91Ó°ÊÓ

In the study An X-Ray Fluorescence Method for Analyzing Polybutadiene-Acrylic Acid (PBAA) Propellants, Quarterly Reports, RK-TR-62-1, Army Ordnance Missile Command, an experiment was conducted to determine whether or not there is a significant difference in the amount of aluminum achieved in the analysis between certain levels of certain processing variables. The data given in the table were recorded. $$ \begin{array}{cccccc} &\ {\text { Phys. Mixing Blade Nitrogen }} \\ \text { Obs.} &{ State } & \text { Time } &\text { Speed } & {Condition}& { Aluminum}{ } \\ \hline \mathbf{1} & 1 & 1 & 2 & 2 & 16.3 \\ 2 & 1 & 2 & 2 & 2 & 16.0 \\ 3 & 1 & 1 & 1 & 1 & 16.2 \\ 4 & 1 & 2 & 1 & 2 & 16.1 \\ 5 & 1 & 1 & 1 & 2 & 16.0 \\ 6 & 1 & 2 & 1 & 1 & 16.0 \\ 7 & 1 & 2 & 2 & 1 & 15.5 \\ 8 & 1 & 1 & 2 & 1 & 15.9 \\ 9 & 2 & 1 & 2 & 2 & 16.7 \\ 10 & 2 & 2 & 2 & 2 & 16.1 \\ 11 & 2 & 1 & 1 & 1 & 16.3 \\ 12 & 2 & 2 & 1 & 2 & 15.8 \\ 13 & 2 & 1 & 1 & 2 & 15.9 \\ 14 & 2 & 2 & 1 & 1 & 15.9 \\ 15 & 2 & 2 & 2 & 1 & 15.6 \\ 16 & 2 & 1 & 2 & 1 & 15.8 \end{array} $$ The variables are given below. .4: mixing time level \(1-2\) hours level \(2-4\) hoursi B: \(\quad\) blade speed \(\begin{array}{lll}\text { level } & 1-36 & \text { rpin }\end{array}\) level \(2-78\) rpm \(C: \quad\) condition of nitrogen passed over propellant level \(1-\mathrm{dry}\) level \(2-72 \%\) relative humidity \(D: \quad\) physical state of propellant level 1 -uncured level 2 -cured Assuming all three- and four-factor interactions to be negligible, analyze the data. Use a 0.05 level of significance. Write a brief report summarizing the findings.

Step by step solution

01

Understanding The Data

First, understand what the data is representing. Each row in the table represents a different trial of the experiment. There are four factors: physical state of the propellant (A), mixing time (B), blade speed (C), and the condition of nitrogen passed over propellant (D). Each factor has two levels, so there are a total of 16 combinations or trials.

02

Setting Up ANOVA

Set up the Analysis of Variance (ANOVA) table. The ANOVA table should include the source of variation (the 4 factors and their interactions), the sum of squares (SS), the degrees of freedom (df), the mean squares (MS), the F-value, and the p-value.

03

Calculating SS Among and Within

Calculate the sum of squares among groups (SSA, SSB, SSC, SSD, SSAB, SSBC, SBD, SSCD) and the sum of squares within groups (SSE). This is done by finding the total sum of squares (SST), and subtracting the sum of squares among groups.

04

Calculating df, MS and F-value

Next, calculate the degrees of freedom (df), mean squares (MS), and the F-value. The df is the number of categories of each factor minus 1. The MS is the SS divided by the df. The F-value is the ratio of the mean squares between groups to the mean square within groups.

05

Testing The Hypothesis

Using the F-distribution table and the given significance level (0.05), find the critical F-value and reject the null hypothesis if the calculated F-value is greater than the critical F-value.

06

Summarizing Findings

Interpret the results and summarize the findings. If any of the factors' p-values is less than the significance level, this means that the factor has a statistically significant effect on the aluminum content.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Statistical Significance

Understanding statistical significance is fundamental when interpreting ANOVA results. The term refers to the likelihood that the difference observed in an experiment or a study is due to something other than random chance. To assess this, researchers set a threshold called the significance level, typically denoted as \( \alpha \). The common significance level is \( \alpha = 0.05 \), meaning there's a 5% risk of concluding that a difference exists when there isn't one.

In the context of the propellant study, we use the significance level to determine if processing variables like physical state, mixing time, blade speed, and nitrogen condition, actually impact the aluminum content quantitatively measured. If the calculated p-value for a factor is less than \( \alpha \), we would declare a statistically significant difference attributable to that factor. This would suggest that the level of aluminum detected in the propellant analysis changes with different conditions of that particular factor beyond what we'd expect from random variation.

Sum of Squares

The sum of squares (SS) is a measure used in statistics to quantify variations within data. When performing ANOVA, SS is a critical component that breaks down the total variability observed in the data into components attributable to each source of variation. There are several types of SS:

Between-Group Sum of Squares (SSB)

This reflects the variance due to the differences between the group means. In your ANOVA for the propellant study, each factor like mixing time and blade speed contributes to SSB.

Within-Group Sum of Squares (SSW)

Reflects the variance within each group, which could be due to randomness or experimental error.

Total Sum of Squares (SST)

The overall variability in the data. Calculated as the sum of SSB and SSW.

By partitioning SST into SSB and SSW, we can find out how much of the variation in aluminum content is due to the systematic differences in processing variables and how much is due to other factors, including inherent experimental variability.

F-distribution

The F-distribution is the foundation for conducting ANOVA and is used to calculate the F-value, a ratio that compares the variance between group means to the variance within groups. Here's how it works:

Forming the F-ratio

An F-value is calculated by dividing the mean square value of the variance among the groups (MSB) by the mean square value of the variance within the groups (MSW). In equation form, it looks like \( F = \frac{MSB}{MSW} \).

F-distribution Characteristics

The F-distribution is a right-skewed distribution that is defined by two different degrees of freedom: df1 for the numerator and df2 for the denominator. These degrees of freedom directly influence the shape of the F-distribution curve.

Considering the experiment to analyze PBAA propellants, after calculating the F-value for each factor, it's compared to a critical F-value from the F-distribution table. If the calculated F-value is greater than the critical F-value at a specified significance level, the null hypothesis (that there’s no effect) is rejected, indicating that there is a statistically significant difference caused by that factor.