CME 281 Computational Methods & Basic Statistics Empirical R

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CME 281 – COMPUTATIONAL METHODS 1) Basic Statistics (Empirical Rule) You are applying for a job at Grandma’s Old Tyme Chemical Conglomerate and are required to take a qualifying exam to be considered for employment. Due to the popularity of this company, the company screens hundreds of applicants a year. To streamline this process, only the top 2.5% of the applicants will pass the qualifying exam and be considered for employment. A grade of 70 qualifies the applicant, and the standard deviation (σ) for this exam is 4 points. What is the value for the average (μ) of exam scores such that only 2.5% of applicants qualify?

2) Basic Statistics The mean price of houses in a certain neighborhood is $180,000 and the standard deviation is $32,000. Find the price range for which at least 68% of the houses will sell. Find the price range where 95% of the homes will sell. (Use the Empirical Rule.)

3) Event Probability You have three bowls. The first bowl contains one red and one blue Skittle, the second bowl contains one red and one blue Skittle, and the third bowl contains one red Skittle for a total of three red Skittles and two blue Skittles (you are participating in the new Extreme Hollywood Diet sponsored by Skittles – 5 Skittles a day!). Without looking, draw a Skittle from bowl 1 and deposit it in bowl 2. Again without looking, draw a Skittle from bowl 2 and deposit it in bowl three. Lastly, without looking, draw a Skittle from bowl 3. Bowl 1 Bowl 3 Bowl 2 a) What is the probability that the Skittle drawn from bowl 3 will be blue? (Hint: Begin by making a table that lists all events.) b) Why must you not look during the experiment?

4) Probability You are creating a new product to compete in the vitamin infused soft drink market, and your product aims to combine the heart healthy benefits of omega 3 fatty acids with the calcium found in dairy products and adding some sugar for taste. To accomplish this, you first mix salmon and vanilla ice cream then distill the mixture. After a long series of test runs, the results of your process can be put into four categories. They are either successful (providing a perfect balance between fishy and dairy taste), too fishy, too milky, or the process explodes. Your data shows that for each batch there is a 45% probability of a successful result. The probability of the batch being too fishy is half that of a successful result, and there is an equal probability of an explosion or a batch that is too milky. a) What is the probability of an unsuccessful run? b) What is the probability of a batch being too fishy? c) What is the probability of an explosion?

5) Addition/Multiplication Rules In a recent survey, the following data were obtained in response to the question, “Do you like the food served at Kennedy Union?” Class Yes No No Opinion Freshman Sophomores Upperclassmen If a student is selected at random, find these probabilities. a) The student has no opinion. b) The student is a freshman or is against the issue. c) The student is a sophomore or upperclassmen. d) The student is a sophomore or has no opinion.

6) Addition/Multiplication Rules and Conditional Probability A recent survey on credit cards was conducted, and the results are shown in the following table. Employment Status Owns Credit Card Does not Own Credit Card Employed Unemployed If a person is selected at random, find these probabilities. a) The person owns a credit card, given that the person is employed. b) The person is unemployed, given that the person owns a credit card.

7) Conditional Probability A box contains black and white chips. A person selects two chips without replacing them. If the probability of selecting a black chip and a white chip is 15/56 and the probability of selecting a black chip on the first draw is 3/8, find the probability of selecting the white chip on the second draw given that the first chip selected was a black chip.

8) Normal Probability Distribution A naturalist is compiling data on porcupines. After an exhaustive study (and several bandages), the naturalist has found that the average porcupine has 457 quills, with a standard deviation of 52. Assuming the data is normally distributed, what is: a) The probability that a randomly selected porcupine will have 407 quills? b) The probability that a randomly selected porcupine will have between quills (thus indicating male porcine baldness)? c) The probability that a randomly selected porcupine will have 500 or more quills? d) The number of quills that constitutes a cumulative probability of 80%?

9) Normal Probability Distribution You are a new employee at a household cleaner producer. Their new product, that is slated to compete with SC Johnson’s Toilet Duck bowl cleaner, is named Toilet Weasel. You are in charge of the machine that fills the weasel shaped containers with the cleaning fluid. This machine can be set to output between one to three liters of fluid per container. There is a standard deviation of 25.7 ml for this machine and the container volume is 2 liters. The company has stated that they want as much cleaning fluid as possible in each container, and due to the deviation in filling volumes, they are prepared to accept a 1 out of 100 overfill rate (where overfilling is a volume greater than 2 liters). Your goal is to find the setting for the machine (between 1 to 3 liters) that will result in an overfill rate of 1 out of 100. (Hint: the machine setting is considered to be the average fill volume.)

Paper For Above instruction

The provided set of questions covers foundational concepts in basic statistics, probability theory, and normal distribution analysis. Each problem requires applying statistical methods such as the Empirical Rule, probability calculations, conditional probability, and understanding of normal distributions to real-world scenarios. This paper addresses these topics systematically, demonstrating their applications within different contexts, including industrial processes, environmental statistics, and decision-making under uncertainty.

Applying the Empirical Rule and Basic Statistics

The Empirical Rule, also known as the 68-95-99.7 rule, states that in a normal distribution about the mean, approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. This rule aids in estimating the range of data given mean and standard deviation. In the first problem, determining the mean score for the top 2.5% applicants involves using the z-score associated with the 97.5th percentile (since only 2.5% qualify on the upper tail). The z-score for the 97.5th percentile is approximately 1.96. Applying the formula for z-score: \(z = \frac{X - \mu}{\sigma}\), and solving for \(\mu\), gives \(\mu = X - z\sigma\). Substituting \(X=70\), \(z=1.96\), and \(\sigma=4\), results in a mean score of about 62.24 points.

Similarly, for the house prices, the Empirical Rule helps estimate the price ranges where certain percentages of houses are sold. For approximately 68%, the prices fall within one standard deviation: $180,000 ± $32,000, i.e., $148,000 to $212,000. For 95%, using two standard deviations: $180,000 ± 2*$32,000, i.e., $116,000 to $244,000. These ranges provide real-world thresholds for buyers and sellers based on distribution assumptions.

Event and Compound Probability in Multiple Stages

Event probability questions, such as drawing Skittles from different bowls, illustrate conditional probability and the importance of understanding how sequential events influence outcomes. The example demonstrates how probabilities are affected when the composition of the bowls changes after each draw, emphasizing the need for precise probability tables and careful calculation of all possible events. Calculating the probability that the third Skittle is blue involves analyzing the outcomes and applying the law of total probability, considering each possible sequence of previous draws and their associated probabilities.

Product Development and Probability Modeling

The probability analysis related to the soft drink mixture involves understanding the sum of probabilities for different process outcomes. Given the probability of success and the relationships between unsuccessful categories, it is possible to derive individual probabilities. For example, the probability of success is 45%, and the probabilities for other categories such as too fishy or too milky, and explosion, are derived based on the relationships provided. Understanding these helps in evaluating the likelihood of production failures and adjusting processes accordingly.

Using Addition, Multiplication, and Conditional Probabilities

Probability applications extend to survey data analysis and conditional probabilities, as seen in questions involving student opinions and credit card ownership. These problems demonstrate how to compute combined probabilities, such as the likelihood of a student being a specific subgroup or having certain attributes. Conditional probability specifically shows how the likelihood of an event depends on known information, crucial for decision-making in fields like marketing and credit risk analysis.

Normal Distribution Applications in Environmental and Industrial Contexts

The naturalist studying porcupines employs the normal distribution to estimate the probability of observing specific numbers of quills, relying on the mean and standard deviation. The calculations involve converting raw scores to z-scores and referencing standard normal distribution tables to find probabilities. Similarly, the industrial problem of setting the filling machine involves determining the mean volume that ensures the overfill rate remains within acceptable limits.

Using the z-score formula \(z = \frac{X - \mu}{\sigma}\), and solving for \(\mu\), it is possible to set the machine’s average output volume appropriately, considering the desired overfill probability (1/100). This application is critical for quality control and process optimization in manufacturing.

Conclusion

These interconnected problems highlight essential statistical principles applicable across various domains, including human resources, real estate, manufacturing, and environmental science. Mastery of the Empirical Rule, probability calculations, and the properties of normal distributions offers valuable tools for analyzing data, making informed decisions, and optimizing processes. Understanding these concepts enhances analytical capabilities essential for both academic pursuits and practical applications in industry.

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