Solved Problems

An engineering firm claims that their new bridge design can withstand a maximum load of at least $1000 \text{ kN}$ before deflecting beyond acceptable limits. An independent auditor suspects the actual maximum load is lower. Formulate the null and alternative hypotheses to test the auditor's suspicion.

A quality control inspector is testing the compressive strength of concrete cylinders. The null hypothesis ($H_0$) is that the concrete meets the specified strength. The alternative hypothesis ($H_1$) is that the concrete does not meet the specified strength. Describe a Type I and Type II error in this context and discuss their engineering implications.

A geotechnical engineer takes a sample of $15$ soil specimens to determine their mean shear strength. The population standard deviation of the shear strength is unknown, and the population is assumed to be normally distributed. Should the engineer use a Z-test or a T-test for hypothesis testing?

An environmental engineer conducts a hypothesis test to determine if the concentration of a pollutant in a river exceeds the regulatory limit of $50 \text{ mg/L}$. The calculated p-value for the right-tailed test is $0.034$. The significance level is $\alpha = 0.05$. What is the appropriate decision and interpretation?

The historical data for a specific concrete mix design shows a population mean compressive strength of $30 \text{ MPa}$ with a population standard deviation of $3 \text{ MPa}$. A new admixture is added to improve strength. A random sample of $36$ specimens with the admixture yields a mean strength of $31.2 \text{ MPa}$. Test the claim that the admixture increases the mean strength at a significance level of $\alpha = 0.05$.

A manufacturer claims their reinforcing steel bars have a mean yield strength of at least $500 \text{ MPa}$. An independent tester samples $25$ bars and finds a mean strength of $495 \text{ MPa}$ with a standard deviation of $10 \text{ MPa}$. Test the manufacturer's claim at $\alpha = 0.05$.

A city regulation requires that at least $80\%$ of new residential buildings incorporate a specific energy-efficient insulation. An inspector randomly selects $150$ new buildings and finds that $110$ of them have the required insulation. Test whether the actual proportion is less than $80\%$ at the $0.05$ significance level.

An engineer tests the mean curing time of two epoxy resins. Resin A ($35$ samples) has a mean curing time of $45 \text{ hours}$ with a standard deviation of $4 \text{ hours}$. Resin B ($40$ samples) has a mean of $42 \text{ hours}$ with a standard deviation of $5 \text{ hours}$. Test the hypothesis that the means are different at $\alpha = 0.01$.

Traffic engineers are comparing two intersection designs. For Design 1, $12$ trials yield a mean vehicle delay of $48 \text{ seconds}$ with a standard deviation of $6 \text{ seconds}$. For Design 2, $14$ trials yield a mean delay of $41 \text{ seconds}$ with a standard deviation of $5 \text{ seconds}$. Assume the population variances are equal. Test if Design 1 has a significantly greater mean delay than Design 2 at $\alpha = 0.05$.

To test a new retrofitting technique, $8$ structural beams are tested for maximum deflection (in mm) before and after retrofitting. The differences ($d = \text{After} - \text{Before}$) yield a mean difference of $\bar{d} = -2.5 \text{ mm}$ with a standard deviation of $s_d = 1.2 \text{ mm}$. Test if the retrofitting significantly reduces deflection at $\alpha = 0.05$.

Two suppliers provide bolts for a construction project. Supplier A provides a sample of $200$ bolts, out of which $12$ are defective. Supplier B provides a sample of $250$ bolts, out of which $10$ are defective. Test if there is a significant difference in the proportion of defective bolts between the two suppliers at $\alpha = 0.05$.

A cement packaging machine is supposed to fill bags with a variance of $\sigma^2 = 0.5 \text{ kg}^2$. A random sample of $20$ bags yields a sample variance of $s^2 = 0.8 \text{ kg}^2$. Test if the machine is operating with a variance different from the specified $0.5 \text{ kg}^2$ at $\alpha = 0.05$. Assume the bag weights are normally distributed.