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Chapter 12: Problem 63

Physical properties of six flame-retardant fabric samples were investigated in the article "Sensory and Physical Properties of Inherently Flame-Retardant Fabrics" (Textile Research, 1984: 61-68). Use the accompanying data and a .05 significance level to determine whether a linear relationship exists between stiffness \(x(\mathrm{mg}-\mathrm{cm})\) and thickness \(y(\mathrm{~mm})\). Is the result of the test surprising in light of the value of \(r\) ? $$ \begin{array}{r|rrrrrr} x & 7.98 & 24.52 & 12.47 & 6.92 & 24.11 & 35.71 \\ \hline y & .28 & .65 & .32 & .27 & .81 & .57 \end{array} $$

Short Answer

Expert verified
There is a significant linear relationship between stiffness and thickness.

Step by step solution

01

State the Hypotheses

To test whether a linear relationship exists between stiffness and thickness, we set up the null hypothesis \( H_0: \beta_1 = 0 \), indicating no linear relationship, and the alternative hypothesis \( H_a: \beta_1 eq 0 \), indicating a linear relationship exists.
02

Calculate the Correlation Coefficient

Use the dataset to compute the correlation coefficient \( r \). The formula to calculate \( r \) is:\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \]Substituting the given values, compute \( r \approx 0.905 \).
03

Compute the Test Statistic

The test statistic for testing the significance of the correlation coefficient is calculated as:\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]Where \( n = 6 \) is the number of data points. Substituting values, we find \( t \approx 3.47 \).
04

Determine the Critical Value

For a two-tailed test at \( \alpha = 0.05 \) with \( n-2 = 4 \) degrees of freedom, refer to the t-distribution table to find the critical value: \( t_{critical} \approx \pm 2.776 \).
05

Make the Decision

Compare the computed test statistic \( t = 3.47 \) with the critical value \( t_{critical} = 2.776 \). Since \( |3.47| > 2.776 \), we reject the null hypothesis \( H_0 \).
06

Interpret the Result

Since the test statistic exceeds the critical value, this indicates a significant linear relationship exists between stiffness and thickness at the \(0.05\) level. This result is consistent with the high value of \( r \approx 0.905 \), which suggests a strong positive correlation.

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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 statistical method used to make a decision about the existence of an effect or a relationship in a dataset. In this context, we explore whether there is a linear relationship between two variables: stiffness and thickness of the flame-retardant fabrics.
To begin with, we set up two hypotheses: the null hypothesis (**H**_0_) and the alternative hypothesis (**H**_a_).
  • The null hypothesis (**H**_0_) posits that there is no linear relationship between the variables, which is mathematically represented as \( \beta_1 = 0 \).
  • Conversely, the alternative hypothesis (**H**_a_) suggests that there is some linear relationship, indicated by \( \beta_1 eq 0 \).

By testing these hypotheses, we aim to either accept or reject the null hypothesis using statistical calculations.
Significance Level
The significance level, denoted as \( \alpha \), is a threshold used to decide whether to reject the null hypothesis. In this exercise, we set the significance level at 0.05.
This means there is a 5% risk of concluding that a linear relationship exists when there is none.
  • If our computed test result has a probability less than 0.05 of occurring under the null hypothesis, we reject **H**_0_.
  • Otherwise, we might not have enough evidence to reject **H**_0_, suggesting that any observed effect might be due to random chance.

A 0.05 significance level is a common choice in many scientific fields and indicates a moderate balance between the risks of Type I and Type II errors.
Correlation Coefficient
The correlation coefficient, often represented by \( \rho \) or **r**, measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.
Here's what these values imply:
  • +1 indicates a perfect positive linear relationship.
  • -1 means a perfect negative linear relationship.
  • 0 suggests no linear correlation.
In our study, a calculated correlation coefficient of \( r \approx 0.905 \) signifies a strong positive correlation. This implies that as the stiffness increases, the thickness also tends to increase. It's a critical indicator before running any tests, as it visually cues us about the relationship.
However, to confirm the significance of this correlation, further hypothesis testing is necessary.
Linear Relationship Hypothesis
A linear relationship hypothesis tests whether two variables are linearly related across a set of data points. This forms the core of our investigation here.
  • The presence of a linear relationship means changes in one variable consistently result in changes in the other, plotted as a "line" in a graph.
Our hypothesis test concludes whether to accept or reject this presumed relationship. If the null hypothesis is rejected, it signifies a statistically significant linear connection between stiffness and thickness.
In this case, with the observed data leading to a test statistic that surpasses the critical value, we reject the null hypothesis. Hence, we affirm the existence of a significant linear relationship. This conclusion also aligns with the high correlation coefficient, rendering the result unsurprising.

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