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
kompoz [17]
3 years ago
12

A popular resort hotel has 300 rooms and is usually fully booked. About 7% of the time a reservation is canceled before the 6:00

p.m. deadline with no pen-alty. What is the probability that at least 285 rooms will be occupied? Use the binomial distribution to find the exact valu
Mathematics
1 answer:
kicyunya [14]3 years ago
8 0

Answer:

8.69% probability that at least 285 rooms will be occupied.

Step-by-step explanation:

For each booked hotel room, there are only two possible outcomes. Either there is a cancelation, or there is not. So we use concepts of the binomial probability distribution to solve this question.

However, we are working with a big sample. So i am going to aproximate this binomial distribution to the normal.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

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)}

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)}.

In this problem, we have that:

A popular resort hotel has 300 rooms and is usually fully booked. This means that n = 300

About 7% of the time a reservation is canceled before the 6:00 p.m. deadline with no pen-alty. What is the probability that at least 285 rooms will be occupied?

Here a success is a reservation not being canceled. There is a 7% probability that a reservation is canceled, and a 100 - 7 = 93% probability that a reservation is not canceled, that is, a room is occupied.  So we use p = 0.93

Approximating the binomial to the normal.

E(X) = np = 300*0.93 = 279

\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{300*0.93*0.07} = 4.42

The probability that at least 285 rooms will be occupied is 1 subtracted by the pvalue of Z when X = 285. So

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

Z = \frac{285- 279}{4.42}

Z = 1.36

Z = 1.36 has a pvalue of 0.9131.

So there is a 1-0.9131 = 0.0869 = 8.69% probability that at least 285 rooms will be occupied.

You might be interested in
The wholesale price for a shirt is 3.50 A certain department store marks up the wholesale price by 90% Find the price of the shi
NemiM [27]

Answer:

$6.65

$6.65

$6.65

6 0
3 years ago
Find the sum of a finite geometric series.
blsea [12.9K]
Sum of a finite geometric series=a(r^(n) -1)/(r -1)
or
sum of a finite geometric series=a(1 - r^(n))/(1 - r)
the Smiths would spend a total of $2928.2 on clothing in 4 years
3 0
3 years ago
Read 2 more answers
A company purchased a delivery van for $27,000 with a salvage value of $2,300 on september 1, 2014. it has an estimated useful l
skad [1K]
I believe The answer is $1,900

6 0
3 years ago
A coyote can run up to 43 miles per hour while a rabbit can run up to 35 miles per hour.write two equivalent expression and then
yaroslaw [1]
D=s*t
distaance=speed times time

cd=coyote distance*time=ds*dt
rd=rabbit diatance*time=rs*rt

given
t=6 for all, so dt=rt=6

and ds=43
rs=35


cd=43*6=258miles
rd=35*6=210miles

how much more?
258-210=48

48 more miles
5 0
3 years ago
You are 100m from the base of a tall building. If your feet are exactly 180m from the top of the building, how tall is the build
KatRina [158]

Answer:

the building would be 280 feet (judging on how this is worded.)

Step-by-step explanation:

you are already 100 meters from the base ( bottom) of the building so you add 180 as you are 180 ft from the top of the building. 100 + 180 is equal to 280.

8 0
2 years ago
Other questions:
  • Is answer choice D correct help please ??
    5·2 answers
  • 193 Sam is 4 years younger than Amy. Five years ago he was half her age. How old are they now?
    15·2 answers
  • The length of a rectangle is (x + 2). The
    5·1 answer
  • Chose the correct equation for the line shown on the graph?
    12·1 answer
  • WHAT IS INSIDE QRS <br> HELP ME NOW PLS
    7·2 answers
  • Mateo has 5 pints of cherries . He dividez them into 1/4 pint servings. How many servings doez he make?
    6·1 answer
  • The family is having leftover grilled steak for dinner. There is ½ of a steak left for Julie, her brother, and her mom to share.
    14·1 answer
  • Triangle proportions pls help w/ work
    15·2 answers
  • Help!! i’ll mark u brainlinest
    14·2 answers
  • The size of an exterior angle of a triangle is the supplement of smaller of two opposite interior angles If the greater of the o
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!