Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.1.39

Chapter 3, Problem 3.1.39

Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 0.58%, 0.29%, 0.13%, 0.14%, 0.15%, and 0.19%. Identify the single percentage growth rate that is the same as the six consecutive growth rates by computing the geometric mean of 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019.

Verified step by step guidance

Step 1: Understand the problem. The geometric mean is used to find a single growth rate that represents the average of multiple growth rates. The formula for the geometric mean is: $ \text{Geometric Mean} = \sqrt[n]{x_1 \cdot x_2 \cdot x_3 \cdot \ldots \cdot x_n} $, where $ n $ is the number of values and $ x_1, x_2, \ldots, x_n $ are the values.

Step 2: Identify the values to use in the formula. The given growth rates are 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019. These represent the annual growth factors for the 6-year period.

Step 3: Multiply all the growth factors together. This means calculating $ 1.0058 \cdot 1.0029 \cdot 1.0013 \cdot 1.0014 \cdot 1.0015 \cdot 1.0019 $.

Step 4: Take the 6th root of the product obtained in Step 3. The 6th root can be expressed as raising the product to the power of $ \frac{1}{6} $, i.e., $ \text{Geometric Mean} = (\text{Product})^{1/6} $.

Step 5: Subtract 1 from the geometric mean obtained in Step 4 and multiply by 100 to convert it back to a percentage growth rate. This gives the single percentage growth rate equivalent to the six consecutive growth rates.

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Key Concepts

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

Geometric Mean

The geometric mean is a measure of central tendency that is particularly useful for sets of positive numbers, especially when dealing with rates of change or growth. It is calculated by multiplying all the values together and then taking the nth root of the product, where n is the number of values. This mean is less affected by extreme values compared to the arithmetic mean, making it ideal for financial and economic data.

Rates of Change

Rates of change represent how a quantity changes over time, often expressed as a percentage. In finance, they are crucial for understanding growth rates, such as interest rates or investment returns. The geometric mean is particularly suited for calculating average rates of change over multiple periods, as it accounts for compounding effects, providing a more accurate representation of growth.

Compounding

Compounding refers to the process where the value of an investment increases because the earnings on an investment earn interest as time passes. This concept is fundamental in finance, as it affects how growth rates are calculated over time. The geometric mean effectively captures the impact of compounding by averaging growth rates, allowing for a single growth rate that reflects the cumulative effect of multiple periods.

Related Practice

Textbook Question

What’s Wrong? Education Week magazine published a list consisting of the mean teacher salary in each of the 50 states for a recent year. If we add the 50 means and then divide by 50, we get \$56,479. Is the value of \$56,479 the mean teacher salary for the population of all teachers in the 50 United States? Why or why not?

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Textbook Question

Percentiles. In Exercises 17–20, use the following radiation levels (in W/kg) for 50 different cell phones. Find the percentile corresponding to the given radiation level.

0.48 W/kg

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Textbook Question

Identifying Significant Values with the Range Rule of Thumb. In Exercises 33–36, use the range rule of thumb to identify the limits separating values that are significantly low or significantly high.

U.S. Presidents Based on Data Set 22 “Presidents” in Appendix B, at the time of their first inauguration, presidents have a mean age of 55.2 years and a standard deviation of 6.9 years. Is the minimum required 35-year age for a president significantly low?

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Textbook Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Smart Thermostats Listed below are selling prices (dollars) of smart thermostats tested by Consumer Reports magazine. Are any of the resulting statistics helpful in selecting a smart thermostat for purchase?

250 170 225 100 250 250 130 200 150 250 170 200 180 250

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Textbook Question

The Empirical Rule Based on Data Set 1 “Body Data” in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/) Using the empirical rule, what is the approximate percentage of women with platelet counts

a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

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Textbook Question

Foot Lengths Based on Data Set 9 “Foot and Height” in Appendix B, adult males have foot lengths with a mean of 27.32 cm and a standard deviation of 1.29 cm. Is the adult male foot length of 30 cm significantly low, significantly high, or neither? Explain.