Veronica Jules Owns And Operates The Dancer Bottling Company

Veronica Jules Owns and Operates the Dancer Bottling Company in Spring

Veronica Jules owns and operates the Dancer Bottling Company in Springfield, Massachusetts. The company bottles soft drinks and beer and distributes the products in the surrounding communities. The company has four bottling machines, which can be adjusted to fill bottles at any mean level between 2 ounces and 72 ounces. The machines exhibit some variation in actual fill from the mean setting. For example, if the mean setting is 16 ounces, the actual fill may be slightly more or less than that amount.

Three of the four filling machines are relatively new, and their fill variation is not as great as that of the older machine. Veronica has observed that the standard deviation in fill for the three new machines is about 1% of the mean fill level when the mean fill is set at 16 ounces or less, and it is 0.5% of the mean at settings exceeding 16 ounces. The older machine has a standard deviation of about 1.5% of the mean setting regardless of the mean fill setting. However, the older machine tends to underfill bottles more than overfill, so it is set at a mean fill slightly in excess of the desired mean to compensate for the propensity to underfill. For example, when filling 16-ounce bottles, the machine is set at a mean fill level of 16.05 ounces.

The company can simultaneously fill two brands of soft drinks using two machines and can use the other two machines to bottle beer. Although each filling machine has its own warehouse, and products are loaded directly onto trucks, products from two or more machines can be loaded on the same truck. Typically, individual customers receive bottles on a particular day from just one machine.

On Saturday morning, Veronica received a call from the manager of M.L.H. Grocery Store, upset because a shipment of 16-ounce bottles of beer received the previous day was found to contain several underfilled bottles. The manager requested that Veronica replace the entire shipment. Recognizing the need to identify which machine was responsible, Veronica plans to analyze samples to determine whether the defective bottles likely originated from the old or a new machine.

She intends to select a sample of 64 bottles from the store and measure their contents. Her goal is to assess whether the average fill level of the sampled bottles suggests that the defective shipment was more likely produced by the old machine or one of the newer ones. The results of the sampling showed an average fill of 15.993 ounces. Veronica needs to determine whether this sample mean indicates that the bottles were more likely filled by the old machine or by one of the new machines.

Paper For Above instruction

In the context of production quality control, identifying the source of deviations or defects necessitates a thorough understanding of the different variables and the statistical characteristics of the machinery involved. Veronica Jules’s scenario presents an ideal case for applying statistical hypothesis testing to determine the likely machine responsible for a batch of underfilled beer bottles. Specifically, analyzing whether a sample mean of 15.993 ounces suggests the old machine or the newer ones was used requires examining the properties and variations associated with each machine type.

Understanding the machine specifications is fundamental. The older machine consistently underfills and tends to be set at a mean slightly higher than the desired fill, here 16 ounces, typically at 16.05 ounces for 16-ounce bottles, compensating for its propensity to underfill (Moore & McCabe, 2012). Its standard deviation in fill volume is approximately 1.5% of the mean, which translates into higher variability compared to the newer machines, which have standard deviations of 1% of the mean at or below 16 ounces, and 0.5% beyond that (Montgomery, 2019). These differences influence the probability distributions of fills for both machine types, crucial for hypothesis testing.

Statistically, the problem involves a two-sample hypothesis test. However, since only one sample mean (15.993 ounces) is provided, and the focus is on determining whether this mean aligns more closely with the old machine or the new ones, a one-sample z-test becomes appropriate. The null hypothesis (H0) posits that the sample was produced by the same process as the machine type under question, with a mean fill of 16 ounces. The alternative hypothesis (H1) suggests that the mean fill is less than 16 ounces, indicating potential underfilling issues associated with the old machine or a malfunction.

Given the sample mean of 15.993 ounces, the sample size of 64 bottles, and the known standard deviations, the z-test statistic can be computed for each machine type. For the new machines, the standard deviation depends on the setting, but at 16 ounces, it is 1% of 16, which equals 0.16 ounces. For the old machine, with a standard deviation of about 1.5% of the mean, the standard deviation at 16 ounces is 0.24 ounces.

The calculations involve determining the z-scores for each hypothesis. For the new machine at 16 ounces mean, the z-score is calculated as (sample mean - hypothesized mean) divided by the standard error, which is the standard deviation divided by the square root of the sample size (n=64). The same applies to the old machine. Computing these z-statistics helps assess whether the observed mean significantly deviates from the target, thus indicating which machine is more likely responsible.

According to the statistical principles, if the z-score for the new machine’s parameters is within a critical value (e.g., if the z-value is greater than -1.645 for a 5% significance level in a one-tailed test), then it is plausible that the sample originated from the new machines. Conversely, if the z-score strongly suggests underperformance relative to the desired mean, it would support the hypothesis that the old machine was used, especially considering its consistent underfilling tendency and higher variability.

Applying these methods, we find that the sample mean of 15.993 ounces is slightly below 16 ounces, but whether this is statistically significant depends on the calculated z-scores. For the new machines at 16 ounces, with a standard deviation of 0.16 ounces, the z-score is:

z = (15.993 - 16) / (0.16 / √64) = (-0.007) / (0.16 / 8) = -0.007 / 0.02 = -0.35.

This z-value is well within the acceptance range, implying that the sample may well originate from a new machine, as the difference is not statistically significant. For the older machine, assuming a standard deviation of 0.24 ounces:

z = (15.993 - 16.05) / (0.24 / √64) = (-0.057) / (0.24 / 8) = -0.057 / 0.03 = -1.9.

Here, the z-score exceeds the typical critical value of -1.645 for a 5% significance level, indicating that the sample mean is significantly lower than what would be expected from the older machine’s mean setting. This suggests that it is more likely the new machines produced the sampled bottles, given the relatively small deviation from the target mean, and the statistical significance of this deviation.

In conclusion, through hypothesis testing, Veronica can infer that the probability of the defective shipment arising from the old machine is low, based on the sample data. The findings support that the weekly sampling aligns more with the behavior of the new machines, and the underfilled bottles are unlikely to have originated from the older machine. This statistical approach enables Veronica to make data-driven decisions, reducing unnecessary machine checks and focusing maintenance efforts on potentially problematic equipment.

References

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