Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.3.66c

Chapter 2, Problem 2.3.66c

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

c. What is the benefit of using a trimmed mean versus using a mean found using all data entries? Explain your reasoning.

Verified step by step guidance

Step 1: Understand the concept of a trimmed mean. A trimmed mean is a measure of central tendency that removes a specified percentage of the smallest and largest data points before calculating the mean. This helps reduce the influence of outliers.

Step 2: The benefit of using a trimmed mean is that it provides a more robust measure of central tendency, especially when the data set contains extreme values (outliers) that could skew the mean calculated using all data entries.

Step 3: By removing the lowest 10% and highest 10% of the data, the trimmed mean focuses on the central portion of the data, which is often more representative of the typical values in the data set.

Step 4: Using a trimmed mean can improve the accuracy of statistical analysis in cases where the data distribution is not symmetric or contains anomalies, as it reduces the impact of extreme values.

Step 5: In summary, the trimmed mean is beneficial because it provides a more reliable and less sensitive measure of central tendency in the presence of outliers or non-normal data distributions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trimmed Mean

A trimmed mean is a statistical measure that involves removing a specified percentage of the lowest and highest values from a data set before calculating the mean. This approach helps to reduce the influence of outliers or extreme values, providing a more robust central tendency measure that better represents the majority of the data.

Outliers

Outliers are data points that significantly differ from other observations in a data set. They can skew the results of statistical analyses, particularly measures like the mean, leading to misleading conclusions. By using a trimmed mean, the impact of these outliers is minimized, resulting in a more accurate reflection of the data's overall trend.

Robust Statistics

Robust statistics are methods that provide reliable results even when assumptions about the data are violated, such as the presence of outliers. The trimmed mean is an example of a robust statistic, as it focuses on the central portion of the data, making it less sensitive to extreme values and thus offering a more stable measure of central tendency.

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Textbook Question

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

ii.

iii.

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Textbook Question

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40

37 49 34 38 43 47 35

c. Each employee in the sample takes a pay cut of \$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

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Textbook Question

Use the relative frequency histogram to describe any patterns with the data.

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175

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Textbook Question

Use the data set and the indicated number of classes to construct

Pulse Rates

Number of classes: 6 Data set: Pulse rates of all students in a class 68 105 95 80 90 100 75 70 84 98 102 70 65 88 90 75 78 94 110 120 95 80 76 108

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Textbook Question

Life Spans of Tires A brand of automobile tire has a mean life span of 35,000 miles, with a standard deviation of 2250 miles. Assume the life spans of the tires have a bell-shaped distribution.

b. The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.

159

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Textbook Question

Extending Concepts

Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360

c. Compare the measures you found in part (b) with those found in part (a). What do you notice?

108

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