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Chapter 3: Problem 76

A sample is given. Indicate whether each option is a possible bootstrap sample from this original sample. Original sample: 85,72,79,97,88 . Do the values given constitute a possible bootstrap sample from the original sample? (a) 79,79,97,85,88 (b) 72,79,85,88,97 (c) 85,88,97,72 (d) 88,97,81,78,85 (e) 97,85,79,85,97 (f) 72,72,79,72,79

Short Answer

Expert verified
The possible bootstrap samples are Options (a), (b), (e), and (f). Options (c) and (d) are not possible bootstrap samples.

Step by step solution

01

Analysis of Option (a)

Check each number in this option (79,79,97,85,88). All these numbers are present in the original data set (85,72,79,97,88). Also, the size of the set is the same. Therefore, option (a) is a possible bootstrap sample.
02

Analysis of Option (b)

Check each number in option (72,79,85,88,97). All these numbers are present in the original data set (85,72,79,97,88) and the set鈥檚 size is the same. Therefore, option (b) is a possible bootstrap sample.
03

Analysis of Option (c)

Check each number in option (85,88,97,72). Despite all numbers being present in the original data set (85,72,79,97,88), the size of the set is not the same, as it lacks one number. Therefore, option (c) is not a possible bootstrap sample.
04

Analysis of Option (d)

Check each number in option (88,97,81,78,85). The numbers 81 and 78 from this option are not present in the original data set (85,72,79,97,88), despite the set size being the same. Therefore, option (d) is not a possible bootstrap sample.
05

Analysis of Option (e)

Check each number in option (97,85,79,85,97). All these numbers are present in the original data set (85,72,79,97,88) and the set鈥檚 size is the same. Therefore, option (e) is a possible bootstrap sample.
06

Analysis of Option (f)

Check each number in option (72,72,79,72,79). All these numbers are present in the original data set (85,72,79,97,88). The size of the set is the same. Therefore, option (f) is a possible bootstrap sample.

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Key Concepts

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

Understanding Statistical Analysis
Statistical analysis is a broad and crucial field in data science and research. It involves collecting, reviewing, and interpreting quantitative data to discover trends, patterns, or relationships. At its core, statistical analysis helps us make informed decisions based on data, rather than assumptions or guesses.
Key elements of statistical analysis include:
  • Descriptive statistics: Summarizes data to understand its basic features, such as mean, median, and standard deviation.
  • Inferential statistics: Makes predictions or inferences about a larger population based on a sample.

Statistical analysis can also help determine the reliability and validity of the data, which is crucial when making predictions or testing hypotheses. Understanding these methodologies allows researchers and analysts to approach problems with structured insights derived from well-interpreted data.
Exploring Sampling Techniques
Sampling techniques are methods used to select a subset of individuals or observations from a larger population. These methods aim to ensure that the sample accurately represents the population, which is essential for making valid inferences.
Different sampling techniques include:
  • Random sampling: Every individual or observation has an equal chance of selection. This method minimizes bias and is the simplest form for fair representation.
  • Stratified sampling: The population is divided into strata, or groups, based on a specific characteristic, and samples are drawn from each stratum. This helps maintain balance and precision in the results.
  • Bootstrap sampling: This is a resampling method where samples are drawn with replacement from the original dataset. It allows estimation of the distribution of a statistic by generating multiple copies of the sample, which can help assess variation and bias in statistics.

These techniques help ensure that any statistical analysis conducted is based on a well-formed representation of the population. Choosing the right sampling method is crucial for the accuracy and credibility of any data-driven conclusions.
Comprehending Data Sets
A data set is essentially a collection of data, often presented in tabular form, where each column represents a particular variable, and each row corresponds to an individual or observation within the study. Data sets are fundamental components in carrying out statistical analyses and deriving meaningful insights.
Key aspects of data sets include:
  • Size: Refers to the number of observations or entries. Larger data sets typically provide more reliable and comprehensive insights.
  • Variables: These are the individual features or characteristics that are recorded for each observation. More variables provide a richer context for analysis.
  • Quality: The accuracy and completeness of data within the set. High-quality data is essential for accurate analytics and problem-solving.

Understanding and working with data sets is critical for any statistical analysis. Good data sets enable analysts to identify patterns and trends, test hypotheses, and come up with data-driven solutions to complex problems.

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Most popular questions from this chapter

What Proportion of Adults and Teens Text Message? A study of \(n=2252\) adults age 18 or older found that \(72 \%\) of the cell phone users send and receive text messages. \({ }^{15}\) A study of \(n=800\) teens age 12 to 17 found that \(87 \%\) of the teen cell phone users send and receive text messages. What is the best estimate for the difference in the proportion of cell phone users who use text messages, between adults (defined as 18 and over) and teens? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Performers in the Rock and Roll Hall of Fame From its founding through \(2015,\) the Rock and Roll Hall of Fame has inducted 303 groups or individuals, and 206 of the inductees have been performers while the rest have been related to the world of music in some way other than as a performer. The full dataset is available in RockandRoll. (a) What proportion of inductees have been performers? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion who were performers for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

In Exercises 3.51 to 3.56 , 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. $$ \hat{p}=0.32 \text { and the standard error is } 0.04 \text { . } $$

Mix It Up for Better Learning In preparing for a test on a set of material, is it better to study one topic at a time or to study topics mixed together? In one study, \(^{14}\) a sample of fourth graders were taught four equations. Half of the children learned by studying repeated examples of one equation at a time, while the other half studied mixed problem sets that included examples of all four types of calculations grouped together. A day later, all the students were given a test on the material. The students in the mixed practice group had an average grade of \(77,\) while the students in the one-ata-time group had an average grade of \(38 .\) What is the best estimate for the difference in the average grade between fourth-grade students who study mixed problems and those who study each equation independently? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

In a random sample of 1000 people, 382 people agree, 578 disagree, and 40 are undecided.
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