normal distribution show working

[tha boss]

Extended-response question
A taxi company determined that on an annual basis the distance travelled per taxi is normally distributed with a mean of 92 000 kilometres and a standard deviation of 23 500 kilometres.
a   What is the probability, correct to four decimal places, that a taxi travels less than 75 000 kilometres per year?

b   What is the probability, correct to four decimal places, that a taxi travels more than 80 000 kilometres per year?

c   What is the probability, correct to four decimal places, that a taxi travels between 60 000 and 100 000 kilometres in the year?

d   Find the minimum mileage that could be expected by 95% of taxis, to the nearest km.

e   Fred runs a fleet of 10 taxis. What is the probability that at least four of the taxis travel more than 80 000 kilometres in a year?






Answers to extended-response questions
a   0.2347
b   0.6952
c   0.5466
d   53 346
e   0.9884

[paulsterio]

Hope this helps - this is how i do it
this guide is courtesy of "Stonecold" - who wrote it and "Taiga" who reposted it

How to Solve Normal Distribution questions without using calculator syntax
Written by Stonecold

As everyone knows, VCAA doesn't like it, and it would be a BIG risk to use it as a part of your workings for a question worth more than a mark.  This is what Derrick Ha told us to do at his lectures, and I just wanted to share it with some in depth examples.

Question 1.  Normally distributed variable X has a mean of 5 and s.d. of 2.  Find Pr(3.3<X<7.1) correct to 4 decimal places.

Normally, you would just type the following into the calc:  normcdf(5,2,3.3,7.1) and hit execute. Answer is 0.6555.  For a 1 mark question or multi choice question, this is fine.  However for questions worth more marks, this should be avoided, as VCAA has stated that recording calculator syntax is not relevant working.  By using this method, you could jeopardize marks.  Here is the calculator syntax free alternative.

I'll let you know now, that you are going to need to memorize the two equations of the standard normal distribution and the transformed normal distribution.  It is no big deal, as you can have them in your reference book, but the last thing you want to be doing is fiddling around with that, so in my opinion, it is best to memorize it.

Standard normal distribution is

Transformed normal distibution is

Here are the workings:



No calculator syntax, yields the same answer, which is exactly what you want.

Question 2.  For a standard normally distributed variable, what is the value of z for Pr(Z<z)=0.6 correct to 3 decimal places?

Normally, in the CAS you would just type in invnorm(0.6,0,1) and press enter.   Answer is 0.253.  However, once again, this is calculator syntax and not appropriate working.  This is how you should set out your answer:



Solving for z gives

Whilst you have still used you calculator to get the answers in both of these questions, you have shown the examiner that you understand how the question would be completed using mathematics related to the course, and that you understand that a normal distribution is no different to any other continuous random variable, and that the probability over a given interval is calculated in the very same way.  i.e. By calculating the definite integral over the specified interval.

These workings use mathematical syntax rather than calculator syntax, so you can rest assured  that workings such as these will guarantee you the marks on short answer questions based on the normal distribution.

I would also like to point out the you should still do your calculator workings using the normcdf and invnorm functions, as this is far less risky as it will minimize keying in wrong data, and will also save time.  However in your written workings, complete something similar to the above to avoid losing precious marks on what are rather simple questions.

[azn_dj]

Wow. Is this a bit overboard?
but when you compare to IB, they give you a very large table of values and you have to find your answers from that

[yawho]

No requirement to do it that way unless you are asked to use a given normal distribution function to find probability. Otherwise it is just plain stupid.

[tony3272]

Wow. Is this a bit overboard?
but when you compare to IB, they give you a very large table of values and you have to find your answers from that

Do you mean this sort of thing? It was on the previous methods study design.

[paulsterio]

IB - there's no CAS calculators

[Mao]

There is no point to write down the Gaussian distribution. Just state you are using the CDF.