/*! 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 12 A bad data point: A scientist sa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 12

A bad data point: A scientist sampled data for a natural phenomenon that she has good reason to believe is appropriately modeled by an exponential function N = N (t).Alaboratory assistant reported that there may have been an error in recording one of the data points but is not certain which one. Use the logarithm to find the suspect data point $$ \begin{array}{|c|c|} \hline t & N \\ \hline 0 & 21.3 \\ \hline 1 & 37.5 \\ \hline 2 & 66.0 \\ \hline 3 & 95.3 \\ \hline 4 & 204.4 \\ \hline 5 & 359.7 \\ \hline 6 & 633.1 \\ \hline \end{array} $$

Short Answer

Expert verified
The data point \( N(3) = 95.3 \) is the likely suspect because it departs from the expected exponential pattern.

Step by step solution

01

Understanding Exponential Growth

An exponential function for this scenario can be written as \(N(t) = N_0 \, e^{kt}\), where \(N_0\) is the initial quantity (\(N(0)\)), and \(k\) is the growth rate. This means that the ratio between consecutive values of \(N(t)\) should be approximately constant for an exponential model.
02

Calculating Consecutive Ratios

Compute the ratio \( \frac{N(t+1)}{N(t)} \) for each pair of consecutive \(t\) values:- \( t = 0, 1: \frac{37.5}{21.3} \approx 1.76 \)- \( t = 1, 2: \frac{66.0}{37.5} \approx 1.76 \)- \( t = 2, 3: \frac{95.3}{66.0} \approx 1.44 \)- \( t = 3, 4: \frac{204.4}{95.3} \approx 2.14 \)- \( t = 4, 5: \frac{359.7}{204.4} \approx 1.76 \)- \( t = 5, 6: \frac{633.1}{359.7} \approx 1.76 \)
03

Identify Anomalous Ratio

Compare each ratio with the others. Most ratios are approximately 1.76, but the ratio at \( t = 2, 3 \), which is approximately 1.44, and the ratio at \( t = 3, 4 \), which is approximately 2.14, deviate significantly from the expected pattern.
04

Use Logarithms for Consistency Check

Convert each \(N(t)\) into its logarithmic form for a better linear analysis by calculating \( \log(N(t)) \).Then, check the consistency in differences: - \( \Delta_{0 \rightarrow 1} =\log(37.5) - \log(21.3) \)- \( \Delta_{1 \rightarrow 2} = \log(66.0) - \log(37.5)\)- Continue similarly and spot where the differences in \( \Delta \) values vary significantly. The greatest deviation will confirm the suspect point identified earlier from ratios.
05

Confirm the Suspect Data Point

The significant deviation of the data points at \( t = 2, 3 \) and \( t = 3, 4 \) in ratios and logarithmic transformation suggests that the point \( N(3) = 95.3 \) or \( N(4) = 204.4 \) might be erroneous, with most evidence pointing more towards \( t=3 \).

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.

Logarithms
Logarithms are incredibly useful in data analysis, especially when dealing with exponential functions. In this context, a logarithm transforms a rapidly growing exponential function into a linear one, making it easier to analyze the data.
When you take the logarithm of an exponential function, the transformation can reveal underlying patterns that are not immediately obvious in the base data. This is because logarithms convert multiplicative relationships into additive ones. For example, with the function \[ N(t) = N_0 \, e^{kt} \], taking the natural logarithm of both sides helps transform it to \( \log(N(t)) = \log(N_0) + kt \). This creates a relationship similar to a straight line, making deviations stand out.
  • Logarithms help ensure the differences between consecutive data points are consistent.
  • When a data point does not fit the pattern, its logarithmic change will be atypical.
Data Analysis
Data analysis is the process of inspecting, cleaning, and modeling data to discover useful information. In the case of the exponential function scenario, data analysis involves examining the ratios between consecutive data points to identify potential anomalies.
Begin by calculating the growth factor between successive values. This is done by dividing each data point by its predecessor to observe whether a consistent pattern emerges. In exponential growth, such ratios should be similar.
  • You compare each ratio with the others to identify any irregularity that may suggest an error.
  • A significant deviation in one of these ratios can point to a bad data point.
Effective data analysis often involves transforming data, like using logarithms, to achieve clear insights which might not be present in the raw values.
Error Detection
Error detection is essential in any data-driven field as it ensures the accuracy and reliability of the dataset used for analysis. In this exercise, identifying a potential data error involves looking for abnormalities in an otherwise consistent set of exponential growth data points.
When a dataset fits an exponential model, all data points should roughly follow the derived exponential expression. If a data point deviates significantly, it indicates a potential recording error.
  • Check for unexpected changes in ratios of consecutive terms.
  • Use logarithmic transformations to spot inconsistencies.
A key part of error detection is validating these findings by comparing both the ratio method and the logarithmic transformation method. Consistency (or lack thereof) in both processes can confirm the error.
Exponential Growth
Exponential growth occurs when the rate of growth is proportional to the current quantity, leading to outcomes where quantities increase rapidly. It is represented mathematically by \(N(t) = N_0 \, e^{kt}\), where \(N_0\) is the initial amount and \(k\) is a constant representing the growth rate.
Such functions are characterized by a constant ratio between successive data points over equal periods, making them predictable when plotted on a logarithmic scale.
  • Exponential growth is common in natural processes such as population growth and radioactive decay.
  • The presence of consistent ratios signals true exponential growth, while deviations can suggest data errors.
Understanding exponential growth is crucial for correctly interpreting data, especially when identifying anomalies in a dataset that is presumed to follow this model.

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

Unit conversion with exponential decay: The exponential function \(N=500 \times 0.68^{t}\), where \(t\) is measured in years, shows the amount, in grams, of a certain radioactive substance present. a. Calculate \(N(2)\) and explain what your answer means. b. What is the yearly percentage decay rate? c. What is the monthly decay factor rounded to three decimal places? What is the monthly percentage decay rate? d. What is the percentage decay rate per second? (Note: For this calculation, you will need to use all the decimal places that your calculator can show.)

Research project: For this project, you should collect and analyze data for a population of M\&M's TM. Start with four candies, toss them on a plate, and add one for each candy that has the M side up; record the data. Repeat this seven times and see how close the data are to being exponential. For a detailed description, go to http://college.hmco.com/PIC/crauder4e.

Rocket flight: The velocity \(v\) attained by a launch vehicle during launch is a function of \(c\), the exhaust velocity of the engine, and \(R\), the mass ratio of the spacecraft. 36 The mass ratio is the vehicle's takeoff weight divided by the weight remaining after all the fuel has been burned, so the ratio is always greater than 1. It is close to 1 when there is room for only a little fuel relative to the size of the vehicle, and one goal in improving the design of spacecraft is to increase the mass ratio. The formula for \(v\) uses the natural logarithm: $$ v=c \ln R \text {. } $$ Here we measure the velocities in kilometers per second, and we assume that \(c=4.6\) (which can be attained with a propellant that is a mixture of liquid hydrogen and liquid oxygen). a. Draw a graph of \(v\) versus \(R\). Include mass ratios from 1 to \(20 .\) b. Is the graph in part a increasing or decreasing? In light of your answer, explain why increasing the mass ratio is desirable. c. To achieve a stable orbit, spacecraft must attain a velocity of \(7.8\) kilometers per second. With \(c=4.6\), what is the smallest mass ratio that allows this to happen? (Note: For such a propellant, the mass ratio needed for orbit is usually too high, and that is why the launch vehicle is divided into stages. The next exercise shows the advantage of this.)

Population growth: A population of animals is growing exponentially, and an ecologist has made the following table of the population size, in thousands, at the start of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { in thousands } \end{array} \\ \hline 2003 & 5.25 \\ \hline 2004 & 5.51 \\ \hline 2005 & 5.79 \\ \hline 2006 & 6.04 \\ \hline 2007 & 6.38 \\ \hline 2008 & 6.70 \\ \hline \end{array} $$ Looking over the table, the ecologist realizes that one of the entries for population size is in error. Which entry is it, and what is the correct population? (Round the ratios to two decimal places.)

Cleaning contaminated water: A tank of water is contaminated with 60 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is \(22 \%\) less salt in the tank than at the beginning of the hour. Let \(S=S(t)\) denote the number of pounds of salt in the tank \(t\) hours after the flushing process begins. a. Explain why \(S\) is an exponential function and find its hourly decay factor. b. Give a formula for \(S\). c. Make a graph of \(S\) that shows the flushing process during the first 15 hours, and describe in words how the salt removal process progresses. d. In order to meet EPA standards, there can be no more than 3 pounds of salt in the tank. How long must the process continue before EPA standards are met? e. Suppose this cleanup procedure costs \(\$ 8000\) per hour to operate. How much does it cost to reduce the amount of salt from 60 pounds to 3 pounds? How much does it cost to reduce the amount of salt from 3 pounds to \(0.1\) pound?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.