Solved Problems

The compressive strength of a certain type of concrete is known to have a mean of $4000 \text{ psi}$ and a standard deviation of $500 \text{ psi}$. A random sample of $36$ specimens is taken. What is the probability that the average strength of this sample is greater than $4100 \text{ psi}$?

A manufacturer claims that $5\%$ of their bolts are defective. If a civil engineer purchases a random sample of $400$ bolts, what is the probability that more than $7\%$ of them are defective?

A machine filling bags of cement has a known variance in weight of $\sigma^2 = 0.25 \text{ kg}^2$. A quality control engineer takes a random sample of $n = 15$ bags and calculates the sample variance $s^2$. Assuming the weights are normally distributed, what is the probability that the sample variance exceeds $0.40 \text{ kg}^2$?

The time it takes for a water filter to process a batch of water follows an exponential distribution with a mean of $12 \text{ minutes}$ and a standard deviation of $12 \text{ minutes}$. A random sample of $40$ batches is monitored. What is the probability that the average processing time for these $40$ batches is less than $10 \text{ minutes}$?

A civil engineer is testing two different brands of steel rebars. Brand A has a mean tensile strength of $70 \text{ ksi}$ with a standard deviation of $4 \text{ ksi}$. Brand B has a mean tensile strength of $68 \text{ ksi}$ with a standard deviation of $3 \text{ ksi}$. If random samples of $n_A = 50$ from Brand A and $n_B = 40$ from Brand B are tested, what is the probability that the sample mean for Brand A is at least $3 \text{ ksi}$ greater than that of Brand B?

Two concrete mix designs, Mix 1 and Mix 2, are being evaluated for surface cracking. Historically, $10\%$ of slabs using Mix 1 and $6\%$ of slabs using Mix 2 develop cracks. If $150$ slabs of Mix 1 and $200$ slabs of Mix 2 are poured, what is the probability that the proportion of cracked slabs in Mix 1 is at least $8\%$ higher than in Mix 2?

A transportation engineer measures the thickness of the asphalt layer on a newly paved road. A random sample of $12$ core measurements has a mean of $4.8 \text{ inches}$ and a sample standard deviation of $0.3 \text{ inches}$. Assuming the asphalt thickness is normally distributed, what is the probability that the sample mean is less than $4.6 \text{ inches}$ if the true population mean is $5.0 \text{ inches}$?

A materials lab is comparing the variability of compressive strength tests from two different technicians. Technician 1 performs $10$ tests with a sample variance of $s_1^2 = 180 \text{ psi}^2$. Technician 2 performs $16$ tests with a sample variance of $s_2^2 = 120 \text{ psi}^2$. Assuming the true variances are equal ($\sigma_1^2 = \sigma_2^2$) and the populations are normally distributed, what is the value of the F-statistic, and does it exceed the critical value for $\alpha = 0.05$ (upper tail)?