Solved Problems

Classify the following variables by type (Quantitative or Qualitative) and level of measurement:

1. The compressive strength of a concrete cylinder (in $\text{psi}$).
2. The type of defect in a welded joint (porosity, crack, undercut).
3. The rating of a bridge on a scale of $1$ to $10$.
4. The year a building was constructed.

Determine whether the following data collected by a civil engineer are discrete or continuous variables:

1. The number of potholes on a $10 \text{ km}$ stretch of highway.
2. The exact volume of water in a reservoir.
3. The number of rebar pieces used in a concrete slab.
4. The weight of a truck passing a weigh station.

A traffic engineer wants to determine the average speed of all vehicles crossing a specific bridge over the course of a year. Because measuring every single vehicle is not feasible, the engineer records the speed of $500$ vehicles over a one-week period. Identify the population and the sample in this scenario.

Determine whether each of the following statements represents descriptive or inferential statistics:

1. "The average compressive strength of the $30$ tested concrete cylinders is $4500 \text{ psi}$."
2. "Based on the test results of $30$ cylinders, the entire batch of $10,000 \text{ m}^3$ of concrete is expected to meet the minimum strength requirement of $4000 \text{ psi}$ with $95\%$ confidence."

A contractor receives a delivery of $800$ steel beams. To verify quality, $15$ beams will be selected for destructive testing. Determine the probability of any individual beam being selected, and describe how a simple random sample can be achieved.

A civil engineering firm wants to study traffic volume on a highway. They decide to measure the number of cars passing a specific point every $15 \text{ minutes}$, starting exactly at $8:00 \text{ AM}$ and ending at $4:00 \text{ PM}$ (inclusive). Identify the sampling technique and calculate the total number of samples taken during this $8$-hour period.

A geotechnical engineer must sample soil from a $50,000 \text{ m}^2$ site. The site has three distinct geological zones: Zone A covers $20,000 \text{ m}^2$, Zone B covers $20,000 \text{ m}^2$, and Zone C covers $10,000 \text{ m}^2$. The engineer plans to take $50$ total samples. Determine the number of samples that should be taken from each zone to achieve a proportionally stratified sample.

A construction company evaluates the quality of a delivery of $500$ rebars. An inspector arrives and selects the $10$ rebars closest to the entrance of the storage yard for tensile testing. All $10$ rebars pass, so the inspector approves the entire batch. Identify the sampling method, explain the potential bias, and mathematically demonstrate the consequence if the first $10$ rebars are from a good batch but $20\%$ of the remaining population is defective.