AboutDr. Nyayapati Swami Expertise I can help you in solving first and second order differential equations. Questions must be at the Undergraduate level. Do not expect me to do all your homework.. If you have a homework question with no clues on how to go about, I will only give you some pointers on solving them.
Experience Ph.D. in Mathematics with more than 15 years of teaching. In addition to undergraduate calculus, I taught many more advanced subjects like Complex Analysis, General Topology, Numerical Analysis, Operations Research, Graph Theory, Mathematical Analysis, Mathematical Economics, Optimisation Theory.
Education/Credentials Ph.D. (University of Toledo, USA)
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Followup To
Question -
Hi Dr Nyayapati,
I would just like you help me with these 3 questions that i'm stuck on below. The ones you sent me back, i have tried the others ones too. thank you
1. the quality control managers of cookies is inspecting a batch of chocolate cookies that has just been baked. if the production process is operating properly, the average number of chococolate chips per cookies is 6.76 with a standard deviation fof 2.6 chociolate chips. what percent of the cookies will have less than 4 chocolate chips? 9.52%
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2. New houses are being built and sold at record paces in the GTA this year. It was reported on the radio last week that on the average one new home was sold every six minutes. In February 3,215 new houses were sold in the GTA region, of which 80% were located in the '905' area code municipalities.
The prices of new homes being built and sold this year are also at record high levels. The average price of a 2-bedroom home was $212,000 with a standard deviation of $5,300. The mean and standard deviation of the prices of 3-bedroom homes were $276,000 and $10,700, respectively. For 4-bedroom homes the corresponding figures were $328,000 and $24,800. All the price data dsets were normally distributed.
The figures also showed that 32% of homes being built this year had 2 bedrooms, 45% had 3 bedrooms, 15% had 4 bedrooms, and the remainder were equal amounts of 5 and 6 bedroom homes. The time to build a 3-bedroom home averaged 205 days, with a standard deviation of 12 days. The sizes of 3-bedroom homes were normally distributed with a mean of 2,540 ft^2 and a standard deviation of 175 ft^2.
In Misssissauga, just west of Winston Churchill Blvd., a large new community of 2400 houses is being built on a 3 square kilometer (3,000,000 m^2) area. ONly 10% of the houses will be bungalows, i.e. one storey. All others will be 2-storey houses. The bungalows are to be randomly distributed thoughout the community.
(Note: All figures above, except those in the 1st paragraph, also apply to the Mississauga community.)
a) What is the probability that a 2-bedroom home will cost more than $225,000?
b) On one street in the Mississauga development there are 23 houses. What is the probability that at most five of them have 4-bedrooms?
c) One developer is building fifty-three 3-bedroom homes in the new Mississauga community. What is the probability that the mean time to construct these homes will be less than 200 days?
d)What is the probability that 12 new homes will be sold in an hour?
e)One developer in the new Mississauga community has a special deal this weekend on 3-bedroom houses. Ther will be a $20,000 price discount on the largest 15% of houses. How big a 3-bedroom house do you have to buy to get this discount?
f) What is the expected number of bedrooms in new homes being built this year?
g) What is the probability that there would be at least 3 bungalows in a 10,000 m^2 area in the new development in Mississauga?
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10. The production process used to produce 500 ft. spools of electrical cable operates continuously, 24 hours a day, and is capable of producing approximately 6 spools per day. The time required to produce one spool is normally distributed with a mean of 4.23 hours and a standard deviation of 0.45 hours. Defects occur in the wire at an average rate of 1 per 1000 feet. As a result, 61% of all spools have no defects. The machine used to produce the wire, needs to be adjusted an average of 4 times per day with a standard deviation of 2 times per day.
a) What is the probability that in a week's production, i.e. 42 spools, there will be at least 35 spools without defects?
b) What is the probability that a spool can be produced in less than 4 hours?
c)What is the probability that the wire producing machine will need to be adjusted at most once in an 8 hour shift?
d)What is the probability that the average time to produce the next 15 spools will be more than 4.1 hours?
e)Seventy-eight (78) percent of spools will be produced within what range of time, centred at the mean?
Answer -
QUESTION 1: You have just the mean 6.76 and standard deviation 2.6. From this it is not possible to draw any conclusion unless the probability distribution is given. You may have to make an assumption that it has a normal distribution. Without this solution is not possible.
QUESTION 2(a): Here price has normal distribution. So you can find the probability:
Let Z = [X - 212000]/5300.
If X = 225000 then Z = [225000 - 212000]/5300 = 2.45
Hence P(X > 225000) = P(Z > 2.45) = 0.0071
(Using the normal distribution tables)
QUESTION 2(b): Use binomial distribution with p = 0.15.
Answer = 23C0 (0.15^0)(0.85^23)+ 23C1 (0.15^1)(0.85^22)
+ 23C2 (0.15^2)(0.85^21) + 23C3 (0.15^3)(0.85^20)
+ 23C4 (0.15^4)(0.85^19) + 23C5 (0.15^5)(0.85^18)
= 0.881.
(You can also use a normal approximation to the binomial)
QUESTION 2(c): Question is not clearly stated. Fifty-three 3 bedroom homes are being constructed. It is not mentioned whether these homes are constructed all at the same time or one after the other. [If they are constructed all at one time, then the time taken to construct Fifty-three houses will be just the time to construct one house!]
QUESTION 2(d): Use Poisson distribution.
Answer = e^(-10)[10^12]/12! = 0.095.
QUESTION 2(e): NOT CLEAR! The question asks how big a 3-bedroom house...... What does BIG mean here?
QUESTION 2(g):
NOT CLEAR! Question says houses are distributed randomly throughout. Can I assume that there will be exactly 8 houses in a 10,000 m^2 area? If yes, then you can use binomial distribution with n = 8 and p = 0.10.
The last question, I will go through later.
Cheers
Dr Swami
Answer 10. The production process used to produce 500 ft. spools of electrical cable operates continuously, 24 hours a day, and is capable of producing approximately 6 spools per day. The time required to produce one spool is normally distributed with a mean of 4.23 hours and a standard deviation of 0.45 hours. Defects occur in the wire at an average rate of 1 per 1000 feet. As a result, 61% of all spools have no defects. The machine used to produce the wire, needs to be adjusted an average of 4 times per day with a standard deviation of 2 times per day.
a) What is the probability that in a week's production, i.e. 42 spools, there will be at least 35 spools without defects?
ANSWER: USE BINOMIAL DISTRIBUTION,
42C0 (0.39^0)(0.61^42) + 42C1 (0.39^1)(0.61^41) + 42C2 (0.39^2)(0.61^40)
+ 42C3 (0.39^3)(0.61^39) + 42C4 (0.39^4)(0.61^38)
+ 42C5 (0.39^5)(0.61^37) + 42C6 (0.39^6)(0.61^36)
+ 42C7 (0.39^7)(0.61^35)
SIMPLIFY THIS TO GET YOUR ANSWER.
b) What is the probability that a spool can be produced in less than 4 hours?
ANSWER: USE NORMAL DISTRIBUTION WITH MEAN = 4.23, SIGMA = 0.45:
P(X < 4) = P[ Z < (4 - 4.23)/0.45 ] = P[ Z < -0.23/0.45 ] = 0.3046 [USING NORMAL TABLES]
c)What is the probability that the wire producing machine will need to be adjusted at most once in an 8 hour shift?
THIS CANNOT BE ANSWERED AS THE DISTRIBUTION IS NOT STATED.
d)What is the probability that the average time to produce the next 15 spools will be more than 4.1 hours?
ANSWER: USE SAMPLING DISTRIBUTION:
P [ SAMPLE MEAN > 4.1 ] = P[ Z > {4.1 - 4.23} / {0.45/ROOT(5)} ] = P[ Z > - 1.12 ] = P[ Z < 1.12 ] = 0.8686.
e)Seventy-eight (78) percent of spools will be produced within what range of time, centred at the mean?
ANSWER: FROM NORMAL TABLES, CRITICAL VALUE = 1.23,
RANGE OF TIME IS 4.23 - (1.23)(0.45) = 3.68 TO 4.23 + (1.23)(0.45) = 4.78
