/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 An auto dealer pays an average o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 4

An auto dealer pays an average of $$\$ 8750$$ with a standard deviation of $$\$ 1200$$ for used car trade-ins and sells new cars for an average of $$\$ 28,500$$ with a standard deviation of $$\$ 3100 .$$ Assuming independence of trade-in and new car prices for a customer, what is the standard deviation of the revenue the dealer should expect to make if a customer trades in a used car and buys a new one? (A) \(\sqrt{3100+1200}\) dollars (B) \(\sqrt{3100+1200}\) dollars (C) \(\sqrt{3100^{2}+1200^{2}}\) dollars (D) \(\sqrt{3100^{2}+1200^{2}}\) dollars (E) \(3100-1200\) dollars

Short Answer

Expert verified
The standard deviation of the revenue is \( \sqrt{3100^{2}+1200^{2}} \) dollars, approximately 3323.6 dollars.

Step by step solution

01

Identify Given Information

The problem gives the mean and standard deviation for two independent variables: - Trade-in price: mean = 8750, standard deviation = 1200 - New car price: mean = 28500, standard deviation = 3100We need to find the standard deviation of the revenue, which is the difference between these two prices.
02

Understand the Formula

The standard deviation of the difference (or sum) of two independent variables is calculated using the formula: \[ \text{SD(revenue)} = \sqrt{\text{(SD(trade-in))}^2 + \text{(SD(new car))}^2} \]
03

Substitute in the Values

Plug in the given standard deviations into the formula: \[ \text{SD(revenue)} = \sqrt{(1200)^2 + (3100)^2} \]
04

Calculate the Squares

Compute the squares of each standard deviation: \[ (1200)^2 = 1440000 \] \[ (3100)^2 = 9610000 \]
05

Sum the Squares

Add the squares of the standard deviations together: \[ 1440000 + 9610000 = 11050000 \]
06

Find the Square Root

Take the square root of the sum to obtain the standard deviation of the revenue: \[ \sqrt{11050000} \approx 3323.6 \]
07

Choose the Correct Option

Looking at the provided choices, the correct option is (C) \( \sqrt{3100^{2}+1200^{2}} \) dollars.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Independent Variables
In this context, the term 'independent variables' refers to the trade-in value of a used car and the price of a new car. Independent variables are variables whose variations do not depend on one another. Here, it means that the trade-in value of a used car is not affected by the new car's price and vice versa.
Revenue Standard Deviation
The revenue standard deviation helps an auto dealer understand the variability or risk associated with the revenue made from selling a new car and taking a trade-in. In mathematical terms, standard deviation is a measure of the dispersion or spread of a set of values. A higher standard deviation means more variability. In this problem, the revenue comes from the combination of a trade-in value and a new car's price.
Sum of Squares
To find the combined standard deviation of two independent variables, you need to find the 'sum of squares'.
  • Square each of the standard deviations: e.g. The standard deviation for the trade-in value is 1200, so the square is: (1200)^2 = 1,440,000
  • For the new car price, the standard deviation is 3100, so the square is: (3100)^2 = 9,610,000
This calculation is essential to find the new combined standard deviation.
Square Root
The final step in finding the standard deviation is taking the square root of the sum of squares. This is because we鈥檙e dealing with variances (which are squared deviations), and to get back to standard deviation, we need the square root. In this case, you sum the squares of the standard deviations and then take the square root:
  • Sum of the squares: 1,440,000 + 9,610,000 = 11,050,000
  • Square root of the sum: \( \sqrt{11050000} \approx 3,323.6 \)
So, the standard deviation of the revenue is approximately 3323.6 dollars.
Trade-In Value
The trade-in value is the amount an auto dealer offers when you trade your old car for a new one. In this exercise, the trade-in value has an average of \(8750 and a standard deviation of \)1200. This standard deviation indicates how much variation there is from the average trade-in value. So, the trade-in value is an independent variable in the dealer's revenue calculation.
New Car Price
The new car price is the selling price of the new car, another crucial component in calculating revenue. Here, the new car's average price is \(28,500, with a standard deviation of \)3100. Just like the trade-in value, this standard deviation shows the variability or risk in the pricing of new cars. Combining these two metrics allows the dealer to predict revenue more accurately.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An inspection procedure at a manufacturing plant involves picking three items at random and then accepting the whole lot if at least two of the three items are in perfect condition. If in reality \(90 \%\) of the whole lot are perfect, what is the probability that the lot will be accepted? (A) \(3(0.1)^{2}(0.9)\) (B) \(3(0.9)^{2}(0.1)\) (C) \((0.1)^{3}+(0.1)^{2}(0.9)\) (D) \((0.1)^{3}+3(0.1)^{2}(0.9)\) (E) \((0.9)^{3}+3(0.9)^{2}(0.1)\)

In a 1974 "Dear Abby" letter, a woman lamented that she had just given birth to her eighth child and all were girls! Her doctor had assured her that the chance of the eighth child being a girl was less than 1 in 100 . What was the real probability that the eighth child would be a girl? (A) 0.0039 (B) 0.5 (C) \((0.5)^{7}\) (D) \((0.5)^{8}\) (E) \(\frac{(0.5)^{7}+(0.5)^{8}}{2}\)

It is estimated that 1 out of 10 adults over the age of 50 use recreational marijuana. In a random sample of seven adult over the age of \(50,\) what is the distribution for the number who use recreational marijuana? (A) Binomial with \(n=10\) and \(p=\frac{1}{10}\) (B) Binomial with \(n=7\) and \(p=\frac{1}{10}\) (C) Binomial with \(n=7\) and \(p=\frac{1}{7}\) (D) Geometric with \(p=\frac{1}{10}\) (E) Geometric with \(p=\frac{1}{7}\)

Suppose that, for any given year, the probabilities that the stock market declines, that women's hemlines are lower, and that both events occur are, respectively, \(0.4,0.35,\) and \(0.3 .\) Are the two events independent? (A) Yes, because \((0.4)(0.35) \neq 0.3 .\) (B) No, because \((0.4)(0.35) \neq 0.3\). (C) Yes, because \(0.4>0.35>0.3\). (D) No, because \(0.5(0.3+0.4)=0.35\). (E) There is insufficient information to answer this question.

Suppose that one out of 20 apples from a particular orchard is wormy. What are the mean and standard deviation for the number of apples to be sampled from this orchard before finding a wormy apple? (A) Mean \(=5,\) standard deviation \(=1\) (B) Mean \(=5,\) standard deviation \(=1-0.05\) (C) Mean \(=10,\) standard deviation \(=\sqrt{\frac{(0.05)^{2}}{1-0.05}}\) (D) Mean \(=20,\) standard deviation \(=\sqrt{380}\) (E) Mean \(=20,\) standard deviation \(=\sqrt{\frac{(0.05)^{2}}{1-0.05}}\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.