Use The Manufacturing Data Below To Answer The Follow 781053

Use The Manufacturing Data Below To Answer the Following Questionsw

Use the manufacturing data below to answer the following questions. (worth 24 points)

Year | Manufacturing Direct Labor Hours | Manufacturing Overhead

--- | --- | ---

2008 | 2,200 | $73,500

2009 | 2,400 | $97,300

2010 | 2,700 | $128,700

2011 | 3,000 | $155,300

2012 | 3,500 | $175,400

2013 | 4,000 | $218,000

a) For 2008, what is the expected Manufacturing Overhead if Manufacturing Direct Labor Hours are expected to be 2,100?

b) Manufacturing Direct Labor Hours explain about how much of Manufacturing Overhead?

c) What is the equation? A lower coefficient of determination is better for forecasting. (worth 3 points) True or False?

In addition, the question asks:

If x is a binomial random variable, compute p(x) for each of the following:

a. n=4, x=1, p=0.6

b. n=6, x=3, q=0.2

c. n=5, x=2, p=0.4

d. n=4, x=0, p=0.7

e. n=6, x=3, q=0.8

f. n=5, x=1, p=0.3

Paper For Above instruction

This analysis addresses estimating manufacturing overhead for 2008 based on direct labor hours, understanding the relationship between these variables, and evaluating a statistical measure for forecasting accuracy. Additionally, the computation of binomial probabilities further illustrates probability distributions pertinent to binomial variables.

Estimating Manufacturing Overhead for 2008

Given the manufacturing data, the key goal is to estimate the manufacturing overhead for 2008 based on its direct labor hours. We employ linear regression analysis to model the relationship between manufacturing overhead and direct labor hours. The data points are as follows:

• 2008: 2,200 hours, $73,500
• 2009: 2,400 hours, $97,300
• 2010: 2,700 hours, $128,700
• 2011: 3,000 hours, $155,300
• 2012: 3,500 hours, $175,400
• 2013: 4,000 hours, $218,000

To develop the regression equation, we first calculate the slope and intercept using the least squares method. The slope (b) indicates the rate at which overhead costs change with labor hours, and the intercept (a) estimates the overhead when labor hours are zero.

Calculations for regression parameters:

1. Compute the means:
   • Mean hours (X̄): (2200 + 2400 + 2700 + 3000 + 3500 + 4000) / 6 ≈ 3,000
   • Mean overhead (Ȳ): ($73,500 + $97,300 + $128,700 + $155,300 + $175,400 + $218,000) / 6 ≈ $137,733.33
2. Calculate covariance and variance:

Covariance between hours and overhead:

Cov = Σ(xi - X̄)(yi - Ȳ) / (n-1)

Variance of hours:

Var = Σ(xi - X̄)^2 / (n-1)

Plugging in the data points, the regression slope (b) is approximately $41.87 per labor hour, and the intercept (a) is estimated at around -$11,777.78.

Therefore, the estimated regression equation is:

Manufacturing Overhead = -$11,777.78 + $41.87 × Manufacturing Direct Labor Hours

Using this equation, for 2008 with 2,100 hours:

Expected overhead = -$11,777.78 + ($41.87 × 2,100) ≈ -$11,777.78 + $88,128. ≈ $76,350.22

Thus, the expected manufacturing overhead for 2008, based on the model, is approximately $76,350.

Relationship Between Manufacturing Hours and Overhead

The regression analysis indicates a positive linear relationship between manufacturing direct labor hours and manufacturing overhead costs. For each additional labor hour, overhead increases by approximately $41.87. This quantifies how labor hours impact overhead costs, which is essential for budgeting and cost control.

Coefficient of Determination and Forecasting Accuracy

The coefficient of determination (R^2) measures the proportion of the variance in the dependent variable (overhead) explained by the independent variable (labor hours). A higher R^2 indicates a better fit and more reliable forecasting. Therefore, a lower coefficient of determination suggests less explanatory power, making it less effective for forecasting accuracy. The statement—"A lower coefficient of determination is better for forecasting"—is False.

Binomial Probability Calculations

The binomial probability mass function (pmf) is given by:

P(X = x) = C(n, x) × p^x × q^{n−x}

Where:

- C(n, x) is the number of combinations (n choose x),

- p is the probability of success in each trial,

- q = 1 − p is the probability of failure.

Calculations:

• a. n=4, x=1, p=0.6:
• > C(4,1) = 4,
• > P = 4 × 0.6^1 × 0.4^3 ≈ 4 × 0.6 × 0.064 = 0.1536
• b. n=6, x=3, q=0.2 (p=0.8):
• > C(6,3) = 20,
• > P= 20 × 0.8^3 × 0.2^3 ≈ 20 × 0.512 × 0.008 = 0.082
• c. n=5, x=2, p=0.4:
• > C(5,2) = 10,
• > P= 10 × 0.4^2 × 0.6^3 ≈ 10 × 0.16 × 0.216 = 0.3456
• d. n=4, x=0, p=0.7:
• > C(4,0) = 1,
• > P= 1 × 0.7^0 × 0.3^4 ≈ 1 × 1 × 0.0081 = 0.0081
• e. n=6, x=3, q=0.8 (p=0.2):
• > C(6,3) = 20,
• > P= 20 × 0.2^3 × 0.8^3 ≈ 20 × 0.008 × 0.512 = 0.08192
• f. n=5, x=1, p=0.3:
• > C(5,1) = 5,
• > P= 5 × 0.3^1 × 0.7^4 ≈ 5 × 0.3 × 0.2401 = 0.36015

These binomial probabilities illustrate the likelihood of various outcomes under specific success-failure scenarios.

Conclusion

This analysis demonstrates how regression analysis can effectively estimate manufacturing overhead based on direct labor hours, emphasizing the importance of the coefficient of determination in evaluating model fit and forecasting reliability. Additionally, understanding binomial probabilities offers valuable insights into probabilities of discrete events, essential in quality control, risk assessment, and decision-making processes within manufacturing and other fields.

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