91Ó°ÊÓ

The article "Flexure of Concrete Beams Reinforced with Advanced Composite Orthogrids" (J. of Aerospace Engr. 1997: 7-15) gave the accompanying data on ultimate load (kN) for two different types of beams. \begin{tabular}{lccc} Type & Sample Size & Sample Mean & Sample SD \\ \hline Fiberglass grid & 26 & \(33.4\) & \(2.2\) \\ Commercial carbon grid & 26 & \(42.8\) & \(4.3\) \\ \hline \end{tabular} a. Assuming that the underlying distributions are normal, calculate and interpret a \(99 \%\) CI for the difference between true average load for the fiberglass beams and that for the carbon beams. b. Does the upper limit of the interval you calculated in part (a) give a \(99 \%\) upper confidence bound for the difference between the two \(\mu\) 's? If not, calculate such a bound. Does it strongly suggest that true average load for the carbon beams is more than that for the fiberglass beams? Explain.

Step by step solution

01

Define the Objective

We need to calculate a 99% confidence interval (CI) for the difference between the true average ultimate loads of fiberglass and carbon grid beams. Denote the true means by \( \mu_1 \) for fiberglass and \( \mu_2 \) for carbon beams.

02

Set Up the Formula for Confidence Interval

The formula for a confidence interval for the difference between two means \( \mu_1 - \mu_2 \) is given by:\[(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]where \( \bar{x}_1 = 33.4\), \( \bar{x}_2 = 42.8\), \( s_1 = 2.2\), \( s_2 = 4.3\), and \( n_1 = n_2 = 26 \). The \( t \)-value is based on \( 25 \) degrees of freedom (since \( n - 1 \) for each sample).

03

Find the t-value for 99% Confidence Level

For 99% confidence level and \( 25 \) degrees of freedom, the critical \( t \)-value can be found using a t-table or calculator: \( t_{\alpha/2} \approx 2.787.\)

04

Calculate the Standard Error

Calculate the standard error (SE) using:\[SE = \sqrt{\frac{2.2^2}{26} + \frac{4.3^2}{26}} = \sqrt{\frac{4.84}{26} + \frac{18.49}{26}} = \sqrt{0.1862 + 0.7112} = \sqrt{0.8974} \approx 0.9472.\]

05

Compute the Confidence Interval

Now compute the confidence interval:\[CI = (33.4 - 42.8) \pm 2.787 \times 0.9472 = -9.4 \pm 2.640\]This results in the interval:\[-12.040, -6.760\].

06

Interpret the Confidence Interval

The 99% confidence interval for the difference \( \mu_1 - \mu_2 \) is \([-12.040, -6.760]\). This interval suggests that the true average load for the fiberglass beams is less than that for the carbon beams, as the entire interval is negative.

07

Comment on Upper Confidence Bound

The upper limit of the interval \(-6.760\) is not an upper confidence bound. A 99% upper confidence bound would be \(( \bar{x}_1 - \bar{x}_2) + 2.787 \times SE = -9.4 + 2.640 = -6.760\). This bound supports the claim that the true average load for carbon beams exceeds that for fiberglass beams, as it remains negative.

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Key Concepts

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

Normal Distribution

A normal distribution, often referred to as the bell curve, is a fundamental concept in statistics. This distribution describes how data points cluster around a mean (average) value. It is symmetrical, meaning most values are close to the mean, and fewer are found as they move away from it. The mean, median, and mode in a normal distribution all coincide.

In the context of confidence intervals, assuming a normal distribution allows us to make predictions about the population based on sample data. This is why, in the given exercise, we assume that the data for the ultimate load of beams follows a normal distribution. This assumption lets us compute the confidence intervals for the means of the two beam types, giving insight into their comparative strengths.

Standard Deviation

Standard deviation (SD) is a measure of how spread out the values in a data set are around the mean. A small SD means the values are closer to the mean, whereas a large SD indicates the values are more spread out.

In our exercise, the standard deviations for the fiberglass and carbon beams are 2.2 and 4.3 respectively. These figures tell us that the ultimate loads of fiberglass beams are more consistent (less spread out around the mean) compared to those of carbon beams. Understanding SD helps interpret the reliability of the calculated means and indicates potential variability. It's crucial for calculating the confidence interval since it appears in the formula for SE (Standard Error).

T-Distribution

The t-distribution is essential in statistics, specifically when dealing with small sample sizes or an unknown population standard deviation. Unlike the normal distribution, it has heavier tails, which means it is more prone to producing values that fall far from the mean. The shape of the t-distribution depends on the degrees of freedom, which generally equals the sample size minus one.

In the case of our exercise, the t-distribution applies since the sample size is moderate (26 beams for each type). For a 99% confidence interval and 25 degrees of freedom, the t-value was found to be approximately 2.787. This t-value is critical in adjusting the width of the confidence interval, accounting for the sample size and ensuring the interval is accurate even if the population SD is unknown.

Sample Size

Sample size, denoted as 'n,' is the number of observations in a sample. It plays a vital role in statistics, impacting the accuracy and reliability of inferential statistics like confidence intervals. Larger samples tend to provide more reliable results since they better represent the population.

In our scenario, both beam types have a sample size of 26. This moderate size affects the precision of our confidence interval - with larger samples, our estimate for the true average load would be even more precise. However, even with this sample size, the analysis using a t-distribution remains robust, offering a reliable difference between the true mean loads for the types of beams.