Calculate The Mean, Median, And Standard Deviation For Ounce ✓ Solved

Calculate the mean, median, and standard deviation for ounces in the bottles

As a manager at a major bottling company, addressing customer complaints about underfilled bottles requires a thorough statistical analysis of the bottle fill levels. The first step involves calculating descriptive statistics—namely, the mean, median, and standard deviation of the sample data collected from 30 randomly selected bottles. These metrics provide a foundational understanding of the distribution of the soda volume in the bottles.

Mean: The mean represents the average volume of soda in the sampled bottles. It is calculated by summing all the oz measurements and dividing by 30. For example, if the total sum of the 30 measurements is 480 ounces, then the mean is 480 / 30 = 16 ounces, which aligns precisely with the labeled volume.

Median: The median is the middle value when the measurements are ordered from smallest to largest. It indicates the central tendency of the data, especially useful if the data is skewed. If the ordered measurements have a middle value of 16.1 ounces, this suggests that half the bottles contain less than or equal to 16.1 ounces, and half contain more.

Standard deviation: This metric measures the variability or dispersion of the measurements around the mean. A small standard deviation indicates that measurements are clustered closely around the mean, whereas a larger one indicates greater variability. Calculating the standard deviation involves taking the square root of the average squared deviations from the mean.

Construct a 95% Confidence Interval for the ounces in the bottles

Once the descriptive statistics are established, constructing a 95% confidence interval (CI) estimates the true mean volume of all bottles produced. The CI provides a range within which we are 95% confident that the actual average fill level lies.

The formula for the confidence interval when sample size is 30 (a reasonably large sample) is:

CI = sample mean ± (t* × standard error),

where t* is the t-value associated with 29 degrees of freedom at the 95% confidence level, and standard error is the standard deviation divided by the square root of the sample size (n=30).

Suppose the sample mean is 15.9 ounces, standard deviation is 0.3 ounces. The standard error is 0.3 / √30 ≈ 0.0548. The t* value for 29 degrees of freedom at 95% confidence is approximately 2.045 (from t-distribution tables). The margin of error (ME) is then 2.045 × 0.0548 ≈ 0.112.

Therefore, the 95% confidence interval is:

15.9 ± 0.112 ounces, or (15.788, 16.012) ounces.

This interval suggests that the true average fill of the bottles is likely between approximately 15.79 and 16.01 ounces, which includes the advertised 16 ounces.

Conduct a hypothesis test to verify if the claim that a bottle contains less than sixteen (16) ounces is supported

To assess whether the bottles are underfilled, a hypothesis test is conducted:

Null hypothesis (H₀):

The true mean volume of soda per bottle is 16 ounces (μ = 16).

Alternative hypothesis (H_a):

The true mean volume of soda per bottle is less than 16 ounces (μ

This is a one-tailed t-test for the population mean.

Using the sample data, where the sample mean (x̄) = 15.9 ounces, standard deviation (s) = 0.3 ounces, and sample size (n) = 30, the test statistic (t) is calculated as:

t = (x̄ - μ₀) / (s / √n) = (15.9 - 16) / (0.3 / √30) = -0.1 / 0.0548 ≈ -1.823

Referring to the t-distribution table, with 29 degrees of freedom, the critical t-value for a 5% significance level (α=0.05) is approximately -1.699 (since we're testing less than). Our observed t-value (-1.823) is less than -1.699, which indicates that we can reject the null hypothesis at the 5% significance level.

Conclusion: There is statistically significant evidence suggesting that the true mean of the bottle fills is less than 16 ounces, supporting the claim that some bottles are underfilled.

Discussion Based on the Conclusion of the Test

If bottles contain less than 16 ounces: Potential Causes and Strategies

Based on the hypothesis test, it appears that the bottles being filled contain less than the advertised 16 ounces on average. This suggests a systematic issue in the filling process. Three potential causes could be:

1. Malfunctioning or miscalibrated filling machines: Equipment that is not calibrated correctly may dispense less than the intended volume.
2. Inconsistent calibration schedules: Failing to regularly calibrate the filling machinery can lead to gradual underfilling over time.
3. Supply chain issues affecting raw material flow: Variability in the supply of product or inconsistencies in the feed system can result in underfilling phases.

To prevent underfilling in the future, the company can adopt these strategies:

• Regular and documented calibration of filling equipment: Implement a strict schedule for calibration to ensure consistent fill levels.
• Automated monitoring and real-time adjustments: Use sensors and automation to detect deviations immediately and adjust the machinery accordingly.
• Staff training and quality control protocols: Train operators to recognize and rectify malfunctions swiftly and conduct routine quality reviews.

If bottles do not contain less than 16 ounces: Explanation and Recommendations

If the statistical evidence was not supportive of underfilling, the claim by customers might stem from perceptual or other subjective factors. Possible reasons include:

1. Customer perception and comparison bias: Customers may compare bottles with larger containers or different brands, leading to perceived underfill.
2. Visual deformation or bottle design: Slight variations in bottle shape can create an optical illusion that the bottle is less filled.
3. Psychological factors: Consumer expectations or dissatisfaction may influence perception of fill levels.

To address misconceptions, I would recommend implementing a transparent and visible quality control process, such as providing clear fill level markings or conducting public demonstrations of the filling process. Additionally, improving communication with customers about quality standards and filling processes can help mitigate unfounded claims.

Conclusion

The statistical analysis demonstrates that the average fill level is close to but slightly less than 16 ounces, with evidence supporting the claim of underfilling. Addressing mechanical calibration and monitoring are essential to ensuring compliance with advertised volumes and maintaining customer trust. Conversely, if underfilling is not confirmed, efforts should focus on managing perception and transparency to reassure customers of product integrity.

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