91Ó°ÊÓ

An experiment was carried out to compare various properties of cotton/polyester spun yarn finished with softener only and yarn finished with softener plus 5\% DP-resin ("Properties of a Fabric Made with Tandem Spun Yarns," Textile Res. \(J ., 1996: 607-611)\). One particularly important characteristic of fabric is its durability, that is, its ability to resist wear. For a sample of 40 softener-only specimens, the sample mean stoll-flex abrasion resistance (cycles) in the filling direction of the yarn was \(3975.0\), with a sample standard deviation of \(245.1\). Another sample of 40 softener-plus specimens gave a sample mean and sample standard deviation of \(2795.0\) and \(293.7\), respectively. Calculate a confidence interval with confidence level \(99 \%\) for the difference between true average abrasion resistances for the two types of fabrics. Does your interval provide convincing evidence that true average resistances differ for the two types of fabrics? Why or why not?

Step by step solution

01

Identify the Data

First, we need to identify the relevant data for each fabric type:- For the softener-only specimens: - Sample size, \( n_1 = 40 \) - Sample mean, \( \bar{x}_1 = 3975.0 \) - Sample standard deviation, \( s_1 = 245.1 \)- For the softener-plus specimens: - Sample size, \( n_2 = 40 \) - Sample mean, \( \bar{x}_2 = 2795.0 \) - Sample standard deviation, \( s_2 = 293.7 \).

02

Calculate the Difference in Sample Means

The difference in sample means is calculated as follows:\[ \bar{x}_1 - \bar{x}_2 = 3975.0 - 2795.0 = 1180.0 \]

03

Determine the Standard Error of the Difference

The standard error of the difference in sample means for independent samples is given by:\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]Plugging in the values:\[ SE = \sqrt{\frac{245.1^2}{40} + \frac{293.7^2}{40}} \approx \sqrt{1504.41 + 2154.49} \approx \sqrt{3658.9} \approx 60.5 \]

04

Determine the Critical Value

For a 99% confidence interval, we need the critical value from the t-distribution. Since our sample sizes are equal and reasonably large, we can approximate this using the standard normal (z) distribution. For 99% confidence, the z-value is approximately 2.576.

05

Calculate the Confidence Interval

Now, we can calculate the confidence interval for the difference in means:\[ (\bar{x}_1 - \bar{x}_2) \pm z \times SE = 1180.0 \pm 2.576 \times 60.5 \]Calculating further:\[ 1180.0 \pm 155.888 \]Thus, the confidence interval is:\[ (1024.112, 1335.888) \]

06

Interpret the Confidence Interval

Since the entire 99% confidence interval (1024.112, 1335.888) does not contain zero, it provides convincing evidence that the true average abrasion resistances for the two fabric types are different.

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Key Concepts

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

Sample Mean

The sample mean is a fundamental concept in statistics. It represents the average value of a set of observations or data points. In the context of the given exercise, the sample mean for each group (softener-only and softener-plus specimens) is calculated by summing all the values in the data set and then dividing by the number of observations. This provides an average measure of the abrasion resistance of the yarn samples. For instance, the sample mean for softener-only specimens is 3975.0 cycles, indicating the average number of cycles the yarn could withstand before showing signs of wear.

By using the sample mean, we are able to make predictions about the population mean, which is why it is essential for calculating confidence intervals. The sample mean is denoted by the symbol \( \bar{x} \).

• Softener-only sample mean: 3975.0 cycles
• Softener-plus sample mean: 2795.0 cycles

Sample Standard Deviation

Sample standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the data points differ from the sample mean. In this exercise, the sample standard deviation helps us understand the spread of the abrasion resistance values within each sample of fabric.
The standard deviation is symbolized by \( s \) and is calculated using the difference between each data point and the sample mean, squared, averaged, and then taking the square root.

For example, the standard deviation for the softener-only yarn samples is 245.1 cycles. This value provides insight into the consistency of fabric durability across the sample.

• Softener-only standard deviation: 245.1 cycles
• Softener-plus standard deviation: 293.7 cycles

T-Distribution

The t-distribution is crucial for understanding smaller sample sizes or when the population standard deviation is unknown. In this exercise, it is used to determine the critical value needed to calculate the confidence interval for the difference between two sample means.
The t-distribution is centered around zero, similar to the normal distribution but with heavier tails, which means it takes into account variability and uncertainty in smaller samples.

For larger sample sizes or known population standard deviations, the t-distribution approximates the normal (z) distribution. In this exercise, with a sample size of 40 for each group, resulting in a large enough sample size, we approximate using the z-distribution to calculate the critical value needed for a 99% confidence interval, which is about 2.576. This critical value helps us determine how far apart the sample means have to be for us to claim a significant difference between the two population means.

Standard Error

The standard error of the difference in sample means quantifies the precision with which we can estimate the difference between two population means. It is calculated from the standard deviations and sample sizes of the two groups.
The formula for the standard error of the difference between means takes into account both the spread of each sample and the sample size, giving us a measure of how much dispersion exists in comparing these means:\[SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]
In the context of this exercise, the standard error calculated is approximately 60.5 cycles. This value is crucial because it is used to construct the confidence interval around the difference in sample means, which ultimately helps us assess whether the difference observed is statistically significant, or could have occurred by random chance.