/*! 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 20 Over-under, Part II. Suppose we ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 20

Over-under, Part II. Suppose we fit a regression line to predict the number of incidents of skin cancer per 1,000 people from the number of sunny days in a year. For a particular year, we predict the incidence of skin cancer to be 1.5 per 1,000 people, and the residual for this year is \(0.5 .\) Did we over or under estimate the incidence of skin cancer? Explain your reasoning.

Short Answer

Expert verified
The regression underestimated the skin cancer incidents.

Step by step solution

01

Setting Up the Problem

We are given the predicted value for the number of skin cancer incidents, which is 1.5 per 1,000 people. The residual for this prediction is given as 0.5. A residual is the difference between the observed (actual) value and the predicted value: \(\text{Residual} = \text{Observed} - \text{Predicted}\).
02

Understanding Residuals

Residuals tell us how far off our predictions are from the actual values. A positive residual means that the observed value was higher than the predicted value, indicating that the prediction understates the actual number. Conversely, a negative residual would mean the prediction overstated the actual value.
03

Applying the Information

Given that the residual is 0.5, we substitute it into the residual formula: \(0.5 = \text{Observed} - 1.5\). Solving for the observed value, we find that \(\text{Observed} = 0.5 + 1.5 = 2.0\) per 1,000 people.
04

Conclusion on Over-Under Estimation

The observed value of skin cancer incidents was 2 per 1,000 people, which is higher than the predicted value of 1.5 per 1,000 people. Therefore, the regression line underestimated the incidence of skin cancer.

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.

Understanding the Regression Line
A regression line is a straight line that best fits a set of data points on a graph. This line helps in predicting the value of a dependent variable based on the value of an independent variable. For instance, in our case, the regression line helps predict the incidence of skin cancer based on the number of sunny days in a year. The line minimizes the differences between the observed values and predicted values, making the predictions as accurate as possible. However, it's important to note that the regression line is an estimate, not a guarantee, and may not perfectly predict actual outcomes.
Exploring Predicted Values
Predicted values are the results we get from using the regression line. They represent the expected outcome based on the relationship between the variables. In our exercise, the predicted value for the incidence of skin cancer was 1.5 per 1,000 people. This prediction is based on the assumption of normal conditions as defined by the data used to create the regression line. Predicted values serve as benchmarks, allowing us to understand how well or poorly our model is performing by comparing them to actual observations.
What are Observed Values?
Observed values are the actual outcomes or measurements gathered from real-world data. In our discussion, the observed value refers to the true incident rate of skin cancer in a specific year, which turned out to be 2 per 1,000 people. These values are crucial in assessing the accuracy of predictions made by our regression line, as they allow us to calculate the residuals. Analyzing observed values can give insights into why our predictions might deviate, hinting at possible changes in external factors or misestimations in our model.
Significance of Underestimation
Underestimation occurs when the predicted value is lower than the observed value. In our case, the prediction of 1.5 incidents per 1,000 people was lower than the observed value of 2 per 1,000 people, resulting in an underestimation. Understanding why an underestimation happens is important because it highlights areas where the regression model might need adjustment or refinement. Consistent underestimations could indicate an omitted variable in the model or an increasing trend that has not been accounted for properly.
Understanding the Incidence of Skin Cancer
The incidence of skin cancer refers to the rate at which new cases occur in a population over a period of time. In the exercise, it refers to the number of new skin cancer cases per 1,000 people annually. Factors influencing this rate can include environmental variables like the number of sunny days, exposure to UV radiation, and public health measures. Understanding the incidence rate is crucial for public health planning, resource allocation, and defining preventive measures. By modeling these incidents, we aim to predict future trends and possibly implement interventions to reduce the incidence of skin cancer.

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

Body measurements, Part II. The scatterplot below shows the relationship between weight measured in kilograms and hip girth measured in centimeters from the data described in Exercise 8,13 . (a) Describe the relationship between hip girth and weight. (b) How would the relationship change if weight was measured in pounds while the units for hip girth remained in centimeters?

Body measurements, Part IV. The scatterplot and least squares summary below show the relationship between weight measured in kilograms and height measured in centimeters of 507 physically active individuals. (a) Describe the relationship between height and weight. (b) Write the equation of the regression line. Interpret the slope and intercept in context. (c) Do the data provide strong evidence that an increase in height is associated with an increase in weight? State the null and alternative hypotheses, report the p-value, and state your conclusion. (d) The correlation coefficient for height and weight is \(0.72 .\) Calculate \(R^{2}\) and interpret it in context.

Body measurements, Part III. Exercise 8.13 introduces data on shoulder girth and height of a group of individuals. The mean shoulder girth is \(107.20 \mathrm{~cm}\) with a standard deviation of \(10.37 \mathrm{~cm}\). The mean height is \(171.14 \mathrm{~cm}\) with a standard deviation of \(9.41 \mathrm{~cm} .\) The correlation between height and shoulder girth is 0.67 (a) Write the equation of the regression line for predicting height. (b) Interpret the slope and the intercept in this context. (c) Calculate \(R^{2}\) of the regression line for predicting height from shoulder girth, and interpret it in the context of the application. (d) A randomly selected student from your class has a shoulder girth of \(100 \mathrm{~cm}\). Predict the height of this student using the model. (e) The student from part (d) is \(160 \mathrm{~cm}\) tall. Calculate the residual, and explain what this residual means. (f) A one year old has a shoulder girth of \(56 \mathrm{~cm}\). Would it be appropriate to use this linear model to predict the height of this child?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.