# STatistic project Format: Each

STatistic project

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

3) One problem associated to a probability mass function(binomial or Poisson), preferably one that considers multiplevalues 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 thatbest fits data using transformations and linear correlations of thetransformations, 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 bein (problem statement) + (formal discussion solution) format.

Answer:

Binomial probability mass function

◆ Consider an exam that contain 10 multiple choice question with4 possible choice for each question with only one choice is correctand the pass mark of the exam is 60% .Then find

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

b) what is the probability for the student to get exactly 2answers 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 theX have the binomial distribution with n = 10 and probability ofselecting 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^{r}q^{n-r}

P(X=0) = C( 10,0) 0.25^{0} 0.75^{8}

= 0.75^{10}

= 0.0563

b) P( X= 2) = C(10,2) 0.25^{2}0.75^{8}

= 45* 0.25^{2}0.75^{8}

= 0.2816

c) if the student fail exams means they score less than 60% thatsays 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 oftime is normally distributed with mean of 10% and standarddeviation of 5% .What is the probability of losing money over agiven time period. And also find the same with doubled standarddeviation.

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 Z0.025hence 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 = 0.975then

Z = 1.96