Quantitative Methods For Business Saibt Assignment 2 S1 2015
quantitative Methods For Business Saibtassignment 2 S1 2015due Monday
The assignments are to be submitted online by 1:00pm on the due date, with a hard copy handed in during class. Late submissions without an extension will incur a penalty of 5% per working day. A completed cover sheet must be attached, and pages should be organized, stapled in the top left corner, and show your name and page number. Handwritten solutions are acceptable if neat; poor presentation attracts a 10% penalty. No plastic sleeves or folders are permitted, and Excel printouts should be pasted into the assignment. All graphs and tables must include your USER ID in their titles.
Paper For Above instruction
The assignment involves developing mathematical models, formulating constraints, and conducting data analysis related to advertising planning, investment data, and statistical analysis of financial data.
Question 1: Advertising Planning for a Public Service Seminar (Total 30 marks)
You are tasked with creating a linear programming model to optimize a promotional budget for a series of public service seminars sponsored by the local Chamber of Commerce. The advertising channels considered are television, radio, and newspapers, each with associated audience reach and costs. Budget and advertisement number constraints exist, along with specific requirements for the proportion of advertisements in each medium.
Specify the decision variables representing the number of advertisements for each medium. Formulate the objective function to maximize total audience reach, including the relevant constraints on advertisement types, total budget, and media utilization proportions. Use Solver to determine the optimal advertising mix, maximum audience, and budget expenditure. Analyze how changes to the radio audience per ad and budget limit affect the optimal solution.
Question 2: Financial Data Analysis and Histograms (Total 20 marks)
You are provided with two years of weekly pricing data for a management investment fund. Your task is to analyze this data by creating histograms, calculating descriptive statistics (including Quartiles 1 and 3), and identifying outliers based on the 1.5× IQR rule. Describe the shape, symmetry, modality, and presence of outliers of the distribution. Provide reasons for your choice of summary measures for the dataset, and interpret these measures in context.
Question 3: Regression Analysis of Price Trends (Total 25 marks)
Using the provided historical price data, develop a regression model to analyze the trend over time. Produce a scatter plot with a regression line, and include the equation and R-squared value. Interpret the slope and intercept, explaining what they imply about the price trend. Use the model to predict future prices (e.g., for January 2003). Evaluate the confidence in your prediction and discuss whether the investment is advisable based on the trend analysis.
Question 4: Normal Distribution and Confidence Intervals for Credit Card Balances (Total 25 marks)
Based on the Australian credit card balances data, assume balances are normally distributed with a mean of $3,127 and a standard deviation of $1,500. Calculate: the probability that an account balance is below $2,500; the balance threshold for the top 25% of accounts; the probability that the average balance of a random sample of 100 accounts exceeds $3,200; construct a 99% confidence interval for the mean balance with a sample mean of $3,180; and determine the necessary sample size to achieve a margin of error of $120 for the confidence interval.
Paper For Above instruction
Question 1: Advertising Planning for a Public Service Seminar
The problem involves maximizing the total audience reach within a fixed promotional budget, considering constraints on the number of advertisements across different media types. The decision variables are the number of advertisements for television (x₁), radio (x₂), and newspapers (x₃). The objective function seeks to maximize total audience reach, expressed as the sum of audiences per advertisement multiplied by the number of ads in each medium.
Decision variables:
Let x₁ = number of television ads
Let x₂ = number of radio ads
Let x₃ = number of newspaper ads
Objective function:
Maximize Z = 4000x₁ + 3000x₂ + 2000x₃
where 4000, 3000, and 2000 are audiences in thousands for TV, radio, and newspapers respectively.
This quantity is to be maximized to reach the largest possible audience.
Resource constraint:
Total cost cannot exceed $25,000:
2500x₁ + 600x₂ + 500x₃ ≤ 25,000
Other constraints:
Maximum number of ads:
x₁ ≤ 10 (TV ads)
x₂ ≤ 15 (Radio ads)
x₃ ≤ 15 (Newspaper ads)
Non-negativity:
x₁, x₂, x₃ ≥ 0
Media proportion constraints:
Radio ads not exceeding 60% of total ads:
x₂ ≤ 0.6(x₁ + x₂ + x₃)
Television ads account for at least 5% of total ads:
x₁ ≥ 0.05(x₁ + x₂ + x₃)
Using Solver, the optimal values of x₁, x₂, and x₃ maximize audience reach while respecting all constraints. The maximum audience, total expenditure, and the impact of changing parameters (e.g., doubling radio audience, increasing budget) can be tested by adjusting your model accordingly.
Question 2: Financial Data Analysis and Descriptive Statistics
The weekly prices collected over two and a half years for a financial instrument in the provided dataset are analyzed through histograms and descriptive statistics. The histogram, generated in Excel, visualizes the distribution of prices across the specified range, allowing assessment of symmetry, modality, and outliers. Descriptive statistics—mean, median, Quartiles 1 and 3, standard deviation—provide numerical summaries of the data.
Applying the 1.5× IQR rule to identify outliers involves calculating the Interquartile Range (IQR = Q₃ - Q₁) and checking for values below Q₁ - 1.5×IQR or above Q₃ + 1.5×IQR. Outliers are observations lying outside these bounds, indicating potential anomalies or measurement errors.
The distribution's shape can be described as symmetric if the mean and median are similar, with unimodality if a single peak exists. Skewness indicates asymmetry, while the presence of outliers suggests extreme values. Summary measures such as median and quartiles are robust against outliers, whereas mean and standard deviation are sensitive to extreme values.
Question 3: Trend Analysis via Regression
A linear regression analysis models the relationship between time and price to evaluate the trend. The regression plot displays data points and the trend line, capturing the overall movement over time. The regression equation takes the form:
Price = a + b × Time
where the slope (b) indicates the average change in price per unit time, and the intercept (a) indicates the estimated price at time zero. The coefficient of determination (R-squared) measures the proportion of variance in prices explained by time.
The slope provides insight into whether prices are increasing or decreasing— a positive slope suggests growth, while a negative indicates decline. Using the regression equation, future prices at specific dates can be forecasted, such as January 2003 data point, by substituting the corresponding time value.
Confidence in the prediction depends on the model's R-squared, residual analysis, and assumption validity. If the data shows a strong linear trend with high R-squared, predictions are more reliable. Otherwise, model limitations should be acknowledged.
Question 4: Normal Distribution and Confidence Intervals for Credit Card Balances
Given that credit card balances are normally distributed with mean = $3,127 and standard deviation = $1,500, probability calculations utilize properties of the normal distribution. To find the probability a balance is less than $2,500, convert to a z-score:
z = (2500 - 3127) / 1500 ≈ -0.418
Using standard normal tables, P(X
To find the balance that at least 25% of accounts exceed, determine the 75th percentile (inverse of the cumulative distribution):
z_{0.75} ≈ 0.674, then Balance = 3127 + 0.674 × 1500 ≈ $4,138. To state that at least 25% of accounts have a balance above this amount, balances above $4,138 comprise the top quartile.
The probability that the mean of a sample of size 100 exceeds $3,200$ is calculated using the standard error:
SE = 1500 / √100 = 150
z = (3200 - 3127) / 150 = 0.480, so P̄ > 3200 ≈ 0.315. This means there's approximately a 31.5% chance that the sample mean exceeds $3200.
Constructing the 99% confidence interval with sample mean $3,180$ involves critical z-value ≈ 2.576:
Margin of Error = 2.576 × 150 / √100 = 2.576 × 15 = $38.64
CI: (3180 - 38.64, 3180 + 38.64) ≈ ($3141.36, $3218.64)
To achieve a margin of error of $120, solve for n:
120 = 2.576 × 150 / √n ⇒ √n = (2.576 × 150) / 120 ≈ 3.222
n ≈ 3.222² ≈ 10.39, so a sample size of at least 11 is needed.
References
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- Bowerman, B. L., O'Connell, R. T., & Murphree, E. S. (2005). Introduction to Business Analytics. McGraw-Hill.
- Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
- Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer.
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