91Ó°ÊÓ

The derailment of a freight train due to the catastrophic failure of a traction motor armature bearing provided the impetus for a study reported in the article "Locomotive Traction Motor Armature Bearing Life Study" (Lubrication Engr., Aug. 1997: 12-19). A sample of 17 high-mileage traction motors was selected, and the amount of cone penetration \((\mathrm{mm} / 10)\) was determined both for the pinion bearing and for the commutator armature bearing, resulting in the following data: \begin{tabular}{lccccccc} & \multicolumn{6}{c}{ Motor } \\ \cline { 2 - 7 } & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) & \(\mathbf{5}\) & \(\mathbf{6}\) \\ Commutator & 211 & 273 & 305 & 258 & 270 & 209 \\ Pinion & 226 & 278 & 259 & 244 & 273 & 236 \\ & & & \multicolumn{5}{c}{ Motor } \\ & \multicolumn{7}{c}{\(\mathbf{9}\)} \\ \cline { 2 - 8 } & \(\mathbf{7}\) & \(\mathbf{8}\) & \(\mathbf{9}\) & \(\mathbf{1 1}\) & \(\mathbf{1 2}\) \\ Commutator & 223 & 288 & 296 & 233 & 262 & 291 \\ Pinion & 290 & 287 & 315 & 242 & 288 & 242 \end{tabular} \begin{tabular}{lccccc} & \multicolumn{5}{c}{ Motor } \\ \cline { 2 - 6 } & \(\mathbf{1 3}\) & \(\mathbf{1 4}\) & \(\mathbf{1 5}\) & \(\mathbf{1 6}\) & \(\mathbf{1 7}\) \\ Commutator & 278 & 275 & 210 & 272 & 264 \\ Pinion & 278 & 208 & 281 & 274 & 268 \end{tabular} Calculate an estimate of the population mean difference between penetration for the commutator armature bearing and penetration for the pinion bearing, and do so in a way that conveys information about the reliability and precision of the estimate. [Note: A normal probability plot validates the necessary normality assumption.] Would you say that the population mean difference has been precisely estimated? Does it look as though population mean penetration differs for the two types of bearings? Explain.

Step by step solution

01

Calculate Differences for Each Motor

Calculate the difference in cone penetration between the commutator armature bearing and the pinion bearing for each of the 17 motors. Subtract the pinion penetration from the commutator penetration to find each difference. \( d_i = ext{Commutator Penetration}_i - ext{Pinion Penetration}_i \) for each motor \( i \).

02

Compute Mean Difference

Once you have all 17 differences \( d_1, d_2, \dots, d_{17} \), compute the mean difference \( \bar{d} \) by taking the average of these differences: \( \bar{d} = \frac{1}{17} \sum_{i=1}^{17} d_i \).

03

Compute the Standard Deviation of Differences

Calculate the standard deviation of the differences \( s_d \) using the formula: \[ s_d = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(d_i - \bar{d})^2} \] where \( n \) is the number of data points, which is 17 here.

04

Calculate Standard Error of the Mean Difference

Calculate the standard error of the mean difference \( SE_{\bar{d}} \) using the standard deviation found above: \[ SE_{\bar{d}} = \frac{s_d}{\sqrt{n}} \] where \( n = 17 \). This helps us understand the precision of the mean difference estimate.

05

Construct the Confidence Interval

Construct the confidence interval for the mean difference at a given confidence level (typically 95%) using the formula: \( \text{CI} = \bar{d} \pm t^* \cdot SE_{\bar{d}} \). Here, \( t^* \) is the critical value from the t-distribution for \( n-1 \) degrees of freedom.

06

Analyze Confidence Interval and Conclusion

Analyze the confidence interval. If the confidence interval does not contain zero, it suggests a significant mean difference between the two types of bearings. The width of the confidence interval also indicates the precision of the estimate; a narrower interval is more precise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Difference

The mean difference is a key concept in statistical analysis, used to identify the average amount by which one group differs from another in a dataset. In this exercise, we calculated the mean difference in cone penetration between commutator armature bearings and pinion bearings. This involved taking the penetration values from each pair and finding the difference. Then, we calculated the average of these differences.

The calculation process starts with determining each individual difference, denoted as \( d_i = \text{Commutator Penetration}_i - \text{Pinion Penetration}_i \). Once you have all the differences, the mean difference \( \bar{d} \) is computed by averaging these values: \( \bar{d} = \frac{1}{17} \sum_{i=1}^{17} d_i \). This mean difference gives us an idea of the overall tendency for commutator bearings to differ in their penetration from pinion bearings. A positive mean difference would suggest that commutator penetres more than pinion on average, and vice versa.

Calculating the mean difference helps us understand whether there is a general pattern or trend in the data, but it is also essential to analyze the reliability and precision of these results.

Standard Deviation

Standard deviation quantifies the amount of variation or spread in a set of data. It's a crucial component in understanding the dispersion of individual differences we calculated between commutator and pinion penetrations.

To find the standard deviation of these differences, we use the formula: \[s_d = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(d_i - \bar{d})^2}\] Here, \( n \) is the number of motors, 17, and \( d_i \) are each of the differences. This formula helps capture how spread out the differences are around the mean difference.

If the standard deviation is high, it indicates a wide range of penetration differences among the motors, leading to less precision in our estimates. A low standard deviation, conversely, suggests the differences are tightly clustered around the mean difference, adding confidence to our analysis. In the context of bearings, understanding this spread helps us assess whether variations are significant or merely noise in the measurements.

Confidence Interval

A confidence interval gives us a range of values, based on our sample data, that is likely to contain the population mean difference. It's a way of quantifying the precision of our mean difference estimate.

In this case, constructing a confidence interval involves the formula: \[\text{CI} = \bar{d} \pm t^* \cdot SE_{\bar{d}}\]where \( \bar{d} \) is the mean difference, \( SE_{\bar{d}} \) is the standard error of the mean difference, and \( t^* \) is the critical value from the t-distribution for 16 degrees of freedom at your chosen confidence level (typically 95%).

The confidence interval's lower and upper bounds help us infer if the true mean difference is likely other than zero, indicating potential significant differences between the bearings. If zero is not within the interval, it suggests a statistically meaningful difference exists. A narrow confidence interval reflects higher precision in estimating this difference, giving us stronger confidence that the estimated mean difference mirrors the true population mean.

Standard Error

The standard error is a measure that indicates the precision of the sample mean difference as an estimate of the population mean difference. It is connected closely to both the standard deviation and the sample size.

For our dataset, it is calculated using the formula: \[SE_{\bar{d}} = \frac{s_d}{\sqrt{n}}\]where \( s_d \) is the standard deviation of the differences, and \( n \), the number of observations, is 17. The standard error decreases with larger sample sizes or lower standard deviations, which means our mean difference estimate becomes more precise.

In practical terms, a smaller standard error indicates that our sample mean difference is likely to be a more accurate reflection of the true mean difference. This figure plays a pivotal role when constructing confidence intervals, as it essentially sets the distance from the mean difference to its confidence bounds. Understanding standard error empowers us to assess how much our mean difference might fluctuate by chance alone, thus highlighting the estimated reliability of our findings.