/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 111 Consider the variable \(x=\) tim... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 111

Consider the variable \(x=\) time required for a college student to complete a standardized exam. Suppose that for the population of students at a particular university, the distribution of \(x\) is well approximated by a normal curve with mean 45 minutes and standard deviation 5 minutes. a. If 50 minutes is allowed for the exam, what proportion of students at this university would be unable to finish in the allotted time? b. How much time should be allowed for the exam if you wanted \(90 \%\) of the students taking the test to be able to finish in the allotted time? c. How much time is required for the fastest \(25 \%\) of all tudents to complete the exam

Short Answer

Expert verified
a. The proportion of students who would be unable to finish the exam in 50 minutes can be calculated using the Z-score and the standard normal distribution table. b. The time that should be allowed for the exam, given we want \(90\%\) of all students to be able to finish in time can be calculated after determining the corresponding Z-score for \(90\%\), and using it in the formula \(X = Z\sigma + \mu\). c. To find the time required for the fastest \(25\%\) of students to complete the exam, get the Z-score for \(25\%\), and compute using same formula \(X = Z\sigma + \mu\).

Step by step solution

01

Determine the Proportion Who Would Be Unable to Finish on Time

To find the proportion of students who would be unable to finish in 50 minutes, we need to calculate the z-score for 50 minutes using formula \(Z = (X - \mu) / \sigma \), where \(X=50\), \(\mu=45\) minutes and \(\sigma=5\). After calculating the Z-score, we can use the standard normal distribution table to find the proportion. This will give us the probability that a randomly chosen student will finish the exam in 50 minutes. However, we want to find the proportion of students who would be unable to finish in this time, therefore we need to subtract the calculated probability from 1.
02

Determine the Time That Should Be Allowed for Exam

To find out how much time should be allowed for the exam if you wanted \(90\%\) of the students taking the test to be able to finish in the allotted time, we need to find the Z-score corresponding to \(90\%\) from the standard normal distribution table. Then we can use formula \(X = Z\sigma + \mu\) to calculate the time, where Z is the Z-score found from the table, \(\mu = 45\) and \(\sigma = 5\).
03

Determine the Time Required for the Fastest Students

To find the time required for the fastest \(25\%\) of students to complete the exam, we first need to find the Z-score corresponding to \(25\%\) from the standard normal distribution table, then again using the formula \(X = Z\sigma + \mu\), we can calculate the time needed.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Z-score Calculation
The z-score is a crucial concept when dealing with normal distributions. It allows us to determine how far a specific point is from the mean of the distribution, measured in standard deviations. To calculate a z-score, we use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:
  • \(X\) is the score or value you are examining.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation of the distribution.
After finding the z-score, we consult a standard normal distribution table to find the probability related to that z-score.For instance, in our task: calculating the z-score for a time of 50 minutes gives us an insight into how unusual or common it is for students to complete a test in that time under the assumed normal distribution. A positive z-score indicates the value is above the mean, while a negative score shows it's below.
Standard Deviation
Standard deviation is a statistical measure that represents the spread or dispersion of a set of data points. In the context of a normal distribution, it tells us how much the values typically vary from the mean. For example, our given data shows a standard deviation of 5 minutes for test completion times, which suggests most completion times cluster around the mean (45 minutes). Understanding the size of the standard deviation helps in assessing whether a particular score is typical or unusual. In practical terms:
  • A smaller standard deviation means the data points are close to the mean.
  • A larger standard deviation indicates more spread out data.
This information is essential when calculating times for a range of students, such as determining how much additional time allows most students to finish their exams on time.
Probability Analysis
Probability analysis involves determining the likelihood of specific events occurring, based on the z-scores and the standard normal distribution.Let's break it down further:
  • To calculate the proportion of students unable to finish a 50-minute test, we find the z-score and then interpret it through the standard normal distribution table. This gives us the probability of completion within 50 minutes, and subtracting from 1 shows those unable to finish.
  • For allowing 90% of students to complete the test, we look for a z-score with that cumulative probability. The corresponding exam time is found by rearranging the standard z-formula to \(X = Z\sigma + \mu\).
  • When considering the fastest 25% of students, we again use probability analysis to find the z-score, which helps us determine the maximum time required for this subset of students to complete the exam, ensuring they are counted among the fastest finishers.
Understanding these probabilities is key to tailoring exam durations to student performance levels effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A chemical supply company currently has in stock 100 pounds of a certain chemical, which it sells to customers in 5 -pound lots. Let \(x=\) the number of lots ordered by a randomly chosen customer. The probability distribution of \(x\) is as follows: $$ \begin{array}{lcccc} x & 1 & 2 & 3 & 4 \\ p(x) & 0.2 & 0.4 & 0.3 & 0.1 \end{array} $$ a. Calculate and interpret the mean value of \(x\). b. Calculate and interpret the variance and standard deviation of \(x\).

A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the random variable \(x=\) actual capacity of a randomly selected tank has a distribution that is well approximated by a normal curve with mean 15.0 gallons and standard deviation 0.1 gallon. a. What is the probability that a randomly selected tank will hold at most 14.8 gallons? b. What is the probability that a randomly selected tank will hold between 14.7 and 15.1 gallons? c. If two such tanks are independently selected, what is the probability that both hold at most 15 gallons?

Suppose that fuel efficiency (miles per gallon, mpg) for a particular car model under specified conditions is normally distributed with a mean value of \(30.0 \mathrm{mpg}\) and a standard deviation of \(1.2 \mathrm{mpg}\). a. What is the probability that the fuel efficiency for a randomly selected car of this model is between 29 and \(31 \mathrm{mpg} ?\) b. Would it surprise you to find that the efficiency of a randomly selected car of this model is less than \(25 \mathrm{mpg} ?\) c. If three cars of this model are randomly selected, what is the probability that each of the three have efficiencies exceeding \(32 \mathrm{mpg}\) ? d. Find a number \(x^{*}\) such that \(95 \%\) of all cars of this model have efficiencies exceeding \(x^{*}\left(\right.\) i.e., \(\left.P\left(x>x^{*}\right)=0.95\right)\).

Suppose that for a given computer salesperson, the probability distribution of \(x=\) the number of systems sold in 1 month is given by the following table: $$ \begin{array}{lcccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ p(x) & 0.05 & 0.10 & 0.12 & 0.30 & 0.30 & 0.11 & 0.01 & 0.01 \end{array} $$ a. Find the mean value of \(x\) (the mean number of systems sold). b. Find the variance and standard deviation of \(x\). How would you interpret these values? c. What is the probability that the number of systems sold is within 1 standard deviation of its mean value? d. What is the probability that the number of systems sold is more than 2 standard deviations from the mean?

Suppose that the distribution of typing speed in words per minute (wpm) for experienced typists using a new type of split keyboard can be approximated by a normal curve with mean 60 wpm and standard deviation 15 wpm ("The Effects of Split Keyboard Geometry on Upper Body Postures, Ergonomics [2009]: 104-111). a. What is the probability that a randomly selected typist's speed is at most 60 wpm? Less than 60 wpm? b. What is the probability that a randomly selected typist's speed is between 45 and 90 wpm? c. Would you be surprised to find a typist in this population whose speed exceeded 105 wpm? d. Suppose that two typists are independently selected. What is the probability that both their speeds exceed 75 wpm? e. Suppose that special training is to be made available to the slowest \(20 \%\) of the typists. What typing speeds would qualify individuals for this training? (Hint: See Example 6.23\()\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.