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Chapter 5: Problem 17

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline 1.5 & {2.4} & {3.6} & {2.6} & {1.6} & {2.4} & {2.0} \\ \hline 3.5 & {2.5} & {1.8} & {2.4} & {2.5} & {3.5} & {4.0} \\\ \hline 2.6 & {1.6} & {2.2} & {1.8} & {3.8} & {2.5} & {1.5} \\\\\hline {2.8}&{1.8} &{4.5}&{1.9} &{1.9}& {3.1}& {1.6}\\\\\hline\end{array}$$ $$\text{Table 5.4}$$ The sample mean \(=2.50\) and the sample standard deviation \(=0.8302.\) The distribution can be written as \(X \sim U(1.5,4.5).\) In this distribution, outcomes are equally likely. What does this mean?

Short Answer

Expert verified
In a uniform distribution, each value within the specified range (1.5 to 4.5) is equally probable.

Step by step solution

01

Understanding Uniform Distribution

A uniform distribution, denoted as \(X \sim U(a, b)\), is a type of probability distribution where all outcomes are equally probable. The distribution is defined by two parameters: \(a\) (the minimum value) and \(b\) (the maximum value). For this problem, \(a = 1.5\) and \(b = 4.5\), indicating that the square footage values between 1.5 and 4.5 (in thousands) are equally likely to occur.
02

Interpretation of Equally Likely Outcomes

In the context of this problem, 'equally likely outcomes' means that each home's square footage has an equal chance of being any number between 1.5 and 4.5 (in thousands of square feet). This implies no specific value is favored, and any square footage within the given range is just as likely to occur as any other.

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Key Concepts

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

Statistics Education
Statistics education helps us understand data through various methods. One important aspect is understanding distributions.
These allow us to see how data is spread across a range of values. By learning about different types of distributions, such as uniform distribution, students can better predict outcomes and make decisions based on data.
When teaching statistics, it's important to emphasize:
  • Data collection and analysis methods
  • How different types of distributions work
  • Practical applications of statistical concepts
Understanding the uniform distribution, for example, can help students grasp how some outcomes have equal chance of occurring. By focusing on real-life examples, like home square footage, statistics education becomes more engaging and relatable.
Probability Distribution
A probability distribution is a way to describe how likely certain outcomes are within a data set. There are many types of probability distributions, each with its unique properties.
One basic type is the uniform distribution. Probability distributions are key for:
  • Determining the likelihood of various outcomes
  • Explaining randomness in data-related events
  • Providing a mathematical framework for prediction
By understanding probability distributions, students can better understand randomness and variability. It allows them to calculate the probability of outcomes, such as predicting the likelihood of a home's square footage being in a specific range in this example.
Uniform Probability
Uniform probability refers to a situation where all results are equally likely to occur. It's often used to model scenarios where there is no favorite or biased result.
For example, rolling a fair die is a uniform probability event; each number has an equal chance of appearing. In the context of square footage, uniform probability means:
  • Every value between 1.5 and 4.5 is equally likely
  • No preference or weighting towards any particular square footage
  • A straightforward model to predict outcomes in this dataset
This makes uniform probability easy to understand and apply, especially in cases involving random outcomes, helping students see the fairness in equally distributed scenarios.
Mathematical Analysis
Mathematical analysis helps us break down and understand the components of a distribution. It involves using equations to describe and predict outcomes based on data.
For a uniform distribution, the formulas required include:
  • The probability density function (PDF)
  • The cumulative distribution function (CDF)
  • Mean and standard deviation calculations
The PDF for a uniform distribution, for example, is calculated using the formula: \[ f(x) = \frac{1}{b-a} \quad \text{for} \; a \leq x \leq b \]The mean of uniform distribution can be found using: \[ \text{Mean} = \frac{a+b}{2} \]By using these calculations, students can assess the probability distribution effectively. Thus, mathematical analysis transforms raw data into meaningful insights, allowing for accurate predictions and reasoning.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises. A distribution is given as \(X \sim \operatorname{Exp}(0.75).\) What is \(m?\)

Use the following information to answer the next 16 exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14. What is being measured here?

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline 1.5 & {2.4} & {3.6} & {2.6} & {1.6} & {2.4} & {2.0} \\ \hline 3.5 & {2.5} & {1.8} & {2.4} & {2.5} & {3.5} & {4.0} \\\ \hline 2.6 & {1.6} & {2.2} & {1.8} & {3.8} & {2.5} & {1.5} \\\\\hline {2.8}&{1.8} &{4.5}&{1.9} &{1.9}& {3.1}& {1.6}\\\\\hline\end{array}$$ $$\text{Table 5.4}$$ The sample mean \(=2.50\) and the sample standard deviation \(=0.8302.\) The distribution can be written as \(X \sim U(1.5,4.5).\) What is \(P(x < 3.5 | x < 4) ?\)

At a 911 call center, calls come in at an average rate of one call every two minutes. Assume that the time that elapses from one call to the next has the exponential distribution. a. On average, how much time occurs between five consecutive calls? b. Find the probability that after a call is received, it takes more than three minutes for the next call to occur. c. Ninety-percent of all calls occur within how many minutes of the previous call? d. Suppose that two minutes have elapsed since the last call. Find the probability that the next call will occur within the next minute. e. Find the probability that less than 20 calls occur within an hour.

Use the following information to answer the next eight exercises. A distribution is given as \(X \sim U(0,12)\). What is the probability density function?
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