/*! 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 38 Data on the IQ test scores and r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 38

Data on the IQ test scores and reading test scores for a group of fifth-grade children give the following regression line: predicted reading score \(=-33.4+0.882(\mathrm{IQ}\) score \()\) (a) What's the slope of this line? Interpret this value in context. (b) What's the \(y\) intercept? Explain why the value of the intercept is not statistically meaningful. (c) Find the predicted reading score for a child with an IQ score of 90 .

Short Answer

Expert verified
(a) 0.882; Each IQ point increases reading score by 0.882. (b) -33.4; It's not meaningful as IQ can't be 0. (c) 45.98.

Step by step solution

01

Identify the slope of the line

The given regression line is \( \text{predicted reading score} = -33.4 + 0.882 \times (\text{IQ score}) \). The slope of the line is the coefficient of the IQ score variable, which is 0.882. This means that for each one-point increase in IQ score, the predicted reading score increases by 0.882 points.
02

Identify the y-intercept

The equation of the regression line shows that the \( y \)-intercept is -33.4. This is the predicted reading score when the IQ score is 0, which is not a realistic scenario because IQ has a minimum positive value. Hence, the \( y \)-intercept is not statistically meaningful in this context.
03

Calculate predicted reading score with IQ of 90

To find the predicted reading score for a child with an IQ score of 90, substitute the IQ value into the regression equation: \[ \text{predicted reading score} = -33.4 + 0.882 \times 90 \] Calculate the result: \[ \text{predicted reading score} = -33.4 + 79.38 = 45.98 \] Thus, the predicted reading score for a child with an IQ score of 90 is 45.98.

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.

Slope Interpretation
In a regression analysis, the slope is an important measure that tells us how the dependent variable typically changes with a one-unit increase in the independent variable. Here, the given regression line is \( \text{predicted reading score} = -33.4 + 0.882 \times (\text{IQ score}) \). The slope of this line is 0.882.

Understanding the Slope
  • The slope, 0.882, describes the expected change in the predicted reading score for each additional point in the IQ score.
  • This means that if a child鈥檚 IQ score increases by 1 point, their predicted reading score will increase by 0.882 points.
  • This positive slope indicates a direct relationship between IQ scores and reading scores: as IQ increases, so does the predicted reading score.
Understanding slope helps us see the strength and direction of the relationship between the two variables. It鈥檚 key in predicting outcomes.
Y-intercept
The y-intercept in a regression equation is the predicted value of the dependent variable when the independent variable is zero. In the equation \( \text{predicted reading score} = -33.4 + 0.882 \times (\text{IQ score}) \), the y-intercept is -33.4.

Exploring the Y-intercept
  • The y-intercept, -33.4, represents the predicted reading score if a child has an IQ of zero.
  • However, IQ scores cannot realistically be zero as they usually start from a positive value in actual measurements.
  • Therefore, the y-intercept in this context is not statistically meaningful or relevant, but it's a necessary part of the mathematical formula that fits the data best.
Recognizing the irrelevance of the y-intercept in such scenarios is crucial in not misinterpreting the regression analysis results.
Prediction
In regression analysis, prediction involves using the regression line to estimate the value of the dependent variable for a given value of the independent variable. In this exercise, we are tasked with predicting the reading score for a child with an IQ score of 90.

Making Predictions
  • To make a prediction, substitute the child's IQ into the regression equation: \( \text{predicted reading score} = -33.4 + 0.882 \times 90 \).
  • Calculate: \( \text{predicted reading score} = -33.4 + 79.38 = 45.98 \).
  • Thus, a fifth grader with an IQ of 90 is predicted to have a reading score of 45.98.
This example demonstrates how we can utilize a regression equation to make informed predictions based on statistical data. By understanding and applying the slope and y-intercept, we can accurately forecast outcomes.

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

We expect that students who do well on the midterm exam in a course will usually also do well on the final exam. Gary Smith of Pomona College looked at the exam scores of all 346 students who took his statistics class over a 10 -year period. \({ }^{22}\) Assume that both the midterm and final exam were scored out of 100 points. (a) State the equation of the least-squares regression line if each student scored the same on the midterm and the final. (b) The actual least-squares line for predicting finalexam score \(y\) from midterm-exam score \(x\) was \(\hat{y}=46.6+0.41 x .\) Predict the score of a student who scored 50 on the midterm and a student who scored 100 on the midterm. (c) Explain how your answers to part (b) illustrate regression to the mean.

Exercise 6 (page 159 ) examined the relationship between the number of new birds \(y\) and percent of returning birds \(x\) for 13 sparrowhawk colonies. Here are the data once again. $$\begin{array}{lrrrrrrrrrrrrr}\hline \text { Percent return: } & 74 & 66 & 81 & 52 & 73 & 62 & 52 & 45 & 62 & 46 & 60 & 46 & 38 \\\\\text { New adults: } & 5 & 6 & 8 & 11 & 12 & 15 & 16 & 17 & 18 & 18 & 19 & 20 & 20 \\\\\hline \end{array}$$ (a) Use your calculator to help make a scatterplot. (b) Use your calculator's regression function to find the equation of the least-squares regression line. Add this line to your scatterplot from (a). (c) Explain in words what the slope of the regression line tells us. (d) Calculate and interpret the residual for the colony that had \(52 \%\) of the sparrowhawks return and 11 new adults.

The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). $$\begin{array}{lccccc}\hline \text { Speed: } & 20 & 30 & 40 & 50 & 60 \\\\\text { Mileage: } & 24 & 28 & 30 & 28 & 24 \\\\\hline \end{array}$$ (a) Make a scatterplot to show the relationship between speed and mileage. (b) Calculate the correlation for these data by hand or using technology. (c) Explain why the correlation has the value found in part (b) even though there is a strong relationship between speed and mileage.

In its recent Fuel Economy Guide, the Environmental Protection Agency gives data on 1152 vehicles. There are a number of outliers, mainly vehicles with very poor gas mileage. If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg. The Chevrolet Malibu with a four-cylinder engine has a combined gas mileage of 25 mpg. What percent of all vehicles have worse gas mileage than the Malibu?

Measurements on young children in Mumbai, India, found this least-squares line for predicting height \(y\) from \(\operatorname{arm} \operatorname{span} x:\) $$\hat{y}=6.4+0.93 x$$ Measurements are in centimeters \((\mathrm{cm})\). Suppose that a tall child with arm span \(120 \mathrm{~cm}\) and height \(118 \mathrm{~cm}\) was added to the sample used in this study. What effect will adding this child have on the correlation and the slope of the least-squares regression line? (a) Correlation will increase, slope will increase. (b) Correlation will increase, slope will stay the same. (c) Correlation will increase, slope will decrease. (d) Correlation will stay the same, slope will stay the same. (e) Correlation will stay the same, slope will increase.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.