Question

STatistic project

Format: Each group will create your own (not created online or in any book) set of study problems for the final exam, in addition to the solution to those problems. The problems should be in depth, creative, and elaborate in nature in order to encapsulate all topics associated with the problem category. If the problem you have is related to data, you can generate a random set of data to use for your solution. The problems that the group project must entail each of the following

3) One problem associated to a probability mass function (binomial or Poisson), preferably one that considers multiple values and not just one.

4) Two problems associated to normal distributions of values: one being for probability, one being an inverse problem.

8) One problem associated to finding a non-linear model that best fits data using transformations and linear correlations of the transformations, and using the model for predictions.

Same groups will apply. Distribute the problems (12 total) evenly amongst the members. The structure of the report should be in (problem statement) + (formal discussion solution) format.

Answer 1

Binomial probability mass function

◆ Consider an exam that contain 10 multiple choice question with 4 possible choice for each question with only one choice is correct and the pass mark of the exam is 60% .Then find

a) what is the probability for the student to get no answer correct ?

b) what is the probability for the student to get exactly 2 answers correct ?

c) what is the probability for the student to fail the exam ?

Answers

a) Let X be the number of questions student answer correctly the X have the binomial distribution with n = 10 and probability of selecting correct answer p = 1/4 = 0.25

Probability of failure

q = 1-p

= 1- 0.25

= 0.75

The we know probability mass function as

P( X= r) = C( n,r) p^rq^n-r

P(X=0) = C( 10,0) 0.25⁰ 0.75⁸

= 0.75¹⁰

= 0.0563

b) P( X= 2) = C(10,2) 0.25²0.75⁸

= 45* 0.25²0.75⁸

= 0.2816

c) if the student fail exams means they score less than 60% that says their correct answer is less than 6

P(X <6) =  

= 0.0563 + 0.1877 + 0.2816 + 0.2503 + 0.1460 + 0.0584

= 0.9803

Normal distribution

◆ suppose the return of investment in stock over a period of time is normally distributed with mean of 10% and standard deviation of 5% .What is the probability of losing money over a given time period. And also find the same with doubled standard deviation.

We know probability of losing money can be find as

P( X<= 0) = P((X-10)/5 <= (0-10)/5)

= P(z<= -2)

= 0.02275

With doubled standard deviation

P(X<=0) = P((X-10)/10 <= (0-10)/10)

= P( z<= -1)

= 0.1587

Inverse problem

Find za for where area A= 0.025 ?

P(z>z0.025) = 1- P(Z<= Z0.025)

Then we can understand that

P( Z<= Z0.025) = 1- P( Z>Z0.025)

= 1- 0.025

= 0.975

Where second equality follows from the definition of Z0.025 hence our problem is equalent to find

Z0.025 such that P( Z<= Z0.025) = 0.975

We can find the Z value respect to the probability = 0.975 then

Z = 1.96