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Chapter 3: Problem 135

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: \(8 \quad 10\) 72 \(13 \quad 8\) \(\begin{array}{ll}10 & 50\end{array}\) 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 shape of the bootstrap distribution might be skewed or irregular due to the outlier in the data set. Depending on the observed skewness or irregularity, it might not be appropriate to construct a confidence interval from this distribution. The presence of an outlier (72) in the small data set is what causes the particular shape of the distribution - being substantially distant from the other data points, it significantly influences the standard deviation whenever it is included in a sample.

Step by step solution

01

Generating the Bootstrap Distribution

Firstly, with technology tools like StatKey or others reflecting statistical modelling, generate a bootstrap distribution for the standard deviation of the given data: \(8, 10, 72, 13, 8, 10, 50\). The bootstrap distribution is created by taking numerous samples from the data, with replacement, and calculating the statistic (in this case, standard deviation) from each sample.
02

Observing the Shape of the Distribution

Next is to observe the shape or pattern of the bootstrap distribution. It might not follow a symmetric, bell-shaped curve, mainly due to the presence of an outlier in the small data set.
03

Discussing the Appropriateness of a Confidence Interval

Analyze whether it is appropriate to construct a confidence interval from this distribution. If the bootstrap distribution is not symmetric, or if it is substantially skewed, it may not be suitable to form a confidence interval from it since standard methods often require the sampling distribution to be approximately symmetric.
04

Understanding the Shape of Distribution

Explain why the distribution might have the shape it does. This can largely be due to the presence of an outlier (72) in the data set. Being a single data point that lies far from the other values, the outlier would have a significant impact on the standard deviation whenever it is included in a sample, thereby influencing the shape of the bootstrap distribution.

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Key Concepts

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

Understanding Standard Deviation
The standard deviation is a fundamental concept in statistics, representing the amount of variation or dispersion in a set of data. When you calculate the standard deviation, you're essentially finding out how spread out the numbers are from the mean (average).

Imagine each data point as a spot on a line. If all the data points cluster closely around the mean, the standard deviation is small. Conversely, if the data points are spread out widely, the standard deviation is large.
  • Calculation: To compute standard deviation, first calculate the mean of the data set, then find the squared differences from the mean, and finally take the square root of the average of these squared differences.
  • Role in Statistics: Standard deviation is pivotal when analyzing data sets as it helps assess how typical or unusual a score is. A high standard deviation indicates more variability in the data, whereas a low standard deviation suggests data values are more similar to the mean.
Standard deviation helps you understand data variability, an essential factor when interpreting a bootstrap distribution.
Confidence Interval Basics
A confidence interval provides a range of values that is likely to contain the population parameter with a certain level of confidence, often expressed as a percentage like 95% or 99%. It acts like a net that captures the true mean or proportion we are estimating for a population.

To construct a confidence interval, you usually need to know the mean of your sample, the standard deviation, and the sample size. The confidence interval then spreads a certain distance from the sample mean based on the statistical formula which includes the standard deviation.
  • Interpreting Confidence Intervals: A 95% confidence interval means that if you were to take 100 different samples and build a confidence interval from each, about 95 of the intervals will contain the true population mean.
  • Symmetry Requirement: For typical confidence interval calculations using standard statistical methods, it is necessary that the data follows a roughly symmetric distribution. However, in the case of our small data set with an outlier, this symmetry might be disrupted.
Confidence intervals are a powerful tool in statistics, providing a way to make predictions about populations based on sample data.
Dealing with Outliers in Data
Outliers are observations that deviate significantly from the other data points. In our exercise, the number 72 serves as an outlier within the data set, far exceeding most of the other values.

When analyzing data, outliers can significantly impact the results. They can skew the data distribution, affect the calculation of the mean, and inflate the standard deviation.
  • Impact on Shape: The inclusion of outliers can result in a skewed distribution especially when the sample size is small, as seen with our bootstrap distribution. This impacts our assumptions and calculations, like standard deviation and confidence intervals.
  • Handling Outliers: Various strategies exist to handle outliers, such as removing them (if justified), adjusting their values, or using statistical methods robust against outliers.
Understanding the role of outliers is crucial, particularly when evaluating the appropriateness of a confidence interval or other statistical measures.

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Most popular questions from this chapter

Use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. What Proportion Go to Maximum Pressure? We see that 75 of the 264 people in the study allowed the pressure to reach its maximum level of \(300 \mathrm{mmHg}\), without ever saying that the pain was too much (MaxPressure=yes). Use this information to estimate the proportion of people who would allow the pressure to reach its maximum level.

Correlation between age and heart rate for patients admitted to an Intensive Care Unit. Data from the 200 patients included in the file ICUAdmissions gives a correlation of 0.037 .

Use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. Does Exertion Boost Pain Tolerance? Use the pain tolerance ratings after the activity to estimate the difference in mean pain tolerance between those who just completed a high exertion activity and those who completed a low exertion activity.

Exercises 3.71 to 3.73 consider the question (using fish) of whether uncommitted members of a group make it more democratic. It has been argued that individuals with weak preferences are particularly vulnerable to a vocal opinionated minority. However, recent studies, including computer simulations, observational studies with humans, and experiments with fish, all suggest that adding uncommitted members to a group might make for more democratic decisions by taking control away from an opinionated minority. \({ }^{36}\) In the experiment with fish, golden shiners (small freshwater fish who have a very strong tendency to stick together in schools) were trained to swim toward either yellow or blue marks to receive a treat. Those swimming toward the yellow mark were trained more to develop stronger preferences and became the fish version of individuals with strong opinions. When a minority of five opinionated fish (wanting to aim for the yellow mark) were mixed with a majority of six less opinionated fish (wanting to aim for the blue mark), the group swam toward the minority yellow mark almost all the time. When some untrained fish with no prior preferences were added, however, the majority opinion prevailed most of the time. \({ }^{37}\) Exercises 3.71 to 3.73 elaborate on this study. What Is the Effect of Including Some Indifferent Fish? In the experiment described above under Fish Democracies, the schools of fish in the study with an opinionated minority and a less passionate majority picked the majority option only about \(17 \%\) of the time. However, when groups also included 10 fish with no opinion, the schools of fish picked the majority option \(61 \%\) of the time. We want to estimate the effect of adding the fish with no opinion to the group, which means we want to estimate the difference in the two proportions. We learn from the study that the standard error for estimating this difference is about \(0.14 .\) Define the parameter we are estimating, give the best point estimate, and find and interpret a \(95 \%\) confidence interval. Is it plausible that adding indifferent fish really has no effect on the outcome?

Do You Find Solitude Distressing? "For many people, being left alone with their thoughts is a most undesirable activity," says a psychologist involved in a study examining reactions to solitude. \({ }^{26}\) In the study, 146 college students were asked to hand over their cell phones and sit alone, thinking, for about 10 minutes. Afterward, 76 of the participants rated the experience as unpleasant. Use this information to estimate the proportion of all college students who would find it unpleasant to sit alone with their thoughts. (This reaction is not limited to college students: in a follow-up study involving adults ages 18 to 77 , a similar outcome was reported.) (a) Give notation for the quantity being estimated, and define any parameters used. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated, given that the margin of error for the estimate is \(8 \%\).
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