# ​Math Final Project

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Question description

Final Project

Conceptual Problems: Probability Density Function and Binomial Convergence

Problem 1

a/Use Excel/Megastat to find the discrete probability and cumulative probability of the Binomial distribution with probability of success p = 0.4 and n = 60. Find its mean and variance.

b/Based upon the chart on Excel, what can you conclude about the binomial convergence?

Hint: Use binom.dist function on Excel and sketch the curve.

Problem 2

Use Excel/Megastat to create the density probability plot of normal distribution. Take μ = 60 and σ = 5. Set up the range values of x from 40 to 80.

Hint: Use norm.dist function and sketch the curve with tick = 0.1

Problem 3 (extra credit)

Demonstrate the Central Limit Theorem(CLT) of the sample mean by sampling a 100 uniform distribution data with 50 variables. Verify the result by computing the sample mean, sample variance and sketch the histogram on Excel/Megastat.

Hint: Generate 100 datasets of 50 variables and calculate 50 sample means to determine the distribution of X̅ and SX̅.

Practical Problems: Historical Trends and Patterns in Stock Prices

Choose two publicly traded companies with a sample size of 50 and a time range (long term/short term) for its stock prices so that you can download the data from quote.yahoo.com or google.com/finance. Make sure to STATE the COMPANY HISTORY and WHY YOU PICK THESE 2 STOCKS for comparison.

Problem 4

Use a statistical package (Excel/Megastat) to find the summary measurements: min, max, mean, median, standard deviation of the historical prices of 2 stocks

a/Draw a Box Plot for both stocks that you picked. Identify the highest and lowest stock prices.

b/Repeat this exercise with histogram/bar-charts and scatter plots for each stock.

c/Write a short paragraph describing how the two stock prices are different. Hint: Base your answer on any notable differences you observe in the two Box Plots. Pace University Math 117 Instructor: Vinh Ly Spring 2018 Data Analysis Project April 4th, 2018

Problem 5

Use Megastat/Excel to find the value of the linear correlation coefficient between these 2 stock prices? Is the correlation significant? Explain the reason for your answer.

a) If the correlation is significant, what does it imply about the trend in the predicted stock?

b) Find the equation for the least squares regression (LSR) line.

c) Interpret the meaning of the slope of the LSR line.

d) Based on the equation of LSR line, what is the “best predicted” value for the stock that you treated as a dependent variable?

Problem 6

Run the hypothesis testing for difference in mean prices of 2 stocks that you picked from the sample size of 50. Assume the difference prices between the two and compare the result with equal/unequal variances or matched-paired hypothesis.

Hint: since F = 𝜎12𝜎22, Use F-test to compare the variances of 2 stocks before choosing the right model to test.

Problem 7

Test the hypothesis that the periodic returns follow a normal distribution, as required by certain models of mathematical finance. Predict the periodic returns that you want to be tested for both population stock prices and run the hypothesis testing based upon your prediction.

Hint: Use z points on the table to test for linearity and periodic return r = 𝑃𝑟𝑖𝑐𝑒2−𝑃𝑟𝑖𝑐𝑒1𝑃𝑟𝑖𝑐𝑒1

Problem 8 (extra credits)

Investigate models which predict the stock price at time t as a function of the stock price at previous times (say St-1 and St-2 for example).

Hint: Run a multiple linear regression model (time series model). Forecast are made on St+1 with respect to St, St-1 and St-2.

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