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Chapter 1: Problem 14

Give one example each of sampling with and sampling without replacement.

Short Answer

Expert verified
One example of sampling with replacement is drawing a ball from a box, recording its color, and putting it back before the next draw is made. An example of sampling without replacement would be drawing a ball, recording its color, and setting the ball aside before the next draw is conducted.

Step by step solution

01

Create an example for sampling with replacement

Imagine a box full of blue and red balls. In this scenario, sampling with replacement would occur if a ball is drawn from the box, the color is recorded, and then the drawn ball is put back in the box before the next ball is drawn. Since the ball is being replaced each time, the probabilities for each draw don't change.
02

Create an example for sampling without replacement

Using the same box of blue and red balls. In this example, sampling without replacement would take place if a ball is drawn, the color is noted, and then the ball is set aside (not replaced into the box) before the next ball is drawn. Because the balls are not replaced, the probabilities for each consecutive draw will change.

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Key Concepts

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

Sampling With Replacement
Sampling with replacement is a method where an item is selected from a population and then returned before selecting the next item. Think of it like picking a colored ball from a box, noting its color, and placing it back into the box. This process ensures that the probability of selecting any particular item remains constant.
For example, if a box contains 5 blue balls and 5 red balls, the probability of picking a blue ball stays at 50% on each draw because all items are replaced. This method keeps the overall population unchanged, maintaining uniformity in probabilities.
  • Ensures independent trials
  • Consistent probabilities for each draw
Sampling Without Replacement
In contrast, sampling without replacement involves selecting an item and not returning it to the population before the next selection. This method inherently alters the composition of the population after each draw, affecting probabilities of subsequent picks.
Consider our box of 5 blue and 5 red balls. Once a ball is drawn and not replaced, the total decreases. If you draw a blue ball, the chance of picking another blue ball next reduces. This approach leads to dependent events.
  • Alters population composition
  • Probability changes with each draw
Probability
Probability is the measure of how likely an event is to occur. It is expressed as a fraction, ratio, or percentage representing the chance of a specific outcome.
For instance, if you're drawing balls from the box mentioned earlier, the probability of drawing a blue ball initially could be 50% if there are equal colors. Probabilities help us understand and predict the likelihood of events in sampling methods.
  • Can be expressed as percentage, fraction, or ratio
  • Helps predict outcomes
Probability Changes
Probability changes are crucial when dealing with situations involving sampling without replacement. Here, drawing one item modifies the probabilities of subsequent draws because the overall population is altered.
Imagine you have just picked a red ball from a box with an initial setup of 5 red and 5 blue balls. There are now 4 red balls left, changing the odds significantly for the next draw. Such changes are pivotal in understanding dependent events in probability studies.
  • Immediate impact on subsequent probabilities
  • Crucial for dependent event scenarios
Example Scenarios
To better understand these concepts, let's explore some scenarios:
- Sampling with replacement involves repeatedly rolling a single die. Since the die is unaffected after each roll, the probability of landing on any number remains the same.
- Sampling without replacement could occur in a card game, like drawing cards from a deck. once a card is drawn and kept aside, the probabilities for drawing any subsequent cards change.
These examples show how different sampling methods impact the calculation and consideration of probability in real-world situations.
  • Rolling dice for replacement scenario
  • Drawing playing cards for non-replacement scenario

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Most popular questions from this chapter

A federal govemment think tank wanted to investigate whether a job training program helps the families who are on welfare to get off the welfare program. The researchers at this agency selected 10,000 families at random from the list of all families that were on welfare. Of these 10,000 families, the agency randomly selected 5000 families and offered them free job training. The remaining 5000 families were not offered such job training. After 3 years the two groups were compared in regard to the percentage of families who got off welfare. Is this an observational study or a designed experiment? Explain.

The following table lists the total revenue for each of a few selected companies for the year 2014 (Source: Fortune Magazine). $$ \begin{array}{lc} \hline \text { Company } & \begin{array}{c} \text { Revenue for Fiscal Year 2014 } \\ \text { (millions of dollars) } \end{array} \\ \hline \text { Walmart } & 476,294 \\ \text { Exxon Mobil } & 407,666 \\ \text { Apple } & 170,910 \\ \text { General Motors } & 155,427 \\ \text { Ford Motor } & 146,917 \\ \text { General Electric } & 146,231 \\ \hline \end{array} $$ Explain the meaning of a member, a variable, a measurement, and a data set with reference to this table.

The number of shoe pairs owned by six women is \(8,14,3,7,10\), and 5 , respectively. Let \(x\) denote the number of shoe pairs owned by a woman. Find: a. \(\sum x\) b. \(\left(\sum x\right)^{2}\) c. \(\sum x^{2}\)

The following table lists the number of billionaires for each of a few selected countries as of October \(2014 .\) $$ \begin{array}{lc} \hline \text { Country } & \text { Number of Billionaires } \\ \hline \text { United States } & 492 \\ \text { China } & 152 \\ \text { Russia } & 111 \\ \text { Germany } & 85 \\ \text { Brazil } & 65 \\ \text { India } & 56 \\ \hline \end{array} $$ Are the data presented in this table an example of cross-section or of time- series data? Explain.

Explain the following. a. Random sample b. Nonrandom sample c. Convenience sample d. Judgment sample e. Quota sample
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