About 12.5% of restaurant bills are incorrect. If 200 bills are selected at random, find the probability that atleast 22 will contain an error. Is this likely or unlikely to occur? Use the normal approximation to the binomial to find the probabilities for the specific value(s) of X. aA. n=30, p=0.5, x=18B. n=50, p=0.8, x=44C. n=100, p=0.1, x=12

Question
Asked Oct 29, 2019

About 12.5% of restaurant bills are incorrect. If 200 bills are selected at random, find the probability that atleast 22 will contain an error. Is this likely or unlikely to occur?

 

Use the normal approximation to the binomial to find the probabilities for the specific value(s) of X. a

A. n=30, p=0.5, x=18

B. n=50, p=0.8, x=44

C. n=100, p=0.1, x=12

check_circleExpert Solution
Step 1

It is given that 12.5% of restaurant bills contain error. So, the proportion of bills contain error (p) is 0.125.

Here, the sample size (n) is 200.

Let X be the random variable that is defined as the number of bills containing error which follows binomial distribution with n=200 and p=0.125.

Using normal approximation to the binomial distribution, follows normal distribution with,

Mean ( np
200 x 0.125
=25
Standard deviation (o)= \fnp(1- p)
200x 0.125x (1-0.125)
= 4.68
help_outline

Image Transcriptionclose

Mean ( np 200 x 0.125 =25 Standard deviation (o)= \fnp(1- p) 200x 0.125x (1-0.125) = 4.68

fullscreen
Step 2

The probability that at least 22 will contain an error can be calculated as follows:

p P(X2 22)
Using the continuity correction, P(X 2 22)= P(X >21.5)
X -и
P(X 21.5) P
21.5-25
4.68
=P(Z>-0.75)
P(Z <0.75) (symmetrical property)
According to the standard normal table, the probability value
corresponding to the z-score 0.75 is 0.7734.
Thus, P(X 2 22) P(X > 21.5)= 0.7734
The probability that at least 22 will contain an error in the sample
of 200 bills is 0.7734.
help_outline

Image Transcriptionclose

p P(X2 22) Using the continuity correction, P(X 2 22)= P(X >21.5) X -и P(X 21.5) P 21.5-25 4.68 =P(Z>-0.75) P(Z <0.75) (symmetrical property) According to the standard normal table, the probability value corresponding to the z-score 0.75 is 0.7734. Thus, P(X 2 22) P(X > 21.5)= 0.7734 The probability that at least 22 will contain an error in the sample of 200 bills is 0.7734.

fullscreen
Step 3

Now the second part of the question is solved by using the normal approximation ...

(а) n%330, р30.5, х%318
Mean(u) np
30x0.5
= 15
Standard deviation(o) = /np(1 -p)
= V7.5
2.74
By the continuity correction table,
Р(x
18) P(17.5 < x < 18.5)
Р(х < 18.5) - Р(< < 17.5)
18.5-15
х-и
P
17.5-15
- P (-
2.74
2.74
= P(z < 1.27737) - P(z < 0.91241)
= 0.899264 0.819224
0.08
help_outline

Image Transcriptionclose

(а) n%330, р30.5, х%318 Mean(u) np 30x0.5 = 15 Standard deviation(o) = /np(1 -p) = V7.5 2.74 By the continuity correction table, Р(x 18) P(17.5 < x < 18.5) Р(х < 18.5) - Р(< < 17.5) 18.5-15 х-и P 17.5-15 - P (- 2.74 2.74 = P(z < 1.27737) - P(z < 0.91241) = 0.899264 0.819224 0.08

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour*

See Solution
*Response times may vary by subject and question
Tagged in

Math

Probability

Related Probability Q&A

Find answers to questions asked by student like you

Show more Q&A add
question_answer

Q: A company is hiring a chef, a prep and a waiter. 5 equally qualified applicants applied. John is one...

A: (a)The probability that he will be the Chef is,

question_answer

Q: Consider an exponential model for car crashing. Let T be the time in days between the beginning of t...

A: Introduction:The cdf of an exponentially distributed random variable, T with the given pdf is F (t) ...

question_answer

Q: A group consists of four men and eight women. Six people are selected to attend a conference. a. In ...

A: We are givenMen = 4Women = 8Total = 12We have to select randomly 6 people.

question_answer

Q: A pair of die is rolled and the sum of the numbers is noted. Find the probability that one die resul...

A: Conditional rule:The formula for probability of A given B is,

question_answer

Q: There are 100 people waiting in line for IDs.  Of these, 50 are from Indiana, 45 are freshman, and 2...

A: To count the number of persons satisfying the given conditions

question_answer

Q: Please, help me with a step by step solutions to question and I will be very grateful to you. I don'...

A:   As you have posted 2 different questions, we are answering the first question. In case you require...

question_answer

Q: for 2 draws find: P(K, diamonds) with replacement P(hearts, hearts) without replacement

A: In standard deck of 52 cards, there are 4 kings (K) and 13 diamonds. The sampling is with replacemen...

question_answer

Q: I don’t quite understand how to set up the probabilities for the listed in the photo. Help me please...

A: We are given candies,6 cherry3 orange2 lemonSo, total = 11Randomly take 2 pieces.

question_answer

Q: Larry, Kim , Jim, Ian, and Simone have all been invited to a dinner party. They arrive randomly and ...

A: Factorial of an integer:The factorial of an integer is the product of all positive integers less tha...