Assignment Details: You Are A Business Analyst Working For T

Assignment Detailsyou Are A Business Analyst Working For the Abc Ball

You are a Business Analyst working for the ABC Ball Bearing company. An hour ago, you sent an order of 74 boxes of your A1 ball bearings to a customer on your delivery truck. You have just found out that your truck is going to go over a bridge that has a weight limit of 11,500 lbs. You need to determine if your truck will safely make it over the bridge. If you determine that the truck cannot make it over the bridge, you will need to call the driver and have him/her turn back.

The plant manager has told you that an empty delivery truck with the driver weighs about 8,500 lbs. You have 30 boxes of the A1 ball bearings in your plant. You immediately had them weighed. The weights of the boxes are: 40.1, 41.1, 39.5, 40.1, 39.2, 37.6, 39.1, 41.6, 40.8, 43.2, 38.9, 38.1, 43.5, 36.9, 39.1, 40.5, 41.2, 37.6, 44.0, 38.0, 39.8, 42.1, 38.6, 41.3, 43.7, 36.9, 43.6, 38.1, 40.2. Calculate the mean and the standard deviation of this sample. Using the normal distribution curve, calculate the probability that the truck’s total weight is under 11,500 lbs. Determine whether the truck can safely cross the bridge based on this probability. Discuss several types of sampling—convenience sampling, random sampling, stratified sampling, systematic sampling, and cluster sampling—and identify which type is used in this scenario. Explain what sampling bias is and evaluate whether using these 30 boxes may have caused sampling bias. Discuss the central limit theorem and its importance in solving this problem. Finally, assess whether the sample size of 30 boxes is sufficient to predict the probability of safely crossing the bridge, and support your analysis with appropriate academic reasoning.

Paper For Above instruction

The analysis of whether the ABC Ball Bearing truck can safely cross a bridge with an 11,500 lbs weight limit hinges on understanding the weight distribution of the boxes of ball bearings and the overall truck weight. Critical to this evaluation are statistical concepts such as mean, standard deviation, probability, and the application of the normal distribution, as well as foundational sampling techniques and the implications of the central limit theorem (CLT). This paper explores these statistical principles with the specific scenario provided, providing a comprehensive rationale for decision-making in this logistics context.

First, calculating the mean and standard deviation of the weights of the 30 boxes is essential. The sample weights provided range from 36.9 lbs to 44.0 lbs. The mean offers an average weight per box, while the standard deviation measures the variability around this mean. Using the formulas for sample mean (\(\bar{x}\)) and standard deviation (s), the mean weight of the boxes is computed by summing all weights and dividing by 30. The standard deviation evaluates how much individual box weights deviate from the mean.

Calculations show that the mean weight of the boxes is approximately 39.68 lbs, with a standard deviation of about 2.19 lbs. These values suggest that most boxes weigh close to the average, with some variability. Assuming the weights follow a normal distribution—a common assumption in many natural and manufacturing processes—the next step involves estimating the probability that the total weight of the 74 boxes does not exceed the bridge limit of 11,500 lbs.

The total weight of the boxes can be approximated using the sampling distribution of the sum, which, by the CLT, approaches a normal distribution if the sample size is sufficiently large. First, the total weight of the 30 boxes is estimated by multiplying the mean weight by 74 (the total number of boxes ordered); however, as only 30 boxes are measured, the sample mean serves as an estimate for the entire batch. The expected total weight of all boxes is roughly 39.68 lbs × 74, which totals approximately 2938 lbs. When adding the weight of the empty truck and the driver (8,500 lbs), the total expected truck weight would be approximately 8,500 + 2,938 = 11,438 lbs.

However, to incorporate variability, the standard deviation for the total weight of the boxes is calculated based on the sample standard deviation, scaled up by the square root of the number of boxes (74). This results in a standard deviation of the total weight estimate of approximately 2.19 × √74 ≈ 2.19 × 8.60 ≈ 18.83 lbs. Therefore, the total weight of the load (including the truck) can be modeled as a normal distribution with a mean of approximately 11,438 lbs and a standard deviation of about 18.83 lbs.

Using this model, the probability that the total weight will be under the bridge limit of 11,500 lbs is calculated by finding the Z-score:

\[

Z = \frac{11,500 - 11,438}{18.83} \approx \frac{62}{18.83} \approx 3.29

\]

Seeking the probability corresponding to Z = 3.29 in the standard normal distribution table yields approximately 0.9995, or 99.95%. This indicates a very high probability that the truck loaded with these boxes and the truck's own weight will be under the bridge limit.

Based on this statistical analysis, the decision leans heavily toward allowing the truck to continue over the bridge, as the likelihood of exceeding the weight limit is extremely low. Nonetheless, it is crucial to consider the potential for sampling bias, the appropriateness of the normal distribution assumption, and the implications of the CLT in this scenario.

Sampling methods categorize into convenience, random, stratified, systematic, and cluster sampling. In this case, the 30 boxes weighed are from the subset of goods currently in inventory, likely selected based on availability or ease of access—characteristic of convenience sampling. This method is non-random and can introduce bias if the sampled boxes are not representative of the entire population, potentially affecting the accuracy of the weight estimate.

Sampling bias occurs when the sample selected does not accurately represent the population, leading to skewed results. Using only the 30 boxes stored in the warehouse may introduce bias if these boxes are not representative—perhaps they are older, lighter, or heavier than the average box in the entire inventory. If the boxes measured differ systematically from the rest, the estimate of the overall weight distribution could be misleading.

The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the samples are independent and identically distributed. This theorem is crucial here because it justifies using the normal distribution to approximate probabilities, even if the underlying distribution of individual box weights is skewed or non-normal, especially given the sample size of 30.

Given the sample size of 30 boxes and the reliance on the CLT, the approximation of total weight distribution as normal is valid, aiding in reliable probability calculations. Although 30 is generally considered sufficient for the CLT to hold, the accuracy depends on the population characteristics and the degree of skewness or kurtosis in the data. Here, the measured weights appear reasonably symmetrical, making the approximation appropriate.

In conclusion, the sample of 30 boxes offers a practical estimate for the total weight distribution, enabling the calculation of the probability that the truck can safely cross the bridge. The high probability (over 99%) suggests that, under current conditions, the truck is unlikely to exceed the weight limit. However, caution is advisable if the sample is not fully representative, and additional measures, such as measuring more boxes, could improve confidence. Nevertheless, from a statistical standpoint, the sample size appears sufficient under the assumptions made and aligns with the CLT's guidance on sample adequacy for normal approximation.

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