·1 answer
  • For her 1st birthday, Ruth's grandparents invested $1000 in an 18-year certificate for her that pays 10% compounded annually. Ho
    13·1 answer
  • Write ( 8 a^− 3 )^− 2/3 in simplest form.
    12·1 answer
  • 30 POINTS PLEASE HELP Solve 2 (e)^3t =10 for t. Round to three decimal places
    10·1 answer
  • A DVD with a radius of 6 centimeters rotates at a speed of 243 revolutions per minute. Find the liner speed in centimeters per s
    14·1 answer
  • 46°<br> Х<br> x<br> X =<br> degrees.
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!