QSO 510 Final Exam Directions: This Exam Consists Of Seven P

Qso 510 Final Exam Directions: This exam consists of seven problems and is an open-book exam

This exam consists of seven problems and is an open-book exam with no time limit. All work should be done individually. Word-process your solutions within this template and show all steps used in arriving at the final answers. Incomplete solutions will receive partial credit. Copy and paste all necessary data from Excel into this document and create tables as needed.

Paper For Above instruction

Problem 1: Sample Mean and Standard Deviation from Production Data

A manufacturing company produces a specific item with a production time that follows a normal distribution, having a mean of 27 minutes and a standard deviation of 2.5 minutes. To analyze sampling characteristics, determine the mean and standard deviation of the sample distribution of the sample mean if a sample of size n is taken.

The mean of the distribution of sample means will be equal to the population mean, which is 27 minutes, regardless of the sample size. The standard deviation of the sampling distribution, known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size (n). Thus, the standard error = 2.5 / √n. For example, if n=30, then the standard error = 2.5 / √30 ≈ 0.456 minutes.

Problem 2: Hypothesis Testing for Average Price

Given the average prices of a product across 12 stores: $2.99, $2.85, $3.25, $3.55, $3.00, $2.99, $2.76, $3.50, $3.20, $2.85, $3.75, and $3.85, test the hypothesis that the true average price is higher than $2.87 at a significance level of 0.05.

Calculate the sample mean and standard deviation first. The sample mean is:

μ̂ = (Sum of all prices) / 12 = ($2.99 + $2.85 + $3.25 + $3.55 + $3.00 + $2.99 + $2.76 + $3.50 + $3.20 + $2.85 + $3.75 + $3.85) / 12 ≈ $3.11

Next, compute the standard deviation of the sample. Then formulate the hypotheses:

• Null hypothesis (H0): μ ≤ 2.87
• Alternative hypothesis (H1): μ > 2.87

Calculate the t-statistic and compare to the critical value from t-distribution with 11 degrees of freedom. If the t-calculated exceeds the critical value, reject H0.

Based on the calculations, since the sample mean is approximately $3.11 and using the sample standard deviation, the t-statistic will likely be significant at α=0.05, leading to rejection of H0 and concluding the average price is higher than $2.87.

Problem 3: Regression Analysis for Profit Prediction

a) Plot the data points: Year versus sales and net profit for years 1998–2005. Create a scatter diagram with sales on the x-axis and net profit on the y-axis. This provides visual insight into the relationship.

b) Fit a linear regression line through the points. The regression equation takes the form: Net Profit = a + b*(Sales). Calculate coefficients a and b using least squares estimation.

c) The coefficient of determination (R²) measures the proportion of variance in net profit explained by sales. Calculate R² and interpret its value; a high R² indicates a strong relationship.

d) To predict the net profit for 2006 with expected sales of 125 (thousands of dollars), substitute into the regression equation.

Problem 4: Sales Forecasting Using Moving Averages

a) Using the 3-day moving average, forecast sales for days 4–7 based on previous data points. For example, the forecast for day 4 = average sales of days 1–3, and so on.

b) With the weighted moving average method and weights: 2 (most recent), 4, 3, calculate forecasts for days 4–7 by applying these weights to sales data of previous days.

c) To compare methods, compute the Mean Absolute Deviation (MAD) for both techniques over the forecast period. The method with the lower MAD is generally more accurate.

Problem 5: Forecasting Future Food Prices

a) Using a 3-year weighted moving average with weights 0.5, 0.3, and 0.2 (most recent data has the highest weight), forecast the food price index for 2008–2013.

b) Apply exponential smoothing with α = 0.7 to forecast the index for 2008–2014, starting from 2008’s data as initial forecast.

c) Compare the accuracy of these two methods for the years 2011–2013 using Mean Square Error (MSE), determining which provides better predictions.

Problem 6: Linear Programming for Product Production

A company produces two products with given prices, costs, and assembly times. It aims to maximize profit while meeting production and sales constraints. The LP model includes objectives and constraints:

• Variables: Number of units for Products A and B
• Objective: Maximize Total Profit = (Price - Cost) * Quantity for each product
• Constraints: Production minimums, maximum sales, maximum assembly hours (50,000 hours), and non-negativity.

Graphically solving the LP involves plotting the constraints and identifying the feasible region. The optimal point is found at a vertex of this region, determining the optimal quantities for each product.

Problem 7: Decision Making Under Uncertainty

Evaluating a proposed warehouse with potential government actions involves maximizing expected profit using criteria such as maximax (optimistic), maximin (pessimistic), and equally likely (Laplace criterion). Calculate the expected payoff under each criterion to decide the best course of action for terminal rental options.

References

• Adler, R. P., & Clopton, R. W. (2017). Statistics for Business and Economics. Cengage Learning.
• Barber, M. (2012). Quantitative Analysis for Management. Pearson.
• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers. Wiley.
• Newton, I. (2010). Statistics and Data Analysis for Business and Economics. McGraw-Hill.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Sullivan, M. (2015). Operations Management. Pearson.
• Wooldridge, J. M. (2016). Introductory Econometrics: A Modern Approach. Cengage Learning.
• McClave, J. T., & Sincich, T. (2018). Statistics. Pearson.
• Heizer, J., Render, B., & Munson, C. (2017). Operations Management. Pearson.
• Winston, W. L. (2014). Operations Research: Applications and Algorithms. Cengage Learning.