Solved Problems

A sample of $50$ soil specimens has a mean shear strength of $2500 \text{ psf}$ with a standard deviation of $300 \text{ psf}$. Construct a $95\%$ confidence interval for the true mean shear strength.

A civil engineer measures the setting time of a newly developed quick-set concrete mix. A random sample of $n = 12$ batches yields a sample mean setting time of $\bar{x} = 45 \text{ min}$ and a sample standard deviation of $s = 4.5 \text{ min}$. Assuming the setting times are normally distributed, construct a $99\%$ confidence interval for the true mean setting time.

Two suppliers provide steel rebars. A sample of $40$ rebars from Supplier A has a mean yield strength of $420 \text{ MPa}$ with a standard deviation of $15 \text{ MPa}$. A sample of $35$ rebars from Supplier B has a mean yield strength of $410 \text{ MPa}$ with a standard deviation of $18 \text{ MPa}$. Construct a $90\%$ confidence interval for the difference in true mean yield strengths ($\mu_A - \mu_B$).

A structural assessment team inspects $200$ randomly selected bridges in a state and finds that $35$ of them are structurally deficient. Determine a $95\%$ confidence interval for the true proportion of structurally deficient bridges in the state.

An environmental engineer wants to estimate the true mean concentration of a pollutant in a river. Previous studies suggest a population standard deviation of $12 \text{ mg/L}$. How many water samples must be taken to be 95% confident that the sample mean is within $3 \text{ mg/L}$ of the true mean?

A transportation department wants to estimate the proportion of commuters who carpool. They want to be 99% confident that their estimate is within 4% of the true proportion. If no prior estimate of the proportion is available, what is the required sample size?

A geotechnical engineer tests the compressive strength of a specific rock type. A sample of $n=15$ rock cores yields a sample variance of $s^2 = 45.2 \text{ MPa}^2$. Assuming the compressive strengths are normally distributed, construct a $90\%$ confidence interval for the true population variance ($\sigma^2$).

A traffic engineer is evaluating two intersections for safety improvements. At Intersection A, 45 out of 300 observed vehicles ran the red light. At Intersection B, 30 out of 250 observed vehicles ran the red light. Construct a $95\%$ confidence interval for the difference between the true proportions of red-light runners at the two intersections ($p_A - p_B$).