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Ultra high performance concrete (UHPC) is a relatively new construction material that is characterized by strong adhesive properties with other materials. The article "Adhesive Power of Ultra High Performance Concrete from a Thermodynamic Point of View" \((J\). of Materials in Civil Engr., 2012: 1050-1058) described an investigation of the intermolecular forces for UHPC connected to various substrates. The following work of adhesion measurements (in \(\mathrm{mJ} / \mathrm{m}^{2}\) ) for UHPC specimens adhered to steel appeared in the article: \(\begin{array}{lllll}107.1 & 109.5 & 107.4 & 106.8 & 108.1\end{array}\) a. Is it plausible that the given sample observations were selected from a normal distribution? b. Calculate a two-sided \(95 \%\) confidence interval for the true average work of adhesion for UHPC adhered to steel. Does the interval suggest that 107 is a plausible value for the true average work of adhesion for UHPC adhered to steel? What about 110 ? c. Predict the resulting work of adhesion value resulting from a single future replication of the experiment by calculating a \(95 \%\) prediction interval, and compare the width of this interval to the width of the CI from (b). d. Calculate an interval for which you can have a high degree of confidence that at least \(95 \%\) of all UHPC specimens adhered to steel will have work of adhesion values between the limits of the interval.

Step by step solution

01

Assess Normality

To determine whether the sample data is from a normal distribution, we use a graphical method such as a Q-Q plot. We plot the sample quantiles against the theoretical quantiles of a normal distribution. If the points roughly form a straight line, the data can be assumed to come from a normal distribution.

02

Calculate Sample Statistics

First, calculate the sample mean and standard deviation from the data. The work adhesion values are 107.1, 109.5, 107.4, 106.8, 108.1. The mean \( \bar{x} = \frac{107.1 + 109.5 + 107.4 + 106.8 + 108.1}{5} = 107.78\) and the standard deviation \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = 1.01 \).

03

Two-Sided Confidence Interval

To compute the 95% confidence interval for the mean, use the formula \( \bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}} \), where \( t_{\alpha/2, n-1} \) is the t-critical value for 95% confidence with 4 degrees of freedom. From t-distribution tables, this value is approximately 2.776. Thus, the CI is \( 107.78 \pm 2.776 \times \frac{1.01}{\sqrt{5}} \approx (106.42, 109.14) \). 107 is a plausible value, but 110 is not.

04

Prediction Interval

For a 95% prediction interval use the formula \( \bar{x} \pm t_{\alpha/2, n-1} s \sqrt{1 + \frac{1}{n}} \). Thus, \( 107.78 \pm 2.776 \times 1.01 \times \sqrt{1 + \frac{1}{5}} \approx (105.67, 109.89) \). This interval is wider than the confidence interval, reflecting more uncertainty around individual predictions.

05

Interval for 95% of Data

A statistical method such as creating tolerance intervals would give us the desired interval: use the formula \( \bar{x} \pm k \cdot s \) where \( k \) is a factor depending on the desired coverage percentage, confidence level, and sample size. From tables for a 95%/95% interval with n=5, \( k \approx 4.03 \), so the interval is \( 107.78 \pm 4.03 \times 1.01 \approx (103.70, 111.86) \).

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Key Concepts

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

Confidence Interval

When we create a confidence interval, our goal is to estimate the range in which the true population parameter, such as the average or mean, lies. In the case of the Ultra High Performance Concrete (UHPC) data, we calculated a two-sided 95% confidence interval. This interval helps us understand the range that contains the true average work of adhesion of UHPC adhered to steel with a 95% chance.
The formula used is \( \bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, and \( t_{\alpha/2, n-1} \) is the t-critical value from the t-distribution table.

• The calculated mean of the observations is 107.78.
• The standard deviation is 1.01.
• For 4 degrees of freedom, the t-value is roughly 2.776.

This gives us an interval: \((106.42, 109.14)\). Within this interval, 107 is plausible, but 110 stretches beyond our calculated range, suggesting it might not be the true mean.

Prediction Interval

When we want to predict the value of a single, future observation from the same population, we use a prediction interval. This differs from a confidence interval since it includes another layer of uncertainty, about individual future observations.
In our UHPC case, the prediction interval formula is \( \bar{x} \pm t_{\alpha/2, n-1} s \sqrt{1 + \frac{1}{n}} \). This perfect blend of the sample mean and standard deviation with an additional adjustment for variability considers that individual observations can have more variability than the mean.

• This prediction interval is formed as \((105.67, 109.89)\).
• Notice, it is wider than the confidence interval because it covers a future single observation.

The prediction interval recognizes the natural fluctuations that a single measurement might experience, making it a valuable tool for anticipating real-world variations.

Normal Distribution

The assumption of data being normally distributed is key in statistical inference. A normal distribution is a bell-shaped, symmetric curve describing how data values spread around the mean. In practice, we check this assumption to validate that our statistical methods will yield reliable results.
For the UHPC adhesion values, the normality check can be performed visually using a Q-Q plot.

• A Q-Q plot compares your data to the expected quantiles of a normal distribution.
• If the plot aligns closely with a straight line, the data can be interpreted as being normally distributed.

Checking normality safeguards our confidence intervals, prediction intervals, and other statistical methods from leading us astray by false assumptions.

Tolerance Interval

A tolerance interval is quite different from confidence or prediction intervals. While a confidence interval estimates where a population parameter might be, and a prediction interval forecasts a future observation, a tolerance interval aims to capture a proportion of the population data within a specified confidence level.
For UHPC, we want an interval that includes at least 95% of all possible work of adhesion values for a high confidence level of 95%. The formula used is \( \bar{x} \pm k \cdot s \), where \( k \) is a calculated factor based on desired confidence and coverage.

• For our data with the sample size of 5, \( k \approx 4.03 \).
• Our tolerance interval thus becomes: \((103.70, 111.86)\).

This interval provides a comprehensive range where we confidently expect most real-world UHPC measures to fall, reflecting not just the average but the breadth of variation in actual data.