MATH 314 Practice Exam 02

Exam 02 on 2026-10-15

Assume the following code has already been run.

import numpy as np
import scipy as sp

  1. Suppose your company produces batches of 200 identical op-amp ICs (amplifiers). Historically, each chip passes a final test 96% of the time. Your boss believes that next month, all the op-amps that can be made will be sold. Each op-amp sells for $0.05. If the company factory can produce 100 batches, what is the expected revenue for next month?

  2. In power systems, random voltage spikes happen for a variety of reasons. It is crucial for electrical engineers to understand the frequency of these spikes to design protective measures accordingly. Suppose that, on average, 5.1 voltage spikes are recorded on the grid every hour. What is the probability that there will be fewer than 2 or more than 7 voltage spikes in an hour?

  1. Suppose you’re modeling the time until failure of an LED from a company for which the mean lifetime of their LEDs is 100,000 hours. For how many hours will the longest lasting 2% of their LEDs last?

  2. A pharmaceutical company manufactures tablets containing a precisely measured dose of an active ingredient. Quality control testing indicates that 4% of tablets contain an amount of the active ingredient outside the acceptable range. Suppose 30 tablets are randomdly selected from a batch. Let represent the number of tablets in the sample that fail quality control.

    a. Use Python to instantiate an appropriate random variable.

    b. What is the probability that exactly 2 of the 30 tablets fail quality control?

    c. What is the probability that at least 3 of the 30 tablets fail quality control?

    d. The company rejects a production batch if 4 or more tablets in the sample fail quality control. What is the probability that a randomly selected batch will be rejected, assuming the batch has the historical defect rate of 4%?

  3. An environmental engineering company develops water filtration systems used in wastewater treatment plants. The mean time between consecutive filter failures 250 days. Let represent the time, in days, between consecutive filter failures.

    a. What is the probability that more than 300 days pass between consecutive filter failures?

    b. What is the probability that the time between consecutive filter failures is between 100 and 200 days?

    c. Engineers want to determine the length of time within which 90% of filter failures will occur. How many days does this correspond to?

  4. Marine scientists are studying ocean temperatures at a particular depth in the Pacific Ocean. Measurements collected over several years indicate that the water temperature at this depth is approximately Normally distributed, with a mean of 12 degrees Celsius and a standard deviation of 1.8 degrees Celsius. Let represent the water temperature, in degrees Celsius, recorded at this depth on a randomly selected day.

    a. What is the probability that the water temperature exceeds 14 degrees Celsius?

    b. What is the probability that the water temperature is between 10 and 13 degrees Celsius?

    c. Certain marine organisms cannot survive in temperatures below a critical threshold. Scientists want to identify the temperature below which the coldest 10% of daily measurements fall. What is this temperature?

    d. Engineers discover that the temperature sensor consistently reads 3 degrees Celsius too high. They correct each measurement by subtracting 3 degrees. Let represent the corrected temperature. What are the mean and variance of ?

    e. Some crazy person, let's not mention any names, decides to convert the corrected temperatures from Celsius to Fahrenheit using the formula . Describe the distribution of the temperatures measured in Fahrenheit. Specify the name of the distribution, it's mean and standard deviation.

  5. Describe the Central Limit Theorem in your own words. Specifically, describe which distributions the original random variables must follow. What properties of the expectation and variance must be true? When does the accuracy of the Central Limit Theorem improve?

  6. Describe how we've been visualizing the Central Limit Theorem. Specifically, why do we need more than one mean? What does this tell you about a collection of means?

  7. Astronomers operate a network of cameras designed to detect meteors entering Earth's atmosphere. Historical observations indicate that an average of 6 meteors are detected per hour. Let represent the number of meteors detected during a randomly selected one-hour interval.

    a. What is the probability that exactly 4 meteors are detected during a one-hour interval?

    b. What is the probability that more than 8 meteors are detected during a one-hour interval?

    c. During a particularly important observation period, astronomers monitor the sky continuously for 30 minutes. What is the probability that at least 5 meteors are detected during this 30-minute interval?