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Chapter 6: Problem 6

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The time it takes for a light bulb to burn out. (b) The weight of a T-bone steak. (c) The number of free-throw attempts before the first shot is made. (d) In a random sample of 20 people, the number with type \(A\) blood.

Short Answer

Expert verified
a) Continuous. b) Continuous. c) Discrete. d) Discrete.

Step by step solution

01

- Define Discrete Versus Continuous

Understand the difference between discrete and continuous variables. A discrete variable can take on a countable number of values (like whole numbers), while a continuous variable can take on any value within a range (like real numbers).
02

- Analyze Part (a)

Consider the time it takes for a light bulb to burn out. Time can be measured in infinitely precise measurements (seconds, milliseconds, etc.), so it is a continuous variable. Possible values are any non-negative real number.
03

- Analyze Part (b)

Think about the weight of a T-bone steak. Weight can be measured to any level of precision (grams, kilograms, etc.), making it a continuous variable. Possible values are any non-negative real number.
04

- Analyze Part (c)

Evaluate the number of free-throw attempts before the first shot is made. This is a count of attempts, which makes it a discrete variable. Possible values are 1, 2, 3, and so on.
05

- Analyze Part (d)

Consider the number of people with type A blood in a random sample of 20 people. This is a count of individuals, so it is a discrete variable. Possible values are integers from 0 to 20.

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Key Concepts

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

Discrete Variable
A discrete variable can assume a finite or countably infinite number of distinct values. For example, the number of free-throw attempts before the first shot is made is a discrete variable. You can count these attempts individually as 1, 2, 3, and so on.
Discrete variables typically are whole numbers and can't be broken down into smaller parts without losing their meaning. For instance, you can't have 1.5 free-throw attempts; it either is 1 or 2.
  • Another example of a discrete variable is the number of people with type A blood in a sample of 20 people. Here, the variable takes on integer values ranging from 0 to 20.
Continuous Variable
A continuous variable can take on any value within a certain range. Unlike discrete variables, continuous variables can be finely divided to any level of precision. For example, the time it takes for a light bulb to burn out is a continuous variable because it can be measured in seconds, milliseconds, or even smaller units.
Continuous variables are often associated with measurements. Another commonly cited example is the weight of a T-bone steak. You can measure its weight very precisely, down to grams, milligrams, or further.
  • These variables are inherently more flexible than discrete variables because they can occupy any point in a given interval.
Statistics
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of data. It helps us understand and make inferences about real-world phenomena.
One of the ways to organize data in statistics is by categorizing the variables involved as either discrete or continuous. Understanding the type of variable you are dealing with is crucial for selecting the right statistical methods and tools to analyze your data.
  • For example, different probability distributions apply to discrete versus continuous variables. Binomial and Poisson distributions apply to discrete variables, while normal and exponential distributions apply to continuous variables.

Moreover, statistical tests, such as the Chi-squared test or Z-test, are designed with the variable type in mind. Learning how to classify variables accurately is foundational for a correct statistical analysis.

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Most popular questions from this chapter

Explain the role of \(\lambda\) and \(t\) in the Poisson probability formula.

BlackJack is a popular casino game in which a player is dealt two cards where the value of the card corresponds to the number on the card, face cards are worth ten, and aces are worth either one or eleven. The object is to get as close to 21 as possible without going over and have cards whose value exceeds that of the dealer. A blackjack is an ace and a ten in two cards. It pays 1.5 times the bet. The dealer plays last and must draw a card with sixteen and hold with seventeen or more. The following distribution shows the winnings and probability for a \(\$ 20\) bet. In cases where the dealer and player have the same value, there is a tie (called a "push"). Source: "Examining a Gambler's Claims: Probabilistic Fact-Checking and Don Johnson's Extraordinary Winning Streak" by W.J. Hurley, Jack Brimberg, and Richard Kohar. Chance Vol. 27.1,2014 $$ \begin{array}{cc} \text { Winnings } & \text { Probability } \\ \hline 0 & 0.0982 \\ \hline \$ 30 & 0.0483 \\ \hline \$ 20 & 0.389275 \\ \hline-\$ 20 & 0.464225 \end{array} $$ (a) Compute and interpret the expected value of the game from the player's point of view. (b) Suppose over the course of one hour, a player can expect to be dealt about 40 hands. How much should a player expect to win or lose over the course of three hours?

In \(2006,\) the New York City Police Department stopped a half million pedestrians for suspected criminal involvement. Raw statistics for these encounters suggest large racial disparities \(-\) 89 percent of the stops involved nonwhites. Do these statistics point to racial bias in police officers' decisions to stop particular pedestrians? Do they indicate that officers are particularly intrusive when stopping nonwhites? 50\. Simulation According to the U.S. National Center for Health Statistics, there is a \(98 \%\) probability that a 20 -year-old male will survive to age \(30 .\) (a) Using statistical software, simulate taking 100 random samples of size 30 from this population. (b) Using the results of the simulation, compute the probability that exactly 29 of the 30 males survive to age 30 . (c) Compute the probability that exactly 29 of the 30 males survive to age \(30,\) using the binomial probability distribution. Compare the results with part (b). (d) Using the results of the simulation, compute the probability that at most 27 of the 30 males survive to age 30 . (e) Compute the probability that at most 27 of the 30 males survive to age 30 using the binomial probability distribution. Compare the results with part (d). (f) Compute the mean number of male survivors in the 100 simulations of the probability experiment. Is it close to the expected value? (g) Compute the standard deviation of the number of male survivors in the 100 simulations of the probability experiment. Compare the result to the theoretical standard deviation of the probability distribution. (h) Did the simulation yield any unusual results?

Determine whether the distribution is a discrete probability distribution. If not, state why. $$ \begin{array}{cc} x & P(x) \\ \hline 0 & 0.2 \\ \hline 1 & 0.2 \\ \hline 2 & 0.2 \\ \hline 3 & 0.2 \\ \hline 4 & 0.2 \\ \hline \end{array} $$

The expected number of successes in a binomial experiment with \(n\) trials and probability of success \(p\) is ______.
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