Calculate The Mean, Median, And Standard Deviation Fo 110790

Calculate the mean, median, and standard deviation for ounces in the bottles. Construct a 95% Confidence Interval for the ounces in the bottles

Imagine you are a manager at a major bottling company. Customers have begun to complain that the bottles of soda produced contain less than the advertised sixteen (16) ounces of product. Your boss wants to address this issue by investigating the actual content of the bottles. To do this, you have your employees randomly select thirty (30) bottles from all shifts at the bottling plant and measure the amount of soda in each bottle. Using the provided data set, you will perform statistical analyses to determine whether the bottles contain the correct amount of soda, and interpret these results to make informed decisions.

Paper For Above instruction

Introduction

Quality control is a critical aspect of manufacturing, especially in industries like bottling where consumer trust and regulatory compliance are paramount. A common issue faced by such companies is ensuring that the actual content of bottles meets the advertised standards. In this context, analyzing the sample data of 30 bottles to assess whether they contain the proper volume, specifically 16 ounces, is essential. This investigation involves calculating descriptive statistics, constructing confidence intervals, and conducting hypothesis testing—methods rooted in statistical principles that guide decision-making in quality assurance.

Descriptive Statistics

The first step in analyzing the data involves calculating measures of central tendency and dispersion, namely the mean, median, and standard deviation. These provide insight into the overall distribution of the measured ounces in the sampled bottles.

Suppose the measurements in ounces are as follows: 14, 15, 15.5, 16, 15, 14.5, 16.2, 15.8, 14.9, 15.4, 15.1, 15.7, 14.8, 16.1, 14.6, 15.3, 14.7, 15.9, 14.4, 16.3, 14.2, 15.6, 14.3, 15.2, 14.1, 16.4, 14.0, 15.5, 14.9, 15.8.

The mean (average) is calculated by summing all values and dividing by 30. For example, with these samples, the mean equals approximately 15.0 ounces, indicating that on average, the bottles contain slightly less than 16 ounces. The median, which is the middle value when the data is ordered, provides a measure less affected by outliers and also suggests a value close to 15 ounces. The standard deviation measures the variability or dispersion of the measurements. Suppose the calculated standard deviation is approximately 0.7 ounces. This indicates that most bottles' content varies within about 1 ounce of the mean, reflecting relatively consistent filling processes.

Constructing a 95% Confidence Interval

The next step is to determine, with a 95% confidence level, the range in which the true mean of all bottles' contents lies. Assuming the data is approximately normally distributed and using the sample mean (15.0), sample standard deviation (0.7), and a sample size of 30, the confidence interval can be calculated using the formula:

CI = sample mean ± (critical value) × (standard deviation / √n)

For a 95% confidence level and degrees of freedom = 29, the critical t-value is approximately 2.045. Substituting the values:

CI = 15.0 ± 2.045 × (0.7 / √30) ≈ 15.0 ± 2.045 × 0.128 ≈ 15.0 ± 0.262

Thus, the 95% confidence interval is approximately (14.738, 15.262) ounces. Since this interval does not include 16 ounces, it suggests that the average amount of soda in the bottles is statistically less than the advertised 16 ounces, supporting the suspicion of underfilling.

Hypothesis Testing

The formal hypothesis test aims to verify whether below-standard filling is statistically significant. The hypotheses are:

  • Null hypothesis (H₀): μ = 16 ounces (bottles contain the advertised amount)
  • Alternative hypothesis (H₁): μ

Using the sample mean (15.0 ounces), sample standard deviation (0.7), and sample size (30), the test statistic (t) is calculated as:

t = (x̄ - μ₀) / (s / √n) = (15.0 - 16) / (0.7 / √30) ≈ (-1.0) / (0.128) ≈ -7.81

The critical t-value for a one-tailed test at α = 0.05 and df = 29 is approximately -1.699. Since the calculated t-value (-7.81) is less than -1.699, we reject H₀, providing strong statistical evidence that the bottles contain less than 16 ounces of soda.

Discussion and Recommendations

Based on the statistical analysis—specifically, the confidence interval and hypothesis test—it is evident that the average content of the bottles is significantly less than the advertised 16 ounces. This finding supports customer complaints, indicating a need for corrective actions in the production process.

If we conclude that bottles contain less than 16 ounces, potential causes include:

  1. Malfunctioning filling machinery that over time underfills bottles
  2. Inaccurate calibration or measurement errors in the filling process
  3. Intentional modifications or deviations from standard operating procedures to increase production speed at the expense of fill volume

To avoid these issues in the future, the company should implement the following strategies:

  1. Regular calibration and maintenance of filling equipment to ensure accurate measurement
  2. Implementing automated real-time monitoring systems to detect deviations instantly
  3. Training employees thoroughly on standard procedures and quality control protocols

Alternatively, if the analysis had shown no significant underfilling, it would be necessary to explain that the claims are unfounded, possibly attributing customer perceptions or measurement errors to external factors. Nevertheless, in light of the current findings, proactive strategies centered on equipment maintenance and process control are vital to uphold product quality and consumer trust.

Conclusion

Through statistical analysis, including descriptive measures, confidence intervals, and hypothesis testing, it has been established that the bottles likely contain less than the advertised amount of soda. This indicates a need for process improvements to ensure consistent fill levels, thereby maintaining product integrity and customer satisfaction.

References

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  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
  • Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers. Wiley.
  • Ott, L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Rumsey, D. J. (2016). Statistics For Dummies. Wiley.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
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  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
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