Course Code BCO127: Applied Management Statistics

Course Code Bco127 Course Name Applied Management Statistics Final

A snack company sells mixed nuts. It claims peanuts make up 13% of the content by weight. A manager wishes to find out if the actual content of peanuts in the packet is what it states on the packet. State the null and alternative hypotheses to check if the packets actually contain 13% peanuts.

A company undertakes regression analysis to determine if there is correlation between number of customers in catchment area (measured in millions) and annual sales (measured in millions of dollars). After charting the data, excel returns a correlation equation of: y = -1.20 + 2.05x. If the number of customers in a catchment area is 3.7 million, what is the predicted level of sales?

The mean production rate for a factory is known to be 31 units per hour, with a standard deviation of 5.5 units. A modification is made to the production line. After the modification, the production rate is seen to rise to 33.8 units per hour, based on a sample of 35 tests. Is there any evidence at the 95% level of significance that productivity has improved?

Your company packs coffee into 10 kg packs. From a sample of 6 packs, you obtain a mean weight of 9.51 kg. What is the 95% confidence interval if it is known that the weight of coffee packs is normally distributed and the standard deviation is 0.55 kg?

A manufacturer of miniature servo-actuators for models claims their motors last for 7500 hours. It is known that the standard deviation is 1000 hours, and that the distribution is normal. If a random sample of 64 motors is taken, with a mean of 7250 hours, is their evidence that the mean is no longer 7500 hours?

You believe that spending on clothes amounts to at least 500 euros per year. You know that clothes spending follows a normal distribution, with a standard deviation of 60. You undertake a random sample of 49 students, and they report spending 480 euros. You wish to test your hypothesis at α=0.05. From your research, is spending on clothes at least 500 euros?

Paper For Above instruction

The following paper addresses six key statistical problems relevant to applied management contexts. Each problem utilizes foundational concepts in hypothesis testing, regression analysis, confidence intervals, and analysis of variance to interpret and analyze real-world data scenarios. This comprehensive discussion aims to demonstrate mastery in designing statistical models, performing analysis, and deriving meaningful conclusions aligned with business and management decision-making processes.

1. Hypotheses Regarding Peanuts Content in Snack Packaging

The initial problem involves testing whether the actual percentage of peanuts in snack packets aligns with the advertised 13%. The null hypothesis (H0) posits that the true proportion of peanuts is 13%, represented as H0: p = 0.13. The alternative hypothesis (Ha) contends that the true proportion differs from 13%, expressed as Ha: p ≠ 0.13. This hypothesis test typically employs a z-test for proportions given the large sample sizes common in manufacturing quality control. The company would collect a sample of packets, determine the proportion of peanuts, and compare it with the claimed proportion using the z-test formula: z = (p̂ - p0) / √[p0(1 - p0)/n], where p̂ is the sample proportion, p0 is the claimed proportion, and n is the sample size.

2. Regression Analysis and Prediction

The second scenario examines the relationship between catchment area size and annual sales through regression analysis. The regression equation provided is y = -1.20 + 2.05x. To predict sales when the catchment size is 3.7 million people, substitute x = 3.7 into the regression equation: y = -1.20 + 2.05(3.7) = -1.20 + 7.585 = 6.385 million dollars. This indicates that, given the model, the expected annual sales for a catchment of 3.7 million is approximately 6.39 million dollars. It exemplifies how predictive modeling facilitates sales forecasting in management contexts, supporting strategic decisions.

3. Testing for Improvement in Production Rate

The third question involves testing whether modifications to a production line have statistically improved output. The known population mean production rate is 31 units/hour with a standard deviation of 5.5 units. Post-modification, the sample mean is 33.8 units/hour based on 35 tests. The null hypothesis posits no change or improvement: H0: μ = 31; the alternative hypothesis suggests an increase: Ha: μ > 31. Applying a z-test for the mean: z = (x̄ - μ0) / (σ/√n) = (33.8 - 31) / (5.5/√35) ≈ 2.8 / (5.5/5.916) ≈ 2.8 / 0.931 ≈ 3.01. Comparing this to critical z-value at α=0.05 (1.645 for one-tailed test), since 3.01 > 1.645, there is statistically significant evidence at the 95% confidence level that productivity has improved due to the modification.

4. Confidence Interval for Coffee Pack Weights

Provisioned data indicates a sample mean of 9.51 kg from 6 packs, with known standard deviation 0.55 kg, and the underlying population is assumed normal. The 95% confidence interval is calculated using the z-distribution: CI = x̄ ± z(σ/√n). With z ≈ 1.96 for 95% confidence, the calculation yields: 9.51 ± 1.96(0.55/√6) ≈ 9.51 ± 1.96*(0.224) ≈ 9.51 ± 0.439. Thus, the confidence interval is approximately (9.07 kg, 9.95 kg). This interval indicates that in the population, the true mean weight per pack is between 9.07 and 9.95 kg with 95% confidence, highlighting the precision of the sample estimate.

5. Testing the Claim of Motor Durability

In assessing the motors' longevity claim, the null hypothesis H0: μ = 7500 hours is tested against the alternative Ha: μ ≠ 7500 hours. The sample mean from 64 motors is 7250 hours with a standard deviation of 1000 hours. Employing a z-test: z = (x̄ - μ0) / (σ/√n) = (7250 - 7500) / (1000/8) = (-250) /125 = -2.0. At α=0.05, the critical z-values are ±1.96. Since |z|=2.0 > 1.96, there is sufficient evidence at the 95% confidence level to reject H0, suggesting that the motors’ average lifespan has significantly decreased from 7500 hours.

6. Hypothesis Testing on Clothing Expenditure

The final inquiry investigates whether students' average clothing spending is at least 500 euros annually. With a sample mean of 480 euros, a known standard deviation of 60 euros, and a sample size of 49, the null hypothesis H0: μ ≥ 500 euros is tested against Ha: μ

Conclusion

This analysis demonstrates the application of various statistical techniques—including hypothesis testing, regression prediction, and confidence intervals—in assessing business-related questions. Each case underscores the importance of statistical rigor in informed decision-making in management. The correct application of formulas, interpretation of results, and acknowledgment of significance levels are critical skills illustrated throughout the discussion.

References

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