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Chapter 3: Problem 65

A sample is given. Indicate whether each option is a possible bootstrap sample from this original sample. Original sample: 17,10,15,21,13,18 . Do the values given constitute a possible bootstrap sample from the original sample? (a) 10,12,17,18,20,21 (b) 10,15,17 (c) 10,13,15,17,18,21 (d) 18,13,21,17,15,13,10 (e) 13,10,21,10,18,17

Short Answer

Expert verified
(c) 10, 13, 15, 17, 18, 21 and (e) 13, 10, 21, 10, 18, 17 are possible bootstrap samples from the original sample.

Step by step solution

01

Check First Option

Check if all values of the first option (10, 12, 17, 18, 20, 21) are present in the original sample (17, 10, 15, 21, 13, 18) and if the option has the same size as the original sample. From the check, it is found that the values 12 and 20 are not present in the original sample, also the size of this option is equal to the original sample size.
02

Check Second Option

Check if all values of the second option (10, 15, 17) are present in the original sample and if the option has the same size as the original sample. From the check, it is found that all values are present in the original sample, but the size of this option is not the same as the original sample size.
03

Check Third Option

Check if all values of the third option (10, 13, 15, 17, 18, 21) are present in the original sample and if the option has the same size as the original sample. From the check, it is found that all values are present in the original sample, and the size of this option is the same as the original sample size
04

Check Fourth Option

Check if all values of the fourth option (18, 13, 21, 17, 15, 13, 10) are present in the original sample and if the option has the same size as the original sample. From the check, it is found that all values are present in the original sample, but the size of this option is not the same as the original sample size
05

Check Fifth Option

Check if all values of the fifth option (13, 10, 21, 10, 18, 17) are present in the original sample and if the option has the same size as the original sample. From the check, it is found that all values are present in the original sample, and the size of this option is also the same as the original size of the sample.

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Key Concepts

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

Sample Size
When working with bootstrap sampling, the concept of sample size plays a crucial role. Bootstrap samples are created by resampling the original sample with replacement, meaning some elements may appear more than once while others may not appear at all.

One essential rule is that the sample size of the bootstrap sample should be identical to the size of the original sample. If the original sample contains six elements, like \(17, 10, 15, 21, 13, 18\), then any potential bootstrap sample must also contain exactly six numbers.

Why is this important?
  • Maintains the statistical properties of the original sample.
  • Ensures consistency in comparisons and analysis.
  • Helps in building an accurate estimation of the population.
Checking this concept was a key part of the exercise, where options were assessed to see if their sizes matched the original sample.
Original Sample
Understanding the original sample is the foundation of bootstrap sampling. It's the initial dataset from which all possible resampling occurs. For instance, the original sample given in this problem is \(17, 10, 15, 21, 13, 18\).

Every bootstrap sample must draw values from this original set. Consider these aspects:
  • Each value in the bootstrap must be present in the original sample.
  • No new values can be introduced that weren't part of the original sample.
  • Values can be repeated, reflecting the replacement method in bootstrapping.
Grasping the original sample helps ensure that each bootstrap sample is relevant and accurately represents potential variations from that specific dataset.
Data Resampling
Data resampling is at the core of the bootstrap method. It's a powerful statistical tool used to estimate the variability of a statistic by randomly sampling with replacement.

Here's how it works:
  • Select a new sample by drawing random numbers from the original sample.
  • Replacement allows the same number to be chosen multiple times.
  • Create many bootstrap samples to analyze different statistical aspects.
Data resampling is crucial for analyzing complex datasets where traditional analytical methods may fall short. It provides a robust way to assess how sample variation might affect conclusions and test statistical hypotheses.

This process was evaluated in the exercise by checking if each option was a legitimate resampling of the original sample while respecting all rules, such as using only existing values within their repeated counts.

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Most popular questions from this chapter

Small Sample Size and Outliers As we have seen, bootstrap distributions are generally symmetric and bell-shaped and centered at the value of the original sample statistic. However, strange things can happen when the sample size is small and there is an outlier present. Use StatKey or other technology to create a bootstrap distribution for the standard deviation based on the following data: \(\begin{array}{llllll}8 & 10 & 7 & 12 & 13 & 8\end{array}\) \(\begin{array}{ll}10 & 50\end{array}\) Describe the shape of the distribution. Is it appropriate to construct a confidence interval from this distribution? Explain why the distribution might have the shape it does.

Daily Tip Revenue for a Waitress Data 2.12 on page 119 describes information from a sample of 157 restaurant bills collected at the First Crush bistro. The data is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a \(90 \%\) confidence interval and one is a \(99 \%\) confidence interval. Interval A: 3.55 to \(4.15 \quad\) Interval B: 3.35 to 4.35 (a) Which one is the \(90 \%\) confidence interval? Which one is the \(99 \%\) confidence interval? (b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both \(90 \%\) and \(99 \%\) confidence. Interpret your results.

Information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. \( \bar{x}_{1}-\bar{x}_{2}=3.0\) and the margin of error for \(95 \%\) confidence is 1.2

A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.36 tells us that 181 of the 273 inductees to the Rock and Roll Hall of Fame have been performers. The data are given in RockandRoll. Using all inductees as your population: (a) Use StatKey or other technology to take many (b) Repeat part (a) using samples of size \(n=20\). random samples of size \(n=10\) and compute the \(\quad\) (c) Repeat part (a) using samples of size \(n=50\). sample proportion that are performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of \(p=0.663\) ? How far away is it?

In estimating the mean score on a fitness exam, we use an original sample of size \(n=30\) and a bootstrap distribution containing 5000 bootstrap samples to obtain a \(95 \%\) confidence interval of 67 to 73 . In Exercises 3.90 to 3.95 , a change in this process is described. If all else stays the same, which of the following confidence intervals \((A, B,\) or \(C)\) is the most likely result after the change: \(\begin{array}{lll}A .66 \text { to } 74 & B .67 \text { to } 73 & \text { C. } 67.5 \text { to } 72.5\end{array}\) Using the data to find a \(99 \%\) confidence interval.
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