Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.4.54c

Chapter 2, Problem 2.4.54c

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.

Verified step by step guidance

Step 1: Understand the problem. The original data set represents annual salaries in thousands of dollars. Each employee takes a pay cut of \$2000, which is equivalent to subtracting 2 from each value in the data set (since the salaries are in thousands). This will create a new data set.

Step 2: Calculate the sample mean for the original data set. The formula for the sample mean is:

x̄ = \sum x

_in, where

x

_i represents each data point and

n

is the number of data points. Compute this for the original data set.

Step 3: Adjust the sample mean for the revised data set. Since subtracting a constant from each data point shifts the entire data set by that constant, the new sample mean is simply the original sample mean minus 2.

Step 4: Calculate the sample standard deviation for the original data set. The formula for the sample standard deviation is:

s = \sqrt{\sum (x}

_i - x̄)2n - 1. Compute this for the original data set.

Step 5: Adjust the sample standard deviation for the revised data set. Subtracting a constant from each data point does not affect the spread of the data, so the sample standard deviation remains the same as the original data set.

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Key Concepts

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

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the values and dividing by the number of observations. In this context, it represents the average salary of employees after a uniform pay cut. The mean is sensitive to changes in data, so a consistent adjustment, like a pay cut, will directly affect its value.

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of values. It indicates how much individual salaries deviate from the sample mean. When all salaries are uniformly adjusted (e.g., a pay cut), the standard deviation remains unchanged, as the relative differences between salaries do not alter.

Uniform Transformation

A uniform transformation refers to applying the same change to all data points in a dataset. In this case, reducing each salary by $2000 is a uniform transformation. This type of adjustment affects the mean but not the standard deviation, as it does not alter the distribution of the data, only shifts it.

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Related Practice

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

Using and Interpreting Concepts

Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

(b) find the interquartile range

56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

138

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Textbook Question

Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula

MAD = Σ |x − x̄| / n.

b. Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16.

198

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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

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.

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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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