ABC BALL BEARING COMPANY

Calculations 

Mean 

Mean = 

Mean = 1205.5/30 

Mean = 40.183 

Therefore mean=40.18 

Standard Deviation 

σ 2 =  ∑ (xi - μ) 2 

σ 2 = ((40.1 - 40.183333333333) 2 + ... + (40.2 - 40.183333333333) 2 )/30 

= 4.1893888888889 

σ =√4.1893888888889 

=2.0467996699455 

Therefore standard deviation=2.05 

To make a decision on whether to call the driver, I should first calculate the Z-score. Z-score is defined as the difference in the number of standard deviations from the total population mean. Z-score is obtained using the formula below: 

Z= 

In this case, Z is the standard score that is being calculated, μ is the sample mean, x represents the observed value, while σ is the data set standard deviation. 

Before calculating the Z-score, the first thing one should do is to calculate the value of x. This can be obtained by subtracting the weight of the empty delivery truck from the bridge limit, including the driver. This is as calculated below: 

Step 1. 

11,500-8,500 = 3,000 

After calculations, the total mass the driver needs for the truck to cross the bridge is 3,000 safely. The delivery truck is burdened with 74 boxes. To get x, the average weight, I have to divide 3000 by 74, as shown in the calculation below: 

Step 2. 

3,000/74 = 40.5 

After the calculations, it has been found that the maximum weight of each box is 40.5. Since x is now known, the next step is to calculate the Z-score. Therefore, x is equal to 40.5, σ, which is the standard deviation, in this case, is equal to 2.05, and μ, which is the sample mean in this scenario, is equal to 40.2. The calculations are as shown below: 

Z= 

z = 40.5 – 40.2 / 2.05 

z = 0.14634146341463 

If the value of Z is rounded to the nearest two decimal places, the value equals 0.15. Despite the value of Z making a cut, this value is a positive number. This value means that a higher chance than the typical possibility of the delivery truck weighing less than the apportioned amount. If the value were a negative value by any chance, then the Z-score would have a lower chance than the typical chance of the delivery truck to weigh less than the appropriated amount. For the delivery truck to weigh less than the appropriated amount, the probability is 0.558. When this value is rounded into two decimal places, the probability becomes 0.56. To represent this probability in a percentage form, multiply the value by 100%, which would therefore become 56%. This, therefore, means that there is a 56% probability for the delivery truck to weigh less than 11,500 pounds. 

When making a decision whether to allow the truck to go across the bridge or not, likelihood plays a significant role. Since a probability of 56% is far about 50%, then this cannot ensure the drive and the truck's safety. Therefore my decision would be to call the driver and have the truck turn back. It is important to ensure the driver's safety and the products. In order to cross the products across the bridge, another delivery or pickup can later be arranged. The probability of falling on the bridge is 44%, and it is not worth it to risk. 

Sampling Methods 

Simple Random Sampling 

If by any chance, a mistake is made and ends up doing a poor job during the sampling phase of the exploration process, we can make the project's veracity to be at risk. To ensure that appropriate outcomes are achieved, researchers use two sampling techniques: probability sampling, which is also known as representative sampling or random sampling, and non-probability sampling ( Berndt , 2020). If a random sampling technique is used, each component in the population has a probability or an equal probability of being designated. The sample is known as representative since it has the characteristics that cut across the whole population. 

Stratified Sampling 

In stratified sampling, the population is subdivided into strata or groups. Then, a random sample is carefully chosen from the population strata based on the percentage of the population it represents. This means that stratified sampling is more accurate compared to random sampling. This sampling method also stipulates the maximum number of samples that are required at maximum. Since the samples would be carefully chosen from the population's subgroups, the total population list must be created to reduce inconveniences. 

Convenient Sampling 

  The convenience sampling technique is widely used since it is affordable, quick, and convenient. Stratified sampling, for example, requires a lot of prior planning, but convenience planning does not require much planning, especially for specific purposes ( Berndt , 2020). A researcher makes use of the available respondents when collecting the data. Therefore this technique is informal, but it is simple. 

Systematic Sampling 

The systematic sampling technique involves using a probability sampling technique in which the researchers select the respondents from the population at regular intervals. For example, suppose the researcher selects the respondents using a random alphabetical order. In that case, they can end up with a representative sample in which the whole population data can be drawn from them. Individuals who the researcher selects may be assigned a number instead of being chosen randomly. 

Cluster Sampling 

The cluster sampling technique, as it is a probability sampling approach, is used to study large populations, especially those covering a wide region. By use of clusters, the researcher makes use of the preexisting units like cities and schools. In this sampling technique, the researcher divides the population into small subgroups known as clusters ( Berndt , 2020). In order to form a sample, the researcher randomly chose individuals from the clusters. The researcher, therefore, makes use of the readily available sample when employing this sampling technique. 

Sampling Bias 

Sampling bias arises when there are differences that are not solely a result of chance. In most cases, sampling bias occurs when values of a value are analytically over-represented or under-represented compared to the true variable distribution ( Berndt , 2020). In this case, the sample size of 30, which is used, is sufficient if a non-probability technique is used, which means that the sample is not representative, and on the other hand, it may cause bias. The central limit theorem then comes in and relaxes the sample distribution. 

Sampling bias in some cases can occur when a sample from the entire population is chosen while the others within the population are not chosen. For instance, in inconvenience sampling, it is not easy to generalize the population survey results as a whole. From this scenario, since only 30 out of 74 boxes were chosen, then there is a high chance that sampling bias was created. Therefore, the central limit theorem was used in this scenario. 

Central Limit Theorem 

When making a decision whether the delivery truck should go ahead and cross the bridge or call the driver to turn back, the central limit theorem plays a significant role. The theorem states that "…if you have a population with mean μ and deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed” (Kwak et al., 2017). The central limit theorem is clear when it states that sample means approximately normal a random sample with mean and standard deviation is drawn from a population if the random samples have a sufficient size. This is absolutely true whether the basis population is either skewed or normal, if the sample size is big enough. The central limit theorem has the two most essential components, which are standard deviation and mean. When figuring out and calculating these two essential components, it is important to check out their role alongside the central limit theorem when trying to figure out whether to let the driver cross the bridge or call the driver and order him to turn back. 

When using the central limit theorem, the larger the population size, the more accurate and better results you would get. A sample of 30 boxes to weigh and compare played an important role since it helped in deciding whether the delivery truck with the load would cross the bridge or not. After carrying out the calculations and carefully looking at the sample, it is evident that a sample of 30 boxes is adequate to make a sound decision regarding whether the delivery truck ought to cross the bridge or not. In this case, it is not worth it to cross the bridge since it is risky. 

Conclusion 

The central limit theorem has played a significant role in this scenario. The sampling technique used is appropriate in making a sound decision, which on the other hand, can cause a sampling bias, but by use of a central limit theorem and Z-score, it concluded that a sample size of 30 is not adequate. In this scenario, the delivery truck plus the products it had carried was almost hitting the weight restriction, and as an analyst, I would not allow it to close the bridge since it was risky. The decision that I would make as a business analyst is to call the driver to turn back and split the products into two delivery trucks, reducing the risk of losing the driver and the products. The calculations made led to the decision not to cross the bridge. 

References 

Berndt, A. E. (2020). Sampling methods.  Journal of Human Lactation ,  36 (2), 224-226. 

Kwak, S. G., & Kim, J. H. (2017). Central limit theorem: the cornerstone of modern statistics.  Korean Journal of Anesthesiology , 70(2), 144.