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Chapter 5: Problem 1

Which of the following are continuous variables, and which are discrete? (a) Number of traffic fatalities per year in the state of Florida (b) Distance a golf ball travels after being hit with a driver (c) Time required to drive from home to college on any given day (d) Number of ships in Pearl Harbor on any given day (e) Your weight before breakfast each morning

Short Answer

Expert verified
(a) Discrete; (b) Continuous; (c) Continuous; (d) Discrete; (e) Continuous.

Step by step solution

01

Understanding Continuous vs. Discrete Variables

Continuous variables are those that can take any value within a given range. These include measurements such as time, distance, and weight, where between any two values, there can be an infinite number of other possible values. Discrete variables, on the other hand, are countable and take on distinct, separate values. These include counts like the number of items or events.
02

Analyzing Variable (a)

The 'Number of traffic fatalities per year in the state of Florida' is a count of distinct events. Since fatalities are counted in whole numbers and cannot be fractional, this variable is discrete.
03

Analyzing Variable (b)

The 'Distance a golf ball travels after being hit with a driver' can take any real value within a range and can include fractions (like 250.5 meters). Thus, this is a continuous variable.
04

Analyzing Variable (c)

The 'Time required to drive from home to college on any given day' is measured in units such as seconds or minutes, which can be divided into smaller units indefinitely. Therefore, it is a continuous variable.
05

Analyzing Variable (d)

The 'Number of ships in Pearl Harbor on any given day' is a count of physical objects and can only take whole numbers. This makes it a discrete variable.
06

Analyzing Variable (e)

'Your weight before breakfast each morning' can be measured to any desired level of precision and can take non-integer values (like 70.5 kg). This makes it a continuous variable.

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Key Concepts

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

continuous variables
Continuous variables are essential components in statistics and data analysis. These variables can assume any value within a specified range. They can be subdivided into infinitely smaller units, allowing for a detailed and precise understanding of changes and relationships in a dataset.

Characteristics of continuous variables include:
  • Ability to take on any value within a range
  • Representing measurements like time, distance, and temperature
  • Being describable using real numbers, including fractions and decimals
For instance, in our example, the distance a golf ball travels or the time it takes to drive somewhere are both continuous variables because they can be measured with great precision and can have infinite outcomes within a range.
discrete variables
In contrast to continuous variables, discrete variables characterize data that can only take specific, distinct values, often whole numbers. These variables are countable and usually represent categories or groups rather than measurements.

Key features of discrete variables are:
  • Values are distinct and separate (e.g., 1, 2, 3)
  • Typically involve counting, not measuring
  • Cannot be subdivided into smaller units
Examples from our scenario include the number of traffic fatalities or the count of ships in a harbor, both of which are recorded in whole numbers and do not allow for fractions or decimals.
statistics education
Statistics education plays a vital role in helping students and professionals effectively understand and interpret data. It involves learning about different types of variables, data collection methods, and analysis techniques, all of which are fundamental for making informed decisions.

In statistics education, understanding the difference between continuous and discrete variables is vital:
  • Continuous variables provide deeper analysis opportunities with precise measurements.
  • Discrete variables help in categorical analysis, where distinct groups are compared.
By grasping these concepts, students can learn to handle real-world data more effectively and make meaningful inferences and predictions.
measurement scales
Measurement scales are critical for classifying and analyzing data in various fields, particularly in statistics. Different types of scales include nominal, ordinal, interval, and ratio, each providing different levels of information about the data.

How the variables are measured can affect the analysis:
  • Nominal scale: used for categories without numerical significance (e.g., types of fruits).
  • Ordinal scale: provides order but not the difference between items (e.g., rankings).
  • Interval scale: offers order and consistent intervals but lacks a true zero (e.g., temperature in Celsius).
  • Ratio scale: offers both a clear zero and equidistant intervals (e.g., weight and distance).
Understanding these scales is necessary for proper data interpretation and ensures that the analysis suits the data type at hand.

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Most popular questions from this chapter

The Denver Post reported that, on average, a large shopping center had an incident of shoplifting caught by security once every three hours. The shopping center is open from 10 A.M. to 9 P.M. (11 hours). Let \(r\) be the number of shoplifting incidents caught by security in an 11 -hour period during which the center is open. (a) Explain why the Poisson probability distribution would be a good choice for the random variable \(r\). What is \(\lambda\) ? (b) What is the probability that from 10 A.M. to 9 P.M. there will be at least one shoplifting incident caught by security? (c) What is the probability that from \(10 \mathrm{A.M}\). to 9 P.M. there will be at least three shoplifting incidents caught by security? (d) What is the probability that from 10 A.M. to 9 P.M. there will be no shoplifting incidents caught by security?

The following is based on information taken from The Wolf in the Southwest: The Making of an Endangered Species, edited by David Brown (University of Arizona Press). Before 1918 , approximately \(55 \%\) of the wolves in the New Mexico and Arizona region were male, and \(45 \%\) were female. However, cattle ranchers in this area have made a determined effort to exterminate wolves. From 1918 to the present, approximately \(70 \%\) of wolves in the region are male, and \(30 \%\) are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (a) Before 1918 , in a random sample of 12 wolves spotted in the region, what is the probability that 6 or more were male? What is the probability that 6 on more were female? What is the probability that fewer than 4 were female? (b) Answer part (a) for the period from 1918 to the present.

Harper's Index reported that the number of (Orange County, California) convicted drunk drivers whose sentence included a tour of the morgue was 569 , of which only 1 became a repeat offender. (a) Suppose that of 1000 newly convicted drunk drivers, all were required to take a tour of the morgue. Let us assume that the probability of a repeat offender is still \(p=1 / 569 .\) Explain why the Poisson approximation to the binomial would be a good choice for \(r=\) number of repeat offenders out of 1000 convicted drunk drivers who toured the morgue. What is \(\lambda\) to the nearest tenth? (b) What is the probability that \(r=0\) ? (c) What is the probability that \(r>1\) ? (d) What is the probability that \(r>2\) ? (e) What is the probability that \(r>3\) ?

When using the Poisson distribution, which parameter of the distribution is used in probability computations? What is the symbol used for this parameter?

In an experiment, there are \(n\) independent trials. For each trial, there are three outcomes, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). For each trial, the probability of outcome \(\mathrm{A}\) is \(0.40\); the probability of outcome \(\mathrm{B}\) is \(0.50\); and the probability of outcome \(\mathrm{C}\) is \(0.10\). Suppose there are 10 trials. (a) Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type \(\mathrm{B}\), and one of type C? Explain. (b) Can we use the binomial experiment model to determine the probability of four outcomes of type \(\mathrm{A}\) and six outcomes that are not of type A? Explain. What is the probability of success on each trial?
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