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Chapter 2: Problem 19

For each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning. (a) Number of pets per household. (b) Distance to work, i.e. number of miles between work and home. (c) Heights of adult males.

Short Answer

Expert verified
(a) Right-skewed; median and IQR preferred. (b) Right-skewed; median and IQR preferred. (c) Symmetric; mean and standard deviation preferred.

Step by step solution

01

Analyzing Number of Pets per Household

In most households, people have either no pets or a small number of pets. Very few households have a large number of pets, often leading to a distribution that is right-skewed. In a right-skewed distribution, the median is a better measure of central tendency because it is less affected by extreme values. Furthermore, the interquartile range (IQR) is preferred over standard deviation to represent variability, as it is more robust to outliers and skewed data.
02

Evaluating Distance to Work

The distance to work from home can vary greatly; however, most people live within a certain range of their workplace, resulting in a few people traveling very long distances. This results in a right-skewed distribution. Therefore, the median is the better measure of central tendency as it is less influenced by long distances. Similarly, the IQR is favored over standard deviation for representing variability, due to its robustness against the skewness.
03

Assessing Heights of Adult Males

Heights generally follow a normal distribution, meaning they are symmetric. In symmetric distributions, the mean is typically the best measure of central tendency, as it duly represents the typical observation. When the distribution is symmetric, the standard deviation serves as an appropriate measure to represent variability since it's not affected by skewness, unlike in skewed distributions.

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Key Concepts

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

Symmetric Distribution
In statistical terms, a symmetric distribution is one where the left and right sides of the graph are mirror images of each other. This means that the data is evenly distributed around a central point.
The most common example of a symmetric distribution is the normal distribution, often referred to as a bell curve. In this type of distribution, the mean, median, and mode are all equal and located at the center of the distribution.
When working with symmetric distributions, various measures of central tendency are used to summarize the data accurately.
  • The mean is typically used as the best measure of central tendency in a symmetric distribution because it takes into account all data points and provides a balanced average.
  • The standard deviation is the preferred measure of variability, as it quantifies the amount of variation or dispersion in the data.
Symmetric distributions, being balanced, provide a clear understanding of the dataset's spread and central tendency.
Right-Skewed Distribution
A right-skewed distribution, also known as positively skewed, is characterized by a longer tail on the right side. This means a larger number of extremes or outliers affect the right part of the graph.
Common examples include income distribution, where fewer people earn very high incomes, causing a skew to the right. In such distributions, measures of central tendency can be heavily influenced by outliers.
  • The median is preferred over the mean, as it is less sensitive to extreme values and reliably represents the dataset's midpoint.
  • For variability, the interquartile range (IQR) is often used, as it is robust to outliers and provides a clear picture of the data's spread.
A right-skewed distribution requires careful interpretation of data, especially when extremes play a significant role.
Measures of Central Tendency
Measures of central tendency provide key insights into where the middle of a dataset lies, effectively summarizing the entire data set.
The three primary measures are:
  • Mean: Often referred to as the average. It is computed by adding all data points and dividing by the number of points. Ideal for symmetric distributions.
  • Median: The middle value when data is arranged in order. It is useful in skewed distributions because it is not swayed by extreme values.
  • Mode: The most frequently occurring data point. In some datasets, particularly categorical data, it provides meaningful insight.
Each measure provides a unique perspective of the central point of the data. Depending on the distribution type, choosing the right measure is crucial for accurate data analysis.
Variability Representation
Variability representation indicates how much the data points in a set differ from each other and the central tendency.
Two standard measures are used to showcase this:
  • Standard Deviation (SD): It reflects the average distance of each data point from the mean, suitable for symmetric distributions as it presumes normality and even spread.
  • Interquartile Range (IQR): This measures the spread of the middle 50% of the data, calculated as the difference between the 25th and 75th percentiles, making it resistant to skewness and outliers.
Choosing the appropriate measure of variability is important, as it significantly influences the interpretation of data distributions, helping to understand deviation, spread, and consistency.

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Most popular questions from this chapter

A factory quality control manager decides to investigate the percentage of defective items produced each day. Within a given work week (Monday through Friday) the percentage of defective items produced was \(2 \%, 1.4 \%, 4 \%, 3 \%, 2.2 \%\). (a) Calculate the mean for these data. (b) Calculate the standard deviation for these data, showing each step in detail.

Rosiglitazone is the active ingredient in the controversial type 2 diabetes medicine Avandia and has been linked to an increased risk of serious cardiovascular problems such as stroke, heart failure, and death. A common alternative treatment is pioglitazone, the active ingredient in a diabetes medicine called Actos. In a nationwide retrospective observational study of 227,571 Medicare beneficiaries aged 65 years or older, it was found that 2,593 of the 67,593 patients using rosiglitazone and 5,386 of the 159,978 using pioglitazone had serious cardiovascular problems. These data are summarized in the contingency table below. \({ }^{49}\) (a) Determine if each of the following statements is true or false. If false, explain why. Be careful: The reasoning may be wrong even if the statement's conclusion is correct. In such cases, the statement should be considered false. i. Since more patients on pioglitazone had cardiovascular problems \((5,386\) vs. 2,593\(),\) we can conclude that the rate of cardiovascular problems for those on a pioglitazone treatment is higher. ii. The data suggest that diabetic patients who are taking rosiglitazone are more likely to have cardiovascular problems since the rate of incidence was \((2,593 / 67,593=0.038) 3.8 \%\) for patients on this treatment, while it was only \((5,386 / 159,978=0.034) 3.4 \%\) for patients on pioglitazone. iii. The fact that the rate of incidence is higher for the rosiglitazone group proves that rosiglitazone causes serious cardiovascular problems. iv. Based on the information provided so far, we cannot tell if the difference between the rates of incidences is due to a relationship between the two variables or due to chance. (b) What proportion of all patients had cardiovascular problems? (c) If the type of treatment and having cardiovascular problems were independent, about how many patients in the rosiglitazone group would we expect to have had cardiovascular problems? (d) We can investigate the relationship between outcome and treatment in this study using a randomization technique. While in reality we would carry out the simulations required for randomization using statistical software, suppose we actually simulate using index cards. In order to simulate from the independence model, which states that the outcomes were independent of the treatment, we write whether or not each patient had a cardiovascular problem on cards, shuffled all the cards together, then deal them into two groups of size 67,593 and \(159,978 .\) We repeat this simulation 1,000 times and each time record the number of people in the rosiglitazone group who had cardiovascular problems. Use the relative frequency histogram of these counts to answer (i)-(iii). i. What are the claims being tested? ii. Compared to the number calculated in part (b), which would provide more support for the alternative hypothesis, more or fewer patients with cardiovascular problems in the rosiglitazone group? iii. What do the simulation results suggest about the relationship between taking rosiglitazone and having cardiovascular problems in diabetic patients?

For each part, compare distributions (1) and (2) based on their means and standard deviations. You do not need to calculate these statistics; simply state how the means and the standard deviations compare. Make sure to explain your reasoning. Hint: It may be useful to sketch dot plots of the distributions. (a) (1) 3,5,5,5,8,11,11,11,13(2) 3,5,5,5,8,11,11,11,20 (b) \((1)-20,0,0,0,15,25,30,30\) (2) -40,0,0,0,15,25,30,30 (c) (1) 0,2,4,6,8,10(2) 20,22,24,26,28,30 (d) (1) 100,200,300,400,500(2) 0,50,300,550,600

Workers at a particular mining site receive an average of 35 days paid vacation, which is lower than the national average. The manager of this plant is under pressure from a local union to increase the amount of paid time off. However, he does not want to give more days off to the workers because that would be costly. Instead he decides he should fire 10 employees in such a way as to raise the average number of days off that are reported by his employees. In order to achieve this goal, should he fire employees who have the most number of days off, least number of days off, or those who have about the average number of days off?

The average on a history exam (scored out of 100 points) was \(85,\) with a standard deviation of 15 . Is the distribution of the scores on this exam symmetric? If not, what shape would you expect this distribution to have? Explain your reasoning.
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