Tips For The Quiz 2 Name Section

Tips For The Quizquiz 2name Sect

Identify and solve statistical and financial problems involving combinations, normal distribution, binomial and Poisson probabilities, confidence intervals, hypothesis testing, classification of cash flows, and cash flow statement preparation based on provided data and scenarios. Provide detailed solutions, calculations, and interpretations with credible references.

Paper For Above instruction

The subject matter of this assignment encompasses a broad spectrum of statistical and financial analysis techniques. The comprehensive understanding and application of such methods are essential for effective decision-making in business and economics. This paper systematically discusses each problem, illustrating solutions with detailed explanations, relevant formulas, and interpretations grounded in statistical theory and financial principles.

Problem 1: Combinatorial Selection

A pollster selects 4 out of 7 available people. The problem asks for the number of different groups of 4 that are possible. This is a classic combination problem, calculated using the binomial coefficient:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Applying the formula:

\[ C(7, 4) = \frac{7!}{4! \times 3!} = \frac{5040}{24 \times 6} = 35 \]

Therefore, there are 35 different possible groups of 4 people from the 7 available.

Problem 2: Normal Distribution and Percentiles

The heights of staff follow a normal distribution with a mean of 70 inches and a standard deviation of 3 inches. To find the percentage of staff shorter than 67 inches and taller than 76 inches, we convert these to z-scores:

\[ z = \frac{X - \mu}{\sigma} \]

For X = 67 inches:

\[ z = \frac{67 - 70}{3} = -1 \]

For X = 76 inches:

\[ z = \frac{76 - 70}{3} = 2 \]

Using standard normal distribution tables:

• P(Z
• P(Z > 2) = 1 - P(Z

Matching with options, the most accurate choice aligns with option (a): 16% & 2.5%, considering approximate rounding and standard table values.

Problem 3: Binomial Probability Calculations

Given a 20% warranty requirement, for 20 vehicles, the binomial distribution models the probability that k vehicles require repairs:

\[ P(X = k) = C(n, k) p^k (1-p)^{n-k} \]

a) Probability that none require warranty service (k=0):

\[ P(0) = C(20, 0) \times 0.2^0 \times 0.8^{20} = 1 \times 1 \times 0.8^{20} \]

Calculating:

\[ P(0) = 0.8^{20} \approx 0.012 \]

b) Probability exactly one vehicle requires warranty:

\[ P(1) = C(20, 1) \times 0.2^1 \times 0.8^{19} = 20 \times 0.2 \times 0.8^{19} \]

Approximating:

\[ P(1) \approx 20 \times 0.2 \times 0.8^{19} \approx 20 \times 0.2 \times 0.013 \approx 0.052 \]

c) For 3 or more vehicles requiring warranty:

\[ P(X \geq 3) = 1 - P(0) - P(1) - P(2) \]

Calculating P(2):

\[ P(2) = C(20, 2) \times 0.2^2 \times 0.8^{18} \approx 190 \times 0.04 \times 0.014 \approx 0.107 \]

Thus,

\[ P(X \geq 3) \approx 1 - 0.012 - 0.052 - 0.107 = 0.829 \]

d) The mean and standard deviation of this binomial distribution:

\[ \mu = np = 20 \times 0.2 = 4 \]

\[ \sigma = \sqrt{np(1-p)} = \sqrt{4 \times 0.8} = \sqrt{3.2} \approx 1.79 \]

Problem 4: Poisson Distribution

Given that the probability of a claim is 0.0005 and that 400 policies are written, the expected number of claims (λ) is:

\[ \lambda = np = 400 \times 0.0005 = 0.2 \]

Probability of exactly 2 claims:

\[ P(X=2) = \frac{\lambda^2 e^{-\lambda}}{2!} = \frac{0.2^2 \times e^{-0.2}}{2} \]

Calculating:

\[ P(2) \approx \frac{0.04 \times 0.8187}{2} \approx 0.0164 \]

Probability of at least 3 claims, from Poisson tables or calculations: 

Using tables, P(X ≥ 3) = 1 - P(0) - P(1) - P(2). P(0) = e^{-λ} = 0.8187, P(1) = λ e^{-λ} = 0.164, P(2) ≈ 0.0164, so:

\[ P(X \geq 3) \approx 1 - 0.8187 - 0.164 - 0.0164 = 0.001 \]

Problem 5: Standard Normal Distribution

Using Z-tables:

• a) P(Z
• b) P(Z>1.4) = 1 - 0.9192 = 0.0808
• c) P(Z
• d) P( -0.50
• e) P(0.50

Problem 6: Confidence Interval for Mean

Given sample data: mean (x̄), sample standard deviation (s), sample size (n) with 95% confidence level:

\[ \text{CI} = \bar{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}} \]

Assuming sample mean and standard deviation are provided, with degrees of freedom df= n-1=9, and t-value approximately 2.262:

Interpretation: The calculated interval estimates the true average weekly child-care cost with 95% confidence, indicating the range in which the population mean is likely to fall.

Problem 7: Hypothesis Test for Proportion

Testing whether the proportion of men on the NJ Turnpike exceeds the national proportion:

Null hypothesis (H0): p = 0.56

Alternative hypothesis (Ha): p > 0.56

Sample proportion: p̂ = 165/256 ≈ 0.6445

Test statistic (z):

\[ z = \frac{p̂ - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]

Calculating:

\[ z = \frac{0.6445 - 0.56}{\sqrt{\frac{0.56 \times 0.44}{256}}} \approx \frac{0.0845}{0.031} \approx 2.73 \]

Using z-tables, critical value at α=0.01 (one-tailed): approximately 2.33. Since 2.73 > 2.33, reject H0, supporting the alternative hypothesis.

Problem 8: Hypothesis Test for Population Mean

Test H0: μ ≥ 18 vs. Ha: μ

Values: 17, 18, 20, 16, 15

Calculate sample mean and standard deviation:

\[ \bar{x} = \frac{17 + 18 + 20 + 16 + 15}{5} = 17.2 \]

\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(17-17.2)^2 + (18-17.2)^2 + (20-17.2)^2 + (16-17.2)^2 + (15-17.2)^2}{4}} \]

Calculations:

\[ s \approx 1.92 \]

Test statistic (t):

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{17.2 - 18}{1.92 / \sqrt{5}} \approx -1.2 \]

Critical t-value at α=0.01, df=4, for one-tailed test, approximately -3.747. Since -1.2 > -3.747, do not reject H0.

Classification of Cash Flows

For each cash inflow/outflow, classify as operating, investing, or financing activities.

• Sale of equipment: Investing
• Sale of stock: Financing
• Payment to suppliers: Operating
• Payment to lenders: Financing
• Sale of investments: Investing
• Purchase of land: Investing
• Dividends paid: Financing
• Sale of goods/services: Operating
• Wages and salaries: Operating
• Lending to other businesses: Investing
• Taxes paid: Operating
• Principal collection on loans: Investing
• Interest and dividends received: Operating
• Issue of bonds: Financing
• Purchase of investments: Investing
• Expenses paid: Operating
• Buyback of stock: Financing

Preparation of Cash Flow Statement

Using provided balance sheets and additional info, the cash flow statement using the indirect method depicts cash flows from operating, investing, and financing activities.

Beginning with net income of $103,000, adjustments for non-cash items like depreciation are made, along with changes in working capital, investments, and financing. For example, increases in accounts receivable and inventory reduce cash, while increases in accounts payable and bonds payable affect cash flows in the respective sections. The sale of land at cost increases cash inflow, and the redemption of bonds causes cash outflows.

This comprehensive process results in an outlined cash flow statement, demonstrating how cash position changed between periods and enabling stakeholders to gauge liquidity status and operational efficiency.

Concluding Remarks

Mastering these statistical calculations, probability models, hypothesis testing, cash flow classifications, and financial statement preparations equips students and professionals with vital analytical skills. These tools support informed decision-making, risk assessment, and strategic planning in diverse business contexts. The critical interpretation of results ensures that theoretical knowledge translates effectively into practical insights, enhancing overall financial literacy and competence.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
• Gordon, M. (2018). Financial Statement Analysis. Wiley.
• McClave, J. T., & Sincich, T. (2017). Statistics. Pearson.
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
• Heizer, J., Render, B., & Munson, C. (2016). Operations Management. Pearson.
• Brigham, E. F., & Houston, J. F. (2019). Fundamentals of Financial Management. Cengage Learning.
• Higgins, R. C. (2012). Analysis for Financial Management. McGraw-Hill Education.
• White, G. I., Sondhi, A. C., & Fried, D. (2003). The Analysis and Use of Financial Statements. John Wiley & Sons.
• Gitman, L. J., & Zutter, C. J. (2015). Principles of Managerial Finance. Pearson.