91Ó°ÊÓ

Fusible interlinings are being used with increasing frequency to support outer fabrics and improve the shape and drape of various pieces of clothing. The article "Compatibility of Outer and Fusible Interlining Fabrics in Tailored Garments" (Textile Res. \(J_{.}, 1997: 137-142\) ) gave the accompanying data on extensibility (\%) at \(100 \mathrm{gm} / \mathrm{cm}\) for both high-quality (H) fabric and poor-quality (P) fabric specimens. \(\begin{array}{rrrrrrrrrr}\mathrm{H} & 1.2 & .9 & .7 & 1.0 & 1.7 & 1.7 & 1.1 & .9 & 1.7 \\ & 1.9 & 1.3 & 2.1 & 1.6 & 1.8 & 1.4 & 1.3 & 1.9 & 1.6 \\ & 8 & 2.0 & 1.7 & 1.6 & 2.3 & 2.0 & & & \\ \mathrm{P} & 1.6 & 1.5 & 1.1 & 2.1 & 1.5 & 1.3 & 1.0 & 2.6 & \end{array}\) a. Construct normal probability plots to verify the plausibility of both samples having been selected from normal population distributions. b. Construct a comparative boxplot. Does it suggest that there is a difference between true average extensibility for high-quality fabric specimens and that for poorquality specimens? c. The sample mean and standard deviation for the highquality sample are \(1.508\) and \(.444\), respectively, and those for the poor-quality sample are \(1.588\) and 530 . Use the two-sample \(t\) test to decide whether true average extensibility differs for the two types of fabric.

Step by step solution

01

Check Normality Using Probability Plot for High-Quality Fabric

To verify if the high-quality fabric dataset can be considered to come from a normal distribution, construct a normal probability plot. Plot the extensibility data on the vertical axis against the normal score on the horizontal axis. If the points follow a straight line approximately, normality is a reasonable assumption.

02

Check Normality Using Probability Plot for Poor-Quality Fabric

Similarly, construct a normal probability plot for the poor-quality fabric dataset. As in Step 1, plot the extensibility data against normal scores, and check for linearity. A roughly linear pattern indicates the data may come from a normal distribution.

03

Construct Comparative Boxplot

Create a boxplot for each dataset (high-quality and poor-quality fabrics) side by side. These boxplots will provide visual comparisons of the medians, interquartile ranges, and potential outliers. Check for overlapping boxes and note any significant differences in the medians.

04

Conduct Two-Sample t-Test

To determine if there is a significant difference in the mean extensibility between high-quality and poor-quality fabrics, perform a two-sample t-test. Use the formula for the test statistic: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] where \(\bar{x}_1 = 1.508\), \(s_1 = 0.444\), \(n_1 = 20\) (assumed sample size inferred from context), \(\bar{x}_2 = 1.588\), \(s_2 = 0.530\), and \(n_2 = 9\). Calculate the degrees of freedom using the formula:\[ u = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}} \] Compare the calculated \(t\)-value with the critical value from the t-distribution table at a chosen significance level (e.g., 0.05). Analyze results to decide if there is enough evidence to conclude a difference.

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Key Concepts

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

Understanding the Normal Probability Plot

The normal probability plot is a diagnostic tool used in statistics to assess whether a dataset is approximately normally distributed. To construct this plot, data values are plotted against the expected values that would be observed if the data follows a normal distribution. A key feature to look for in a normal probability plot is the linearity of the points.

If the points generally follow a straight line, this suggests that the data is normally distributed, making it a simple yet powerful visual check for the normality assumption.

• For the high-quality fabric, you plot the extensibility percentage on the vertical axis and corresponding normal scores on the horizontal axis. A linear pattern would suggest that normality is plausible.
• Similarly, this process is repeated for the poor-quality fabric.
  A clear alignment along a straight line for both datasets would indicate a normal distribution is a reasonable assumption.

Understanding whether the data comes from a normal distribution is vital before applying many statistical tests, as many tests require this assumption to be valid.

Insights from Comparative Boxplots

Comparative boxplots are graphical representations that showcase the distribution of a dataset. By comparing two boxplots side by side, such as those for high-quality and poor-quality fabrics, you can visually assess differences between the two datasets.

A boxplot includes several key components:

• The median, or middle value, which is displayed as a line inside the box.
• The interquartile range (IQR), which shows the range within which the central 50% of data lies.
• Potential outliers, indicated by points outside the range of the whiskers of the box.

When comparing boxplots for the two fabric types, pay particular attention to the extent of overlap between the boxes. If the medians are significantly different and there is minimal overlap, this suggests a difference in central tendency between the groups. This visual insight provides an initial indication of whether the true average extensibilities for the two fabric types might differ significantly.

The Two-Sample T-Test Explained

The two-sample t-test is a statistical procedure used to determine if there is a significant difference between the means of two independent samples. In this scenario, it is used to compare the extensibility between high-quality and poor-quality fabrics.

The test involves the following steps:

• Calculate the mean (\(ar{x}\)), standard deviation (\(s\)), and sample size (\(n\)) for each group.
• Compute the test statistic using:\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.
• The degrees of freedom (\(u\)) must be calculated using a formula that estimates the appropriate value based on the variability and size of the sample groups:

\[ u = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2}{(\frac{s_1^2}{n_1})^2/(n_1-1) + (\frac{s_2^2}{n_2})^2/(n_2-1)} \]

After computing the t-statistic and degrees of freedom, compare the calculated t-value to a critical value from the t-distribution table at your chosen significance level (e.g., 0.05). If the t-value is greater than the critical value, there is enough evidence to conclude a significant difference between the two means. This test is crucial for confirming whether any observed differences between the two fabric types' extensibility are statistically significant and not due to random chance.