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Chapter 3: Problem 70

An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30 , decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects. a. What is the probability that exactly 10 of these are from the second section? b. What is the probability that at least 10 of these are from the second section? c. What is the probability that at least 10 of these are from the same section? d. What are the mean value and standard deviation of the number among these 15 that are from the second section? e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?

Short Answer

Expert verified
Exact probabilities require calculations with a hypergeometric distribution formula for parts a-c. Mean and standard deviation use hypergeometric equations for parts d-e.

Step by step solution

01

Understanding the Total Projects

We know there are a total of 50 projects (20 from the first section and 30 from the second section). We are considering the first 15 graded projects.
02

Calculating Probability for Part a

To find the probability that exactly 10 of the first 15 projects are from the second section, use the hypergeometric distribution formula: \[P(X = k) = \frac{{\binom{m}{k} \cdot \binom{N-m}{n-k}}}{\binom{N}{n}}\]where \(N = 50\), \(m = 30\), \(n = 15\), and \(k = 10\). This calculates as:\[P(X = 10) = \frac{{\binom{30}{10} \cdot \binom{20}{5}}}{\binom{50}{15}}\] Compute this using a calculator.
03

Calculating Probability for Part b

To find the probability that at least 10 of the 15 are from the second section, sum the probabilities from part a that correspond to 10, 11, ..., 15 projects:\[P(X \geq 10) = P(X = 10) + P(X = 11) + \cdots + P(X = 15)\] This involves calculating each probability with the hypergeometric distribution and adding them together.
04

Calculating Probability for Part c

For the probability that at least 10 projects are from the same section, find:\[P(X \geq 10 \text{ from section 2}) + P(Y \geq 10 \text{ from section 1}) - P(X \geq 10 \text{ from section 2 and } Y \geq 10 \text{ from section 1})\]where \(Y = 15 - X\). This involves combinations similar to part b, noting overlap.
05

Finding Mean and Standard Deviation for Part d

The mean \(\mu\) and standard deviation \(\sigma\) for the hypergeometric distribution are given by:\[\mu = \frac{n \times m}{N}\]\[\sigma = \sqrt{\frac{n \times m \times (N-m) \times (N-n)}{N^2 \times (N-1)}}\]where \(N = 50\), \(m = 30\), and \(n = 15\). Calculate these values.
06

Mean and Standard Deviation for Part e

The remaining projects not in the first 15 are \(N - n = 35\). To find the mean and standard deviation, calculate them as for Part d:\[\mu = \frac{35 \times 30}{50} = 21\]\[\sigma = \sqrt{\frac{35 \times 30 \times 20 \times 15}{50^2 \times 49}}\] Compute this for the exact standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
In solving probability problems, especially in educational settings, familiarity with certain probability distributions is crucial. One such distribution is the hypergeometric distribution. This is used when sampling without replacement, and each item in the population falls into one of two categories.
For example, in an exercise dealing with project submissions in two sections, calculating the probability that exactly 10 out of 15 projects graded first are from the second section involves using the hypergeometric formula:
  • The formula is: \[P(X = k) = \frac{{\binom{m}{k} \cdot \binom{N-m}{n-k}}}{\binom{N}{n}}\]
  • Where \(N\) is the total number of items (projects), \(m\) is the total number in the desired category (projects from the second section), \(n\) is the number of items to be drawn (projects graded first), and \(k\) is the number of desired items in the draw.
Understanding these components helps in appropriately using the formula to calculate probabilities like exactly 10 projects being from a specific section, or at least 10 projects being from that section.
This method extends to sequences and can be adjusted as required by the problem's constraints.
Mean and Standard Deviation
The mean and standard deviation are fundamental in quantifying the central tendency and dispersion of a dataset, respectively. When dealing with the hypergeometric distribution, these two measures provide insight into the expected number of successes and the variability of these successes from the mean.
For the projects problem, the mean (\(\mu\)) and the standard deviation (\(\sigma\)) of the number of projects from the second section amongst the first 15 graded are calculated using:
  • The mean formula: \[\mu = \frac{n \times m}{N}\]
  • The standard deviation formula: \[\sigma = \sqrt{\frac{n \times m \times (N-m) \times (N-n)}{N^2 \times (N-1)}}\]
Here, \(N\) is the total number of projects, \(m\) is the number of projects from the second section, and \(n\) is the number of projects selected.
These computations detail how assumptions in probability translate into summary measures, assisting educators in predicting outcomes under specified conditions, which is invaluable in decision-making and assessment strategies.
Statistics in Education
Statistics plays a vital role in assessing educational outcomes and structuring educational methodologies. Through probability and statistical measures, instructors can evaluate the performance of students, not just numerically but also by comprehending variability and predicting potential outcomes.
In education, understanding the spread and central tendency of student performance, such as through project scores, provides deep insight into class dynamics.
  • Probabilistic models like the hypergeometric distribution shed light on how likely certain student groups are to outperform others.
  • Such insights aid teachers in distributing resources and attention effectively, catering to all student needs.
Moreover, summary statistics like mean and standard deviation not only measure but guide educational assessment design and interpret results accurately, making statistics an indispensable tool in educational strategy and planning.

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Most popular questions from this chapter

For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. \(X=\) the number of unbroken eggs in a randomly chosen standard egg carton b. \(Y=\) the number of students on a class list for a particular course who are absent on the first day of classes c. \(U=\) the number of times a duffer has to swing at a golf ball before hitting it d. \(X=\) the length of a randomly selected rattlesnake e. \(Z=\) the amount of royalties earned from the sale of a first edition of 10,000 textbooks f. \(Y=\) the pH of a randomly chosen soil sample g. \(X=\) the tension (psi) at which a randomly selected tennis racket has been strung h. \(X=\) the total number of coin tosses required for three individuals to obtain a match (HHH or TTT)

An individual who has automobile insurance from a certain company is randomly selected. Let \(Y\) be the number of moving violations for which the individual was cited during the last 3 years. The pmf of \(Y\) is $$ \begin{array}{l|cccc} y & 0 & 1 & 2 & 3 \\ \hline p(y) & .60 & .25 & .10 & .05 \end{array} $$ a. Compute \(E(Y)\). b. Suppose an individual with \(Y\) violations incurs a surcharge of \(\$ 100 Y^{2}\). Calculate the expected amount of the surcharge.

A small market orders copies of a certain magazine for its magazine rack each week. Let \(X=\) demand for the magazine, with pmf $$ \begin{array}{c|cccccc} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline p(x) & \frac{1}{15} & \frac{2}{15} & \frac{3}{15} & \frac{4}{15} & \frac{3}{15} & \frac{2}{15} \end{array} $$ Suppose the store owner actually pays \(\$ 1.00\) for each copy of the magazine and the price to customers is \(\$ 2.00\). If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine?

Let \(X=\) the outcome when a fair die is rolled once. If before the die is rolled you are offered either (1/3.5) dollars or \(h(X)=1 / X\) dollars, would you accept the guaranteed amount or would you gamble? [Note: It is not generally true that \(1 / E(X)=E(1 / X)\).

Twenty pairs of individuals playing in a bridge tournament have been seeded \(1, \ldots, 20\). In the first part of the tournament, the 20 are randomly divided into 10 east-west pairs and 10 north-south pairs. a. What is the probability that \(x\) of the top 10 pairs end up playing east- west? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are \(2 n\) pairs, what is the pmf of \(X=\) the number among the top \(n\) pairs who end up playing east-west? What are \(E(X)\) and \(V(X)\) ?
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