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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 poor-quality 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

Calculate Probability Plot Data for H

To construct the normal probability plot for the high-quality fabric, arrange the data in increasing order: \(0.7, 0.8, 0.9, 0.9, 1.0, 1.1, 1.3, 1.3, 1.4, 1.6, 1.6, 1.6, 1.7, 1.7, 1.7, 1.9, 1.9, 2.0, 2.1, 2.3\). Determine the percentiles and corresponding z-scores for these ordered data points, which act as the theoretical quantiles. Plot these data points against the z-scores to form the plot.

02

Calculate Probability Plot Data for P

For the poor-quality fabric, order the data: \(1.0, 1.1, 1.3, 1.5, 1.5, 1.6, 2.1, 2.6\). Calculate the percentiles and match them with their z-scores. Plot these against the data to create the normal probability plot for the poor-quality fabric.

03

Interpret Probability Plots

Look for linearity in both probability plots. If the points roughly follow a straight line, it indicates that the sample data plausibly comes from a normally distributed population. Examine the residuals and outliers indicated in the plots to determine normality.

04

Construct Comparative Boxplot

Draw a boxplot for each fabric type using their data. Display the minimum, first quartile, median, third quartile, and maximum for both the high-quality and poor-quality fabrics. Investigate the boxplot for differences in median lines, spread (interquartile range), and outliers.

05

Interpret Boxplot Comparison

Compare the two boxplots to assess differences in median, spread, and overall distribution shape. Determine if there is a significant visual difference in these aspects between the high-quality and poor-quality samples.

06

Formulate Hypotheses for Two-sample t-test

Define the null hypothesis \(H_0: \mu_H = \mu_P\) (no difference in average extensibility) and the alternative hypothesis \(H_a: \mu_H eq \mu_P\) (there is a difference).

07

Calculate t-test Statistics

Use the formula for the t-statistic: \[ t = \frac{\bar{x}_H - \bar{x}_P}{\sqrt{\frac{s_H^2}{n_H} + \frac{s_P^2}{n_P}}} \]where \(\bar{x}_H = 1.508\), \(s_H = 0.444\), \(n_H = 20\), \(\bar{x}_P = 1.588\), \(s_P = 0.530\), and \(n_P = 8\) are means, standard deviations, and sample sizes for the fabric types. Calculate \(t\).

08

Determine Critical Value and Conclusion

With the degrees of freedom found using a formula or calculator, find the critical value for the two-tailed test at a typical significance level (such as 0.05). Compare the calculated t-statistic to this critical value. Reject the null hypothesis if the t-statistic exceeds the critical value, suggesting a significant difference in means.

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Key Concepts

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

Normal Probability Plot

A normal probability plot is a graphical tool used in statistics to assess whether a set of data is approximately normally distributed. To create a normal probability plot, the data must first be organized in ascending order. The data points are then plotted against a corresponding set of theoretical quantiles, usually derived from a standard normal distribution.

The key to interpreting a normal probability plot is checking the linearity. - If the points in the plot closely follow a straight line, this suggests that the data might come from a normal distribution. - Deviations from a straight line can indicate departures from normality, such as skewness, heavy tails, or outliers.

For the fabrics in this example, creating normal probability plots helps verify whether the high-quality and poor-quality fabric data come from normal populations. To achieve this, the z-scores (standard normal deviates) are paired with the ordered data values. Understanding and applying this method properly equips students to establish normalcy in other datasets beyond textile studies.

Two-Sample t Test

The two-sample t test is a statistical method used to compare the means of two independent groups and decide if they statistically differ from each other. This test is particularly useful for determining if differences in sample means reflect real effects in the population.

In this exercise, we aim to assess whether the average extensibility of high-quality fabric is different from that of poor-quality fabric. - **Null Hypothesis (H_0):** No difference in the means (ar{x}_H = ar{x}_P).- **Alternative Hypothesis (H_a):** There is a difference (ar{x}_H neq ar{x}_P).

The formula for the t-statistic involves comparing the difference in sample means to a weighted measure of variability across the samples. The t-statistic is calculated as\[t = \frac{\bar{x}_H - \bar{x}_P}{\sqrt{\frac{s_H^2}{n_H} + \frac{s_P^2}{n_P}}}\]where \(\bar{x}\) is the sample mean, \(s\) is the standard deviation, and \(n\) is the sample size.

After calculating the t-statistic, it is compared to a critical value derived from the t-distribution. If the absolute value of the t-statistic is greater than the critical value for the chosen significance level (like 0.05), the null hypothesis is rejected, indicating a significant difference between group means.

Comparative Boxplot

A comparative boxplot is a convenient graphical method to visualize and compare the distributions of two or more groups side by side. It displays summary statistics such as median, quartiles, and potential outliers, offering a snapshot of data distribution.

In the context of fabric extensibility: - **Box**: Represents the interquartile range (IQR) where the central 50% of the data lies. - **Median Line**: Depicted inside the box, indicating the central tendency. - **Whiskers**: Extend to the minimum and maximum data points within 1.5 times the IQR from the lower and upper quartiles. - **Outliers**: Points outside the whiskers are often marked as potential outliers.

When constructing a comparative boxplot for high-quality and poor-quality fabrics, looking at median alignment and variability is crucial. - Differences in the median position signify potential shifts in central tendency. - Varying box or whisker lengths suggest differences in data spread.

In this statistics exercise, understanding comparative boxplots helps deduce if there is a visible difference in extensibility between fabric types, even before statistical testing.