Question

The geometric mean (GM) is defined as the nth root of the product of n values. The formula is $$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$ The geometric mean of 4 and 16 is $$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$ The geometric mean of $1,3,$ and 9 is $$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$ The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown. $$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$ or $$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$ His salary is $120 \%$ at the end of the first year and $110 \%$ at the end of the second year. This is equivalent to an average of $14.89 \%,$ since $114.89 \%-100 \%=$ $14.89 \% .$ $$\begin{array}{l}{\text { This answer can also be shown by assuming that }} \\ {\text { the person makes } \$ 10,000 \text { to start and receives two }} \\ {\text { raises of } 20 \% \text { and } 10 \% .}\end{array}$$ $$\begin{array}{l}{\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000} \\ {\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200}\end{array}$$ $$\text { His total salary raise is } \$ 3200 . \text { This total is equivalent to }$$ $$\begin{array}{l}{\$ 10,000 \cdot 14.89 \%=\$ 1489.00} \\ {\$ 11,489 \cdot 14.89 \%=\$ 1710.71}\end{array}$$ $${\$ 3199.71} \approx \$ 3200$$ Find the geometric mean of each of these. a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%. b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.

Name: Problem 37, Chapter 3: Elementary Statistics a Step by Step Approach
Uploaded: 2020-07-13T02:44:17-08:00
Duration: 7 min 14 s
Channel: Rebecca Wickersham

   The geometric mean (GM) is defined as the nth root of the product of n values. The formula is
$$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$
The geometric mean of 4 and 16 is
$$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$
The geometric mean of $1,3,$ and 9 is
$$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$
The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown.
$$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$
or
$$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$
His salary is $120 \%$ at the end of the first year and $110 \%$ at the end of the second year. This is equivalent to an average of $14.89 \%,$ since $114.89 \%-100 \%=$ $14.89 \% .$
$$\begin{array}{l}{\text { This answer can also be shown by assuming that }} \\ {\text { the person makes } \$ 10,000 \text { to start and receives two }} \\ {\text { raises of } 20 \% \text { and } 10 \% .}\end{array}$$
$$\begin{array}{l}{\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000} \\ {\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200}\end{array}$$
$$\text { His total salary raise is } \$ 3200 . \text { This total is equivalent to }$$
$$\begin{array}{l}{\$ 10,000 \cdot 14.89 \%=\$ 1489.00} \\ {\$ 11,489 \cdot 14.89 \%=\$ 1710.71}\end{array}$$
$${\$ 3199.71} \approx \$ 3200$$
Find the geometric mean of each of these.
a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%.
b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%.
c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%.
d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Chapter 3, Problem 37

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Solved by Expert Rebecca Wickersham

07/13/2020

Step 1

The geometric mean is the nth root of the product of n values. In this case, we are dealing with percentages, so we need to convert the percentages to decimals before we can use the formula. Show more…

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The geometric mean (GM) is defined as the nth root of the product of n values. The formula is $$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$ The geometric mean of 4 and 16 is $$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$ The geometric mean of $1,3,$ and 9 is $$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$ The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown. $$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$ or $$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$ His salary is $120 \%$ at the end of the first year and $110 \%$ at the end of the second year. This is equivalent to an average of $14.89 \%,$ since $114.89 \%-100 \%=$ $14.89 \% .$ $$\begin{array}{l}{\text { This answer can also be shown by assuming that }} \\ {\text { the person makes } \$ 10,000 \text { to start and receives two }} \\ {\text { raises of } 20 \% \text { and } 10 \% .}\end{array}$$ $$\begin{array}{l}{\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000} \\ {\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200}\end{array}$$ $$\text { His total salary raise is } \$ 3200 . \text { This total is equivalent to }$$ $$\begin{array}{l}{\$ 10,000 \cdot 14.89 \%=\$ 1489.00} \\ {\$ 11,489 \cdot 14.89 \%=\$ 1710.71}\end{array}$$ $${\$ 3199.71} \approx \$ 3200$$ Find the geometric mean of each of these. a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%. b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.

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Key Concepts

Compound Growth

Compound growth occurs when a quantity grows by a certain percentage over several periods, with each period’s growth applying to the new total rather than the original amount. The geometric mean captures this effect accurately by reflecting the multiplicative accumulation of growth rates over time.

Conversion of Percentage Changes for Growth Calculations

When applying the geometric mean to understand average increases in values such as salary raises, stock prices, or cost increments, it is crucial to convert the percentage changes into their respective growth factors. This conversion allows for a straightforward multiplication of terms, which can then be used to calculate an accurate average growth rate that reflects the compound nature of the changes.

Geometric Mean

The geometric mean is a type of average that is calculated by taking the nth root of the product of n values. It is particularly useful for sets of positive numbers where the values are interrelated multiplicatively or when dealing with exponential growth, as it gives a more accurate representation of the central tendency for such data compared to the arithmetic mean.

Multiplicative Factors

In many practical applications, especially when dealing with growth or percentage changes, raw percentages are first converted to multiplicative factors (for example, a 20% increase becomes a factor of 1.20). This conversion is essential because it accurately represents the compounded impact of successive changes.

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Transcript

00:01 So in this problem, we're going to be looking at the geometric mean, and we're given the new formula for geometric mean.

00:08 So it equals the nth root of x sub 1 times x of 2 times all the way until we get to x sub in.

00:18 So we're dealing with percentages with several of these examples.

00:24 So on part a, we're looking at growth rates for life insurance.

00:31 And we're given for three years, we've grown 35%, 24%, and 18%.

00:42 So what that really means is instead of in a single year, having life insurance be 100%.

00:50 For this first year in these three years, it would be 135%.

00:56 Then for that next year it would be 124%, and then 118%.

01:04 But when we plug these values in, i'm going to convert them to decimals.

01:09 So that just means we're going to move our percent.

01:12 So we're going to do 1 .35, 1 .24, and 1 .18 when we plug this in.

01:23 So we need to know that we have one, two, three values.

01:27 So our in here is three.

01:31 So when we plug this in, we're going to have the cube root.

01:36 And then inside the radical, we're going to have 1 .35 times 1 .24 times 1 .18.

01:50 And when we plug this in, we will get 1 .2547.

01:57 So if we convert this back to a percent, we would just move the decimal two times to the right.

02:05 So we have 125 .47%.

02:12 So what we're going to do here is subtract the 100 % that we're going to do here is subtract the 100 % that we're would originally start with, and we're left with 25 .47%.

02:29 So that is our answer for part a, the geometric mean 435, 24, and 18 % growth rate.

02:40 Then for part b, we have a raise in salary, and we have an 8, 6, 4, and 5 % raise.

02:51 So i'm going to write those out here.

02:55 8, 6, 4, and 5.

02:59 So if we use the same idea here, that's not just 100%.

03:04 You're getting paid 108%.

03:07 So if we take 108 and convert that to a decimal, that would really be 1 .08.

03:17 So we'll do the same thing for 6%, 4 % and 5%...

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