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Chapter 4: Problem 9

Simple random sampling uses a sample of size \(n\) from a population of size \(N\) to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible?

Short Answer

Expert verified
230,300 different random samples are possible.

Step by step solution

01

Understanding Combinations

Simple random sampling without replacement requires us to find the number of combinations possible when selecting a sample. In this context, we need to select 4 accounts from a total of 50 without regard to the order in which they are chosen.
02

Using the Combinations Formula

The number of ways to choose a sample of size 4 from a population of 50 is given by the combination formula, \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n = 50 \) and \( k = 4 \).
03

Applying the Values to the Formula

Substitute \( n = 50 \) and \( k = 4 \) into the formula: \( C(50, 4) = \frac{50!}{4!(50-4)!} \).
04

Simplifying the Factorials

Simplify the expression: \( C(50, 4) = \frac{50 \times 49 \times 48 \times 47}{4 \times 3 \times 2 \times 1} \). The factorials reduce because many terms in \( 50! \) and \( 46! \) will cancel out.
05

Calculating the Result

Calculate the expression: \( 50 \times 49 \times 48 \times 47 = 5,527,200 \). Divide by the factorial of 4: \( 5,527,200 \div 24 = 230,300 \).
06

Conclusion

Conclude that the number of different random samples of four accounts from a population of 50 accounts is 230,300.

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Key Concepts

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

Combinations
Combinations play a crucial role in statistics, especially in the context of simple random sampling. When we talk about combinations, we're interested in the different ways to select a group of items from a larger set where the order does not matter. This is distinguished from permutations, where the order does matter.

For instance, if you're picking a sample of four accounts from a total of 50, the order in which you pick the accounts doesn't affect the sample itself. Thus, we use the combination formula to calculate this number. The combination formula is given by \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where:
  • \( n \) is the total number of items in the population.
  • \( k \) is the number of items to choose.
  • \(!\) (factorial) means a number multiplied by every positive whole number smaller than itself.
Substituting the values for \( n = 50 \) and \( k = 4 \), you can find all the possible samples without concern about the sequence in which you draw each account.
Sample Size
Sample size is a fundamental concept in statistical sampling methods, especially within the realm of simple random sampling. The sample size, often denoted by \( k \), refers to the number of observations or data points you are selecting from a population.

In the original exercise, the sample size is four accounts. This means you're drawing four out of the 50 possible bank accounts to learn something about the entire group (or population). The determination of sample size is crucial as it impacts the accuracy and reliability of the conclusions drawn from your study.
  • Small sample sizes can lead to inaccurate results due to increased variability or underrepresentation.
  • Too large sample sizes, on the other hand, may become impractical and wasteful in terms of resources and time.
A well-chosen sample size can strike a balance, providing the data needed to understand and make inferences about the population effectively.
Population Size
Population size, represented by \( n \), specifies the total number of elements that exist in a set or study. This is the complete set from which a sample is drawn in the sampling process. In the context of the exercise, the population size is 50, meaning there are 50 bank accounts from which you want to take a sample.

The larger the population size, the broader the variety available for sample selection, producing more diverse potential outcomes when taking a sample. However, it's important to note:
  • A larger population allows for a more representative sample, assuming the sample size is also sufficient.
  • Sampling from a smaller population may limit the insights that can be generalized about the population.
It's vital to understand both the population size and the sample size to properly design a study that yields reliable and meaningful inferences about the population.

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Most popular questions from this chapter

In a BusinessWeek/Harris Poll, 1035 adults were asked about their attitudes toward business (BusinessWeek, September 11, 2000). One question asked: "How would you rate large U.S. companies on making good products and competing in a global environment?" The responses were: excellent- \(18 \%\), pretty good \(-50 \%\), only fair- \(26 \%\), poor \(-5 \%\) and don't know/no answer- \(1 \%\) a. What is the probability that a respondent rated U.S. companies pretty good or excellent? b. How many respondents rated U.S. companies poor? c. How many respondents did not know or did not answer?

A company that manufactures toothpaste is studying five different package designs. Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs? In an actual experiment, 100 consumers were asked to pick the design they preferred. The following data were obtained. Do the data confirm the belief that one design is just as likely to be selected as another? Explain. $$\begin{array}{cc} \text { Design } & \text { Number of Times Preferred} \\ 1 & 5 \\ 2 & 15 \\ 3 & 30 \\ 4 & 40 \\ 5 & 10 \end{array}$$

The U.S. population by age is as follows (The World Almanac, 2004 ). The data are in millions of people. $$\begin{array}{lc} \text { Age } & \text { Number } \\ 19 \text { and under } & 80.5 \\ 20 \text { to } 24 & 19.0 \\ 25 \text { to } 34 & 39.9 \\ 35 \text { to } 44 & 45.2 \\ 45 \text { to } 54 & 37.7 \\ 55 \text { to } 64 & 24.3 \\ 65 \text { and over } & 35.0 \end{array}$$ Assume that a person will be randomly chosen from this population. a. What is the probability the person is 20 to 24 years old? b. What is the probability the person is 20 to 34 years old? c. What is the probability the person is 45 years or older?

A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: \(P\left(E_{1}\right)=.10, P\left(E_{2}\right)=.15, P\left(E_{3}\right)=.40,\) and \(P\left(E_{4}\right)=.20 .\) Are these probability assignments valid? Explain.

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw five white balls out of a drum with 55 white balls, and one red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the five white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the five white balls (http://www.powerball.com, March 19, 2006). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the five white balls? c. What is the probability of winning the Powerball jackpot?
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