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
Hoochie [10]
2 years ago
7

A shop sells a particular of video recorder. Assuming that the weekly demand for the video recorder is a Poisson variable with t

he mean 3, find the probability that the shop sells. . (a) At least 3 in a week. (b) At most 7 in a week. (c) More than 20 in a month (4 weeks).
Mathematics
1 answer:
julia-pushkina [17]2 years ago
5 0

Answer:

a) 0.5768 = 57.68% probability that the shop sells at least 3 in a week.

b) 0.988 = 98.8% probability that the shop sells at most 7 in a week.

c) 0.0104 = 1.04% probability that the shop sells more than 20 in a month.

Step-by-step explanation:

For questions a and b, the Poisson distribution is used, while for question c, the normal approximation is used.

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^{-\lambda}*\lambda^{x}}{(x)!}

In which

x is the number of successes

e = 2.71828 is the Euler number

\lambda is the mean in the given interval.

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.

The Poisson distribution can be approximated to the normal with \mu = \lambda, \sigma = \sqrt{\lambda}, if \lambda>10.

Poisson variable with the mean 3

This means that \lambda= 3.

(a) At least 3 in a week.

This is P(X \geq 3). So

P(X \geq 3) = 1 - P(X < 3)

In which:

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

Then

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

P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498

P(X = 1) = \frac{e^{-3}*3^{1}}{(1)!} = 0.1494

P(X = 2) = \frac{e^{-3}*3^{2}}{(2)!} = 0.2240

So

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0498 + 0.1494 + 0.2240 = 0.4232

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 1 - 0.4232 = 0.5768

0.5768 = 57.68% probability that the shop sells at least 3 in a week.

(b) At most 7 in a week.

This is:

P(X \leq 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

In which

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

P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498

P(X = 1) = \frac{e^{-3}*3^{1}}{(1)!} = 0.1494

P(X = 2) = \frac{e^{-3}*3^{2}}{(2)!} = 0.2240

P(X = 3) = \frac{e^{-3}*3^{3}}{(3)!} = 0.2240

P(X = 4) = \frac{e^{-3}*3^{4}}{(4)!} = 0.1680

P(X = 5) = \frac{e^{-3}*3^{5}}{(5)!} = 0.1008

P(X = 6) = \frac{e^{-3}*3^{6}}{(6)!} = 0.0504

P(X = 7) = \frac{e^{-3}*3^{7}}{(7)!} = 0.0216

Then

P(X \leq 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) = 0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680 + 0.1008 + 0.0504 + 0.0216 = 0.988

0.988 = 98.8% probability that the shop sells at most 7 in a week.

(c) More than 20 in a month (4 weeks).

4 weeks, so:

\mu = \lambda = 4(3) = 12

\sigma = \sqrt{\lambda} = \sqrt{12}

The probability, using continuity correction, is P(X > 20 + 0.5) = P(X > 20.5), which is 1 subtracted by the p-value of Z when X = 20.5.

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

Z = \frac{20 - 12}{\sqrt{12}}

Z = 2.31

Z = 2.31 has a p-value of 0.9896.

1 - 0.9896 = 0.0104

0.0104 = 1.04% probability that the shop sells more than 20 in a month.

You might be interested in
Find the area of the triangle.
Maslowich

Answer:

Area of triangle = 1/2*b*h

\frac{1}{2} * 2.8* 7.8\\\frac{1}{2} * 21.84\\\frac{21.84}{2}\\= 10.92

4 0
2 years ago
Read 2 more answers
Z-(-3/4)= 6 1/2 I need help quick
liraira [26]
If Z is the unknown, then
Z-(-3/4)= 6 1/2
Z+3/4= 6 1/2
Z= 6 1/2 - 3/4
Z= 23/4 = 5 3/4
7 0
3 years ago
A baby weighs 8lbs 10oz if her weight doubles in 6 months how much will she weigh in 6 months
Lostsunrise [7]
17 lbs and 4 oz. Because it is just doubling.
8 0
3 years ago
Can someone simplify 9^-2.5?
tresset_1 [31]

Answer: 1/243

Step-by-step explanation: To simplify the problem shown, our first step is to convert the decimal in the exponent, -2.5, to a fraction.

-2.5 is the same thing as -2 and 1/2 so we have 9^-2 and 1/2 and -2 and 1/2 is the same thing as -5/2 so we have 9^-5/2.

The rest of my work is attached in the image provided.

3 0
3 years ago
⚠️⚠️ what’s the M the B and the equation ( Y=Mx+b ) ⚠️⚠️
MAXImum [283]

Answer:

M is the slope and b is the y-intercept. You replace m with the altitude and b with temperature.

Step-by-step explanation:

If you wanted to use 0 and 59 then your equation would say y=0x+59.

I hope that makes sense! Let me know if that's right.

3 0
2 years ago
Other questions:
  • A bathtub in the shape of a rectangular prism is 5 feet long, 3 1/2 feet wide, and 4 1/4 feet high. How much water could the tub
    5·2 answers
  • What is the discriminant of the polynomial below? 2x^2+3x-7
    6·1 answer
  • Solve using the guess-and-test strategy. . . The math club has 57 members who are in sixth or seventh grades. There are twice as
    8·1 answer
  • Suppose A and B are independent events. If P(A) = 0.3 and P(B) = 0.45, what is<br> P(A n B)?
    8·1 answer
  • Plz help ASAP!! Explain your answer! I will mark at brainliest!!!
    12·1 answer
  • In the coordinate plane shown, how far apart are points E and D? Explain how you can use the Pythagorean theorem to determine th
    14·2 answers
  • Determine which function(s) is not a linear function. Select all that apply.
    6·1 answer
  • Jonathan budgeted $250.00 for plants, sets, and seeds for his garden. After reading through his planting guides, he realized he
    14·1 answer
  • HELP PLEASEEE ASAAAAPPPPPPPPPPPP I WILL GIVE BRAINLY TO THE FIRST ONE!!!!!!!!
    9·1 answer
  • What is the domain of f(x) = √x – 4 over the set of real numbers?​
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!