Calculate A T Test For The Following Scenario An Engineer Is

Calculate A T Test For The Following Scenario An Engineer Is Desig

Calculate A T Test For The Following Scenario An Engineer Is Desig

An engineer is designing a stationary parts bin at a work table and considers using an established functional grip reach of 29.55 inches taken from the overall population mean, which was published in a textbook. The engineer is concerned because the workers in the facility are of a smaller stature and decides to test this assumption by taking a small sample of seven individuals. The reach measurements for these individuals are: 27.87, 29.49, 28.34, 28.20, 29.00, 29.56, and 27.95 inches. The task is to perform a t-test to determine if this sample mean significantly differs from the established population mean, and to interpret the resulting p-value in this context. Additionally, the engineer needs to assess whether her hypothesis—that the worker’s reach is less than the population mean—is supported by the data, and to discuss the implications of this statistical result for the design of the parts bin.

Paper For Above instruction

The scenario presented involves conducting a t-test to compare a sample mean to a known population mean, which is a common procedure in inferential statistics used to determine if an observed sample significantly differs from a specified value. Here, an engineer is assessing whether the reach of a specific worker demographic is smaller than the population mean of 29.55 inches by analyzing a sample of seven individuals. This analysis helps inform ergonomic design decisions that ensure safety and efficiency at the work station.

The first step in this analytical process involves calculating the sample mean and standard deviation. The seven reach measurements are: 27.87, 29.49, 28.34, 28.20, 29.00, 29.56, and 27.95 inches. The sample mean (\(\bar{x}\)) is calculated by summing all values and dividing by the number of observations (\(n=7\)):

\[

\bar{x} = \frac{27.87 + 29.49 + 28.34 + 28.20 + 29.00 + 29.56 + 27.95}{7} = \frac{200.41}{7} \approx 28.63 \text{ inches}

\]

Next, the sample standard deviation (s) is calculated by determining the variance: subtracting the mean from each observation, squaring the difference, summing these squared differences, dividing by \(n-1=6\), and taking the square root:

\[

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

\]

Calculating deviations:

- (27.87 - 28.63)^2 = 0.5476

- (29.49 - 28.63)^2 = 0.7360

- (28.34 - 28.63)^2 = 0.0841

- (28.20 - 28.63)^2 = 0.1849

- (29.00 - 28.63)^2 = 0.1369

- (29.56 - 28.63)^2 = 0.8881

- (27.95 - 28.63)^2 = 0.4624

Sum of squared differences: 3.04

Variance: \(3.04/6 \approx 0.507\)

Standard deviation: \(s \approx \sqrt{0.507} \approx 0.71\) inches

With these values, the t-statistic is computed as:

\[

t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{28.63 - 29.55}{0.71 / \sqrt{7}} = \frac{-0.92}{0.268} \approx -3.43

\]

Degrees of freedom (\(df\)): \(n-1=6\). Looking up the critical t-value for a one-tailed test at \(\alpha = 0.05\), we find approximately 1.943. Since \(|t|=3.43 > 1.943\), the result is statistically significant.

The p-value associated with t = -3.43 and df=6 is approximately 0.009, which is less than 0.05, indicating that the sample mean significantly differs from the population mean. Specifically, the data suggests that these workers have a significantly smaller reach, supporting the engineer’s hypothesis.

The concept of the p-value in hypothesis testing represents the probability of obtaining a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. In this case, a low p-value (less than 0.05) indicates strong evidence against the null hypothesis, leading to its rejection and supporting the alternative hypothesis that the worker reach is less than the population mean.

This finding has important practical implications. It suggests that ergonomic designs based solely on general population data may not be suitable for specific worker groups. Customization or adjustments to the parts bin reach should be considered to accommodate smaller-stature workers, enhancing safety and productivity.

In conclusion, the statistical analysis confirms that the workers’ reach differs significantly from the average population reach, highlighting the importance of considering demographic variations in ergonomic design. The engineer's suspicion that the workers’ reach was smaller than the general population is statistically supported, advocating for tailored solutions in work environment modifications.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • McClave, J. T., & Sincich, T. (2018). Statistics (13th ed.). Pearson.
  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
  • Laerd Statistics. (2020). One sample t-test. https://statistics.laerd.com/
  • Urdan, T. C. (2017). Statistics in Plain English. Routledge.
  • Howell, D. C. (2012). Statistical Methods for Psychology. Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Keppel, G., & Wickens, T. D. (2004). Design and Analysis: A Researcher's Handbook. Pearson.
  • Robson, C. (2011). Real World Research. Wiley.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.

Related Assignments