Solved Problems

A civil engineer measures the compressive strength (in $\text{MPa}$) of 5 concrete cylinder samples:

Calculate the sample mean, median, and mode.

Using the same compressive strength data from Problem 1:

Calculate the range, sample variance, and sample standard deviation.

An environmental engineering student's final grade is based on three components: homework ($20\%$), a midterm exam ($30\%$), and a final project ($50\%$). The student scores $85.0$, $78.0$, and $92.0$ on these components, respectively. Calculate the weighted mean score.

The annual growth rates of a city's population over three consecutive years are $2.0\%$, $5.0\%$, and $8.0\%$. Calculate the average annual growth rate using the geometric mean of the growth factors.

A set of traffic speeds (in $\text{km/h}$) recorded on a local road is given below:

Determine the $80^{\text{th}}$ percentile ($P_{80}$) and the first quartile ($Q_1$).

Using the traffic speed data from Problem 5:

Find the interquartile range (IQR) and check if the maximum value ($75 \text{ km/h}$) is an outlier.

A traffic engineer studies the speeds of vehicles on a highway. The frequency distribution of speeds (in $\text{km/h}$) is recorded below:

• $60 - 69 \text{ km/h}$: $5$ vehicles
• $70 - 79 \text{ km/h}$: $18$ vehicles
• $80 - 89 \text{ km/h}$: $42$ vehicles
• $90 - 99 \text{ km/h}$: $27$ vehicles
• $100 - 109 \text{ km/h}$: $8$ vehicles

Calculate the approximate mean speed of the vehicles.

Using the frequency distribution of speeds from Problem 7, calculate the approximate sample variance and standard deviation. The approximate mean is $\bar{x} = 86.0 \text{ km/h}$ and $n = 100$.

A geotechnical engineer compares the variability of two soil properties. Soil A has a mean cohesion of $50 \text{ kPa}$ with a standard deviation of $5 \text{ kPa}$. Soil B has a mean cohesion of $150 \text{ kPa}$ with a standard deviation of $10 \text{ kPa}$. Which soil exhibits higher relative variability?

A structural engineering firm logs the time taken to complete 50 bridge design phases. The median completion time is $120$ hours, but the mean completion time is $145$ hours. What does this indicate about the distribution of the design times?

An urban planner is analyzing household incomes in a newly developed district to determine eligibility for housing subsidies. The income data includes many average-income households but also a small number of extremely high-income households. Which measure of central tendency (mean or median) should the planner use, and why?

Consider two datasets measuring daily water consumption (in thousands of liters) for a factory. Dataset 1 is symmetric with no outliers. Dataset 2 is identical to Dataset 1, except the maximum value is replaced by a massive anomaly caused by a pipe burst. How will this anomaly affect the Interquartile Range (IQR) compared to the Variance?

A steel manufacturing plant produces rebar with a target yield strength of $420 \text{ MPa}$. Historical data shows the distribution of yield strengths is perfectly bell-shaped (normally distributed) with a mean of $425 \text{ MPa}$ and a standard deviation of $5 \text{ MPa}$. According to the Empirical Rule, what percentage of the rebar falls between $410 \text{ MPa}$ and $440 \text{ MPa}$?