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Chapter 11: Problem 60

Numerous factors contribute to the smooth running of an electric motor ("Increasing Market Share Through Improved Product and Process Design: An Experimental Approach," Qual. Engrg., 1991: 361-369). In particular, it is desirable to keep motor noise and vibration to a minimum. To study the effect that the brand of bearing has on motor vibration, five different motor bearing brands were examined by installing each type of bearing on different random samples of six motors. The amount of motor vibration (measured in microns) was recorded when each of the 30 motors was running. The data for this study follows. State and test the relevant hypotheses at significance level .05, and then carry out a multiple comparisons analysis if appropriate. $$ \begin{array}{llllllll} \text { Brand 1: } & 13.1 & 15.0 & 14.0 & 14.4 & 14.0 & 11.6 & 13.68 \\ \text { Brand 2: } & 16.3 & 15.7 & 17.2 & 14.9 & 14.4 & 17.2 & 15.95 \\ \text { Brand 3: } & 13.7 & 13.9 & 12.4 & 13.8 & 14.9 & 13.3 & 13.67 \\ \text { Brand 4: } & 15.7 & 13.7 & 14.4 & 16.0 & 13.9 & 14.7 & 14.73 \\ \text { Brand 5: } & 13.5 & 13.4 & 13.2 & 12.7 & 13.4 & 12.3 & 13.08 \end{array} $$

Short Answer

Expert verified
Perform an ANOVA test: reject \(H_0\) if \(F\)-statistic > critical value. Use post hoc tests if \(H_0\) is rejected.

Step by step solution

01

State the Hypotheses

For this analysis, we will perform an ANOVA test to see if there is a significant difference in motor vibrations across different bearing brands. The null hypothesis (\(H_0\)) states that all mean vibrations are equal: \(\mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5\). The alternative hypothesis (\(H_a\)) states that at least one mean vibration is different: \(\mu_i eq \mu_j\) for some \(i eq j\).
02

Calculate the Group Means

Calculate the mean vibration for each brand: - Brand 1: Mean = 13.68 - Brand 2: Mean = 15.95 - Brand 3: Mean = 13.67 - Brand 4: Mean = 14.73 - Brand 5: Mean = 13.08
03

Calculate the Overall Mean

To find the overall mean vibration across all brands, sum all observations and divide by the total number of observations (30 motors):\[\bar{x}_\text{total} = \frac{13.1 + 15.0 + 14.0 + 14.4 + 14.0 + 11.6 + 16.3 + 15.7 + 17.2 + 14.9 + 14.4 + 17.2 + 13.7 + 13.9 + 12.4 + 13.8 + 14.9 + 13.3 + 15.7 + 13.7 + 14.4 + 16.0 + 13.9 + 14.7 + 13.5 + 13.4 + 13.2 + 12.7 + 13.4 + 12.3}{30} = 14.22\]
04

Calculate Between-Group and Within-Group Variability

The between-group variability measures how much the group means differ from the overall mean, while the within-group variability measures how much individual observations differ within the same group. These are used to calculate the F-statistic in an ANOVA test.
05

Perform the ANOVA Test

Using statistical software or an ANOVA table, calculate the F-statistic by comparing the between-group and within-group variability. Then compare this to the critical F-value at \(\alpha = 0.05\). If the F-statistic exceeds the critical value, reject the null hypothesis.
06

Conduct Multiple Comparisons (if needed)

If the null hypothesis is rejected, perform post hoc tests (e.g., Tukey's HSD) to determine which specific groups' means are significantly different from each other. This step will elaborate on the differences found in the ANOVA test.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a critical aspect of statistical analysis in research. Its primary goal is to determine whether there is enough evidence to reject or not reject a proposed statement, known as a hypothesis. In the context of motor vibration analysis, hypothesis testing starts by setting up two hypotheses. The **null hypothesis** (\(H_0\)) claims that there is no difference in the means of vibration levels across different brands of bearings. In mathematical terms, this is expressed as \(\mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5\).
  • The alternative hypothesis (\(H_a\)) suggests that at least one brand has a different mean vibration level, i.e., \(\mu_i eq \mu_j\) for some \(i eq j\).
The purpose is to test these hypotheses at a significance level of 0.05, which indicates a 5% risk of concluding a difference when there is none. A clear understanding of hypothesis testing helps in making informed decisions based on data analysis.
Experimental Design
Experimental design is the blueprint of a research study, describing the steps required to conduct experiments efficiently. In the exercise regarding motor vibration analysis, we're interested in understanding the impact of different bearing brands on the vibration level of motors.
  • Random samples of six motors were used for each of the five different brands of bearings.
  • The variable measured is vibration, quantified in microns, indicating the extent of vibration produced.
  • The experimental design helps in controlling variables that might influence the vibration, ensuring that the outcome observed is due to the bearing brand.
Proper experimental design enhances the reliability of the conclusions drawn by minimizing bias and variability.
Motor Vibration Analysis
Motor vibration analysis deals with assessing the amount of vibration a motor experiences, which is crucial for its smooth operation and longevity. In this study, vibration is measured in microns, a unit representing one-millionth of a meter.
  • Analyzing motor vibration involves comparing means obtained from different motor bearings.
  • Understanding vibration behavior helps identify which bearing brand contributes less to overall vibration.
  • Minimizing vibration can lead to increased motor efficiency and reduced noise.
As noise and vibrations are undesirable in electric motors, implementing vibration analysis allows optimization of motor components resulting in improved performance.
Post Hoc Tests
Post hoc tests are follow-up analyses conducted after an ANOVA test has shown significant results. They help determine exactly where the differences lie between group means.
  • In the context of the exercise, if the ANOVA test indicates significant differences among the bearing brands, post hoc tests, like Tukey's HSD, are used.
  • The goal is to pinpoint which specific groups differ.
  • Post hoc tests control the family-wise error rate, ensuring that the risk of Type I error (false positive) across multiple comparisons is maintained at the desired level.
These tests are essential in providing deeper insights beyond the initial ANOVA results, enhancing decision-making in experimental studies.

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Most popular questions from this chapter

The article "Computer-Assisted Instruction Augmented with Planned Teacher/Student Contacts" (J. Exper. Ed., Winter 1980-1981: 120-126) compared five different methods for teaching descriptive statistics. The five methods were traditional lecture and discussion (L/D), programmed textbook instruction (R), programmed text with lectures (R/L), computer instruction (C), and computer instruction with lectures (C/L). Forty-five students were randomly assigned, 9 to each method. After completing the course, the students took a 1-h exam. In addition, a 10-minute retention test was administered 6 weeks later. Summary quantities are given. $$ \begin{array}{lcccc} & {\text { Exam }} & {\text { Retention Test }} \\ \text { Method } & \overline{\boldsymbol{x}}_{\boldsymbol{i}} \cdot & \boldsymbol{s}_{\boldsymbol{i}} & \overline{\boldsymbol{x}}_{\boldsymbol{i}} \cdot & \boldsymbol{s}_{\boldsymbol{i}} \\ \hline \text { L/D } & 29.3 & 4.99 & 30.20 & 3.82 \\ \text { R } & 28.0 & 5.33 & 28.80 & 5.26 \\ \text { R/L } & 30.2 & 3.33 & 26.20 & 4.66 \\ \text { C } & 32.4 & 2.94 & 31.10 & 4.91 \\ \text { C/L } & 34.2 & 2.74 & 30.20 & 3.53 \end{array} $$ The grand mean for the exam was \(30.82\), and the grand mean for the retention test was \(29.30\). a. Does the data suggest that there is a difference among the five teaching methods with respect to true mean exam score? Use \(\alpha=.05\). b. Using a \(.05\) significance level, test the null hypothesis of no difference among the true mean retention test scores for the five different teaching methods.

In a study to assess the effects of malaria infection on mosquito hosts ("Plasmodium cynomolgi: Effects of Malaria Infection on Laboratory Flight Performance of Anopheles stephensi Mosquitos," Exp. Parasitol., 1977: 397-404), mosquitoes were fed on either infective or noninfective rhesus monkeys. Subsequently the distance they flew during a 24 -h period was measured using a flight mill. The mosquitoes were divided into four groups of eight mosquitoes each: infective rhesus and sporozites present (IRS), infective rhesus and oocysts present (IRD), infective rhesus and no infection developed (IRN), and noninfective (C). The summary data values are \(\bar{x}_{1}=4.39(\mathrm{IRS}), \quad \bar{x}_{2}=4.52(\mathrm{IRD}), \quad \bar{x}_{3}=\) \(5.49(\mathrm{IRN}), \quad \bar{x}_{4}=6.36(\mathrm{C}), \quad \bar{x}_{. .}=5.19, \quad\) and \(\sum \sum x_{i j}^{2}=911.91\). Use the ANOVA \(F\) test at level \(.05\) to decide whether there are any differences between true average flight times for the four treatments.

In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125 -cc two-stroke motorcycle, five plugs of each brand were tested for the number of miles (at a constant speed) until failure. The partial ANOVA table for the data is given here. Fill in the missing entries, state the relevant hypotheses, and carry out a test by obtaining as much information as you can about the \(P\)-value.

Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated with one of the four coatings and buried in one of the three types of soil for a fixed time, after which the amount of corrosion (depth of maximum pits, in \(.0001\) in.) is determined. The depths are shown in this table: $$ \begin{aligned} &\text { Soil Type (B) }\\\ &\begin{array}{l|lll} & \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline \mathbf{1} & 64 & 49 & 50 \\ \mathbf{2} & 53 & 51 & 48 \\ \mathbf{3} & 47 & 45 & 50 \\ \mathbf{4} & 51 & 43 & 52 \\ \hline \end{array} \end{aligned} $$ a. Assuming the validity of the additive model, carry out the ANOVA analysis using an ANOVA table to see whether the amount of corrosion depends on either the type of coating used or the type of soil. Use \(\alpha=.05\). b. Compute \(\hat{\mu}, \hat{\alpha}_{1}, \hat{\alpha}_{2}, \hat{\alpha}_{3}, \hat{\alpha}_{4}, \hat{\beta}_{1}, \hat{\beta}_{2}\), and \(\hat{\beta}_{3}\)

The accompanying data resulted from an experiment to investigate whether yield from a chemical process depended either on the formulation of a particular input or on mixer speed. $$ \begin{array}{cc|ccc} & & {\text { Speed }} \\ { 3 - 5 } & & \mathbf{6 0} & \mathbf{7 0} & \mathbf{8 0} \\ \hline \text { Formulation } & & 189.7 & 185.1 & 189.0 \\ & & 188.6 & 179.4 & 193.0 \\ & & 190.1 & 177.3 & 191.1 \\ & & 165.1 & 161.7 & 163.3 \\ & & 165.9 & 159.8 & 166.6 \\ & & 167.6 & 161.6 & 170.3 \\ \hline \end{array} $$ A statistical computer package gave \(\mathrm{SS}(\mathrm{Form})=\) \(2253.44, \mathrm{SS}(\) Speed \()=230.81, \mathrm{SS}(\) Form* Speed \()\) \(=18.58\), and SSE \(=71.87\). a. Does there appear to be interaction between the factors? b. Does yield appear to depend on either formulation or speed? c. Calculate estimates of the main effects. d. Verify that the residuals are \(0.23,-0.87,0.63\), \(4.50,-1.20,-3.30,-2.03,1.97,0.07,-1.10\), \(-0.30,1.40,0.67,-1.23,0.57,-3.43,-0.13\), \(3.57\). e. Construct a normal plot from the residuals given in part (d). Do the \(\varepsilon_{i j k}\) 's appear to be normally distributed? f. Plot the residuals against the predicted values (cell means) to see if the population variance appears reasonably constant.
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