/*! 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 21 Are hot dogs that are high in ca... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 21

Are hot dogs that are high in calories also high in salt? The figure below is a scatterplot of the calories and salt content (measured as milligrams of sodium) in 17 brands of meat hot dogs. (a) The correlation for these data is \(r=0.87 .\) Explain what this value means. (b) What effect does the hot dog brand with the lowest calorie content have on the correlation? Justify your answer.

Short Answer

Expert verified
(a) The correlation \( r = 0.87 \) indicates a strong positive relationship. (b) The lowest calorie hot dog likely has a minor effect unless it's an outlier.

Step by step solution

01

Interpreting Correlation Coefficient

The correlation coefficient, denoted by \( r \), measures the strength and direction of the linear relationship between two variables. A value of \( r = 0.87 \) indicates a strong positive linear relationship, meaning that as the calorie content in hot dogs increases, the salt content tends to increase as well.
02

Analyzing Lowest Calorie Hot Dog Impact

To analyze the effect of the lowest calorie hot dog on the correlation, consider that removing or having outlier points typically affects correlation. If this lowest calorie hot dog follows the general trend (even if it's an extreme value), its removal might slightly reduce \( r \) but won't significantly disrupt the strong linear trend indicated by \( r = 0.87 \). However, if the point is an outlier, it might distort \( r \), making the relationship appear stronger than it is. Removing a non-outlier point on the trend usually has minimal impact.

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.

Scatterplot
A scatterplot is a visual tool used in statistics to display the relationship between two numerical variables. In our study of hot dogs, the scatterplot maps each brand's calorie count to its sodium content.
Scatterplots are invaluable because they allow us to visually assess whether there is a relationship between the two variables.
Key aspects of scatterplot analysis include:
  • Identifying patterns or trends (e.g., do points form a steady line, indicating correlation?)
  • Spotting potential anomalies, known as outliers, which may deviate significantly from the overall pattern
  • Determining the direction of the relationship (positive, negative, or none)
In our case, a pattern emerges suggesting that as the calories increase in hot dogs, so does the sodium content.
Linear Relationship
A linear relationship implies that there is a straight-line connection between two variables. In this hot dog study, we are looking at calories and sodium content.
When data points on a scatterplot form a straight-line pattern, it suggests a potential linear relationship. This relationship is quantified by the correlation coefficient.
Some key points about linear relationships are:
  • A positive linear relationship means both variables increase together 鈥� as seen in this hot dog example.
  • A negative linear relationship shows that as one variable increases, the other decreases.
  • A zero correlation coefficient indicates no linear relationship.
Understanding the nature of this relationship helps us make predictions about one variable based on the other.
Outliers
Outliers are data points that differ significantly from other observations. They can affect statistical measurements like correlation.
In the hot dog example, a brand with an unusually low calorie count might be an outlier. Such points may skew the correlation coefficient, giving a misleading impression of the relationship's strength.
Key considerations for outliers:
  • Determine if the outlier represents a meaningful deviation or just random variation.
  • Decide whether to include or exclude outliers based on their impact on the analysis.
  • Outliers can show unique patterns but often are excluded to ensure general trends are accurately measured.
The presence of outliers is crucial to consider when interpreting data and calculating correlation.
Statistical Analysis
Statistical analysis involves applying quantitative techniques to understand data and draw conclusions. The correlation coefficient is a central part of our analysis of the relationship between hot dog calories and sodium.
This analysis provides a structured approach to understanding our scatterplot data.
  • Compute the correlation coefficient to assess the strength of the linear relationship.
  • Use regression analysis to model the relationship and make predictions.
  • Apply confidence intervals and hypothesis tests to validate findings.
Conducting a thorough statistical analysis ensures the reliability of results and supports data-driven decisions, like improving product recipes or targeting health-conscious consumers.

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 a car's highway gas mileage to be related to its city gas mileage. Data for all 1198 vehicles in the government's recent Fuel Economy Guide give the regression line: predicted highway \(\mathrm{mpg}=4.62+1.109(\mathrm{city} \mathrm{mpg})\) (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 highway mileage for a car that gets 16 miles per gallon in the city.

A college newspaper interviews a psychologist about student ratings of the teaching of faculty members. The psychologist says, "The evidence indicates that the correlation between the research productivity and teaching rating of faculty members is close to zero." The paper reports this as "Professor McDaniel said that good researchers tend to be poor teachers, and vice versa." Explain why the paper's report is wrong. Write a statement in plain language (don't use the word "correlation") to explain the psychologist's meaning.

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.

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable \(x\) to be the percent change in a stock market index in January and the response variable \(y\) to be the change in the index for the entire year. We expect a positive correlation between \(x\) and \(y\) because the change during January contributes to the full year's change. Calculation from data for an 18 -year period gives $$\begin{array}{c}\bar{x}=1.75 \% \quad s_{x}=5.36 \% \quad \bar{y}=9.07 \% \\\ s_{y}=15.35 \% \quad r=0.596\end{array}$$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) Suppose that the percent change in a particular January was 2 standard deviations above average. Predict the percent change for the entire year, without using the least-squares line. Show your work.

We expect that a baseball player who has a high batting average in the first month of the season will also have a high batting average the rest of the season. Using 66 Major League Baseball players from the 2010 season, \({ }^{23}\) a least-squares regression line was calculated to predict rest-of- season batting average \(y\) from first-month batting average \(x .\) Note: \(\mathrm{A}\) player's batting average is the proportion of times at bat that he gets a hit. A batting average over 0.300 is considered very good in Major League Baseball. (a) State the equation of the least-squares regression line if each player had the same batting average the rest of the season as he did in the first month of the season. (b) The actual equation of the least-squares regression line is \(\hat{y}=0.245+0.109 x .\) Predict the rest-of-season batting average for a player who had a 0.200 batting average the first month of the season and for a player who had a 0.400 batting average the first month of the season. (c) Explain how your answers to part (b) illustrate regression to the mean.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.