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Chapter 3: Problem 116

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.

Short Answer

Expert verified
The bootstrap distribution's shape is likely to be skewed due to the outlier in the small sample data provided. The presence of an outlier has influenced the distribution. This makes it inappropriate to form a confidence interval as the data is not representative and would result in an inaccurate interval. The distribution shape is formed this way primarily because of the outlier.

Step by step solution

01

Understand the Sample Data

First, understand what the given data represents. It is obvious that two data groups are provided. The first data group contains: 8, 10, 7, 12, 13, and 8, and the second data group contains: 10 and 50. In both datasets, we can notice that 10 is a common number. However, in the second data group, 50 is an outlying number when compared with the rest of the numbers
02

Create Bootstrap Distribution

Use StatKey or another technology to create a bootstrap distribution. The result will vary depending on the number of resamples selected but it should give a representative distribution of the data provided.
03

Describe the Shape of the Distribution

After creating the bootstrap distribution, describe how it looks. The presence of an outlier, especially in a small sample size, can severely skew the distribution, leading to a shape that might be significantly different than a typical bell-curve. It is important to notice if the distribution is skewed towards the outlying number.
04

Determine the Appropriateness of Confidence Interval

Last but not least, determine if it is appropriate to construct a confidence interval from this distribution. Remember, the presence of outliers may have a significant impact on the confidence interval. Therefore, if the distribution is too skewed, it would not be accurate or reliable to construct a confidence interval.
05

Explain the Shape of the Distribution

In the end, explain why the distribution might have the shape it does. It is likely due to the presence of the outlier in the small sample size. This can heavily influence the distribution, pulling the mean towards it and creating a skewed distribution.

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Key Concepts

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

Small Sample Size
When working with data, the size of the sample you use can significantly influence your results. In the context of bootstrap sampling, where you repeatedly sample from your data to gain insights, a small sample size can lead to misleading outcomes.
Small samples are more susceptible to fluctuations caused by individual data points, especially when those points are unusual or extreme.

With larger samples, the influence of any single point is diluted, giving you a more stable and reliable view of the overall population. However, in smaller samples, like in our example, each data point has a greater impact.
This greater impact can cause the bootstrap distribution to be less reliable or representative of the broader situation.
  • Small sample sizes increase variability.
  • Individual data points have more influence.
  • Can lead to skewed distributions.
When analyzing data, it's crucial to consider sample size to ensure your interpretations are sound and your conclusions are valid.
Outliers
Outliers are data points that significantly differ from other observations. In a small sample size, like the one provided, even a single outlier can dramatically affect the results of your analysis. They can skew the distribution, making it difficult to interpret the usual statistical measures correctly.
In bootstrap sampling, this effect can be magnified. Since bootstrapping involves repeated sampling, each time the outlier is included, it can shift the distribution disproportionately.

Outliers can occur for several reasons: measurement error, data entry error, or natural variance. Regardless of the cause, their presence requires careful consideration.
  • Outliers can skew distributions.
  • They require consideration in analysis.
  • Can be due to error or natural variance.
In analyzing data, identifying and understanding the influence of outliers is crucial for ensuring the accuracy and reliability of your conclusions.
Confidence Interval
A confidence interval provides a range within which we expect a population parameter to lie, based on our sample data. However, when constructing a confidence interval, the shape of the bootstrap distribution is crucial.
Ideally, for the most reliable confidence intervals, the distribution should be symmetric and bell-shaped.

However, with small sample sizes and the presence of outliers, as in our example, the distribution can become skewed. This skewness can result in a confidence interval that does not accurately reflect the true parameter of the population because it's heavily influenced by the outlier or small sample variability.
  • Confidence intervals provide a range for population parameters.
  • The shape of the bootstrap distribution affects interval reliability.
  • Skewed distributions can lead to inaccurate intervals.
To ensure your confidence intervals are meaningful and representative, it's essential to consider both the size of your sample and the presence of outliers in the data.

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Most popular questions from this chapter

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

Give information about the proportion of a sample that agrees with a certain statement. Use StatKey or other technology to estimate the standard error from a bootstrap distribution generated from the sample. Then use the standard error to give a \(95 \%\) confidence interval for the proportion of the population to agree with the statement. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. In a random sample of 400 people, 112 agree and 288 disagree.

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, \(^{13}\) 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-at-a-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.

How Often Does the Fish Majority Win? In a school of fish with a minority of strongly opinionated fish wanting to aim for the yellow mark and a majority of less passionate fish wanting to aim for the blue mark, as described under Fish Democracies above, a \(95 \%\) confidence interval for the proportion of times the majority wins (they go to the blue mark) is 0.09 to \(0.26 .\) Interpret this confidence interval. Is it plausible that fish in this situation are equally likely to go for either of the two options?

Predicting Election Results Throughout the US presidential election of \(2012,\) polls gave regular updates on the sample proportion supporting each candidate and the margin of error for the estimates. This attempt to predict the outcome of an election is a common use of polls. In each case below, the proportion of voters who intend to vote for each candidate is given as well as a margin of error for the estimates. Indicate whether we can be relatively confident that candidate A would win if the election were held at the time of the poll. (Assume the candidate who gets more than \(50 \%\) of the vote wins.) (a) Candidate A: 54\% Candidate B: \(46 \%\) Margin of error: \(\pm 5 \%\) (b) Candidate A:52\% Candidate B: \(48 \%\) Margin of error: \(\pm 1 \%\) \(\begin{array}{llll}\text { (c) Candidate A: 53\% } & \text { Candidate B: } 47 \% & \text { Margin }\end{array}\) of error: \(\pm 2 \%\) (d) Candidate A: 58\% Candidate B: 42\% Margin of error: \(\pm 10 \%\)
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