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Chapter 7: Problem 78

Richard鈥檚 Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery. \(X \sim\) _____(____,____) a. \(U(0,4)\) b. \(U(10,2)\) c. \(E \chi p(2)\) d. \(\quad N(2,1)\)

Short Answer

Expert verified
The correct distribution is a. \(U(0,4)\).

Step by step solution

01

Understand the Problem

Richard's Furniture Company delivers furniture between 10 A.M. and 2 P.M., which means their delivery timeframe is 4 hours long. We need to find the distribution that represents how long individuals wait after 10 A.M.
02

Identify the Type of Distribution

The delivery process operates continuously and uniformly between 10 A.M. and 2 P.M. A uniform distribution is ideal for situations where all outcomes within a certain interval are equally likely.
03

Determine Parameters for Uniform Distribution

A uniform distribution is defined as \( U(a, b) \),where \( a \) is the start time, and \( b \) is the end time. Given that deliveries start at 10 A.M., and the wait is measured in hours past 10 A.M., the range is from 0 to 4 hours.
04

Match with Provided Options

The correct parameters for uniform distribution based on the problem are \( U(0, 4) \). This is represented in the given options as choice a. \( U(0, 4) \).

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Key Concepts

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

Discrete Probability Distributions
Discrete probability distributions describe scenarios where outcomes are limited to distinct values. Think of it as counting things, like the number of students passing a test or the number of phone calls a company receives. In mathematics, a probability distribution assigns a probability to each possible outcome in a sample space.
For discrete distributions:
  • Outcomes are counted, not measured.
  • Probabilities are assigned to individual outcomes and must sum up to 1.
  • Examples include the binomial, Poisson, and geometric distributions.
Richard鈥檚 Furniture Company delivery times don't follow a discrete distribution because they cover a continuous four-hour window, not distinct intervals.
Probability Distribution Selection
Selecting the right probability distribution involves understanding the nature of the data. Distributions can be discrete or continuous. When choosing a distribution, consider:
  • The Range of Outcomes: Are the outcomes continuous or discrete?
  • Equity of Occurrence: Do all outcomes in your range have an equal chance of happening?
For continuous and uniform processes like Richard鈥檚 Furniture deliveries, where every minute within a four-hour timeframe is equally likely for a delivery, the uniform distribution fits perfectly. It鈥檚 denoted as \( U(a, b) \), where \(a\) is the start time, and \(b\) is the end time. In this case, all delivery times between 0 and 4 hours after 10 AM are equally probable.
Continuous Random Variables
Continuous random variables deal with outcomes that can take any value within a certain range. They are often measurements, such as delivery times, temperatures, or distances.
Characteristics of continuous variables include:
  • Measured on a continuous scale, not counted.
  • Infinitely many possible values within a range.
  • Examples include normal, exponential, and uniform distributions.
In the scenario with Richard鈥檚 Furniture Company, the waiting time falls within a continuous range from 0 to 4 hours. Since every possible delivery time in this range is equally likely, it perfectly exemplifies a continuous uniform distribution. Unlike discrete variables, we use integrals to calculate probabilities over intervals when dealing with continuous random variables.

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Most popular questions from this chapter

Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 96 days. Use the normal approximation to the binomial to calculate the following probabilities. Round the standard deviation to four decimal places. a. Find the probability that Janice is the driver at most 20 days. b. Find the probability that Roberta is the driver more than 16 days. c. Find the probability that Barbara drives exactly 24 of those 96 days.

An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let X = the object of interest. What is the standard deviation of \(\Sigma X ?\)

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of \(\$ 4.59\) and a standard deviation of \(\$ 0.10 .\) Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. What's the approximate probability that the average price for 16 gas stations is over \(\$ 4.69 ?\) a. almost zero b. 0.1587 c. 0.0943 d. unknown

Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of 1.1547. We randomly survey 64 homes with children. a. In words, \(X=\) ______ b. The distribution is ____. c. In words, \(\Sigma X=\) ______ d. \(\Sigma X \sim\) ____(____,____) e. Find the probability that the total weight of open boxes is less than 250 pounds. f. Find the 35th percentile for the total weight of open boxes of cereal.

A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken. Find the probability that the mean actual weight for the 100 weights is greater than 25.2.
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