/*! 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 50 If the sample correlation coeffi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 50

If the sample correlation coefficient is equal to 1 , is it necessarily true that \(\rho=1 ?\) If \(\rho=1\), is it necessarily true that \(r=1 ?\)

Short Answer

Expert verified
No, if r is equal to 1 it does not necessarily mean that \(\rho\) is equal to 1 because the sample data may not perfectly represent the population. Yes, if \(\rho\) is equal to 1, it means that r should be 1 as the sample must represent the population correlation.

Step by step solution

01

Understanding Correlations

In correlation, '1' often represents a perfect positive correlation. This means that for every positive increase in one variable, there is a positive increase of a fixed proportion in the other. In terms of 'r' and '\(\rho\)', it's possible that both can hit the extreme value of '1' but each under different circumstances.
02

Correlation Coefficient r=1

The sample correlation coefficient 'r' equals 1 when there is a perfect positive relationship in the sample data i.e., an increase in one variable corresponds to a proportional increase in the other. This, however, does not necessarily mean '\(\rho=1\)' because the sample may not perfectly represent the population.
03

Population Correlation Coefficient \(\rho=1\)

The population correlation coefficient '\(\rho\)' equals 1 when there is a perfect positive correlation between two variables across the entire population. If this is the case, any sample taken from the population should also show a perfect positive correlation, thus, 'r' should also be equal to 1.

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.

Sample Correlation Coefficient
The sample correlation coefficient, often represented by 'r', indicates the strength and direction of a linear relationship between two variables in a sample.
  • If 'r' is 1, it suggests a perfect positive correlation in the sample data.
  • This means that if one variable increases, the other variable increases by a fixed proportion.
However, it鈥檚 important to understand that a perfect correlation in a sample does not necessarily mean that this relationship will hold true for the larger population.
'r' is derived from the sample and may change with different samples.
Therefore, while 'r' can indicate a strong linear relationship, it is not a definitive measure for the entire population.
Population Correlation
The population correlation, denoted by '\(\rho\)', measures the degree of association between two variables in an entire population.
  • When \(\rho = 1\), it signifies a perfect positive linear relationship across the whole population.
  • This implies that any sample from the population should theoretically exhibit the same correlation.
However, it鈥檚 crucial to note that real-world data often exhibits some variability.
Even if \(\rho\) is 1, practical factors such as sample size or data collection methods might result in a sample where \(r\) is somewhat less than 1.
But conceptually, if every possible sample were examined, \(r\) should reflect \(\rho\), given a perfect correlation in the population.
Perfect Positive Correlation
A perfect positive correlation occurs when two variables move in complete sync with each other.
  • In statistical terms, this is represented by a correlation coefficient of 1.
  • For every unit increase in one variable, there is a proportional and equivalent increase in the other.
This type of correlation is often represented by a straight line on a graph, showing that the relationship is linear and exact.
While this idea is straightforward in theory, it鈥檚 very rare in real-life data due to numerous influencing factors.
Moreover, both 'r' and '\(\rho\)' can indicate perfect positive correlation, but it's vital to distinguish between their contexts: 'r' is for samples, and '\(\rho\)' is for populations.

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

The article "Technology, Productivity, and Industry Structure" (Technological Forecasting and Social Change [1983]: \(1-13\) ) included the accompanying data on \(x=\) research and development expenditure and \(y=\) growth rate for eight different industries. $$ \begin{array}{rrrrrrrrr} x & 2024 & 5038 & 905 & 3572 & 1157 & 327 & 378 & 191 \\ y & 1.90 & 3.96 & 2.44 & 0.88 & 0.37 & -0.90 & 0.49 & 1.01 \end{array} $$ a. Would a simple linear regression model provide useful information for predicting growth rate from research and development expenditure? Use a .05 level of significance. b. Use a \(90 \%\) confidence interval to estimate the average change in growth rate associated with a 1 -unit increase in expenditure. Interpret the resulting interval.

The accompanying summary quantities resulted from a study in which \(x\) was the number of photocopy machines serviced during a routine service call and \(y\) was the total service time (min): \(n=16 \quad \sum(y-\bar{y})^{2}=22,398.05 \quad \sum(y-\hat{y})^{2}=2620.57\) a. What proportion of observed variation in total service time can be explained by a linear probabilistic relationship between total service time and the number of machines serviced? b. Calculate the value of the estimated standard deviation \(s_{e .}\) What is the number of degrees of freedom associated with this estimate?

The article "Cost-Effectiveness in Public Education" (Chance [1995]: \(38-41\) ) reported that, for a sample of \(n=44\) New Jersey school districts, a regression of \(y=\) average SAT score on \(x=\) expenditure per pupil (thousands of dollars) gave \(b=15.0\) and \(s_{b}=5.3\). a. Does the simple linear regression model specify a useful relationship between \(x\) and \(y\) ? b. Calculate and interpret a confidence interval for \(\beta\) based on a \(95 \%\) confidence level.

Suppose that a simple linear regression model is appropriate for describing the relationship between \(y=\) house price and \(x=\) house size (sq ft) for houses in a large city. The true regression line is \(y=23,000+47 x\) and \(\sigma=5000\). a. What is the average change in price associated with one extra sq \(\mathrm{ft}\) of space? With an additional 100 sq \(\mathrm{ft}\) of space? b. What proportion of \(1800-\) sq-ft homes would be priced over \(\$ 110,000 ?\) Under \(\$ 100,000\) ?

Legumes, such as peas and beans, are important crops whose production is greatly affected by pests. The article "Influence of Wind Speed on Residence Time of Uroleucon ambrosiae alatae on Bean Plants" (Environmental Entomology [1991]: \(1375-1380\) ) reported on a study in which aphids were placed on a bean plant, and the elapsed time until half of the aphids had departed was observed. Data on \(x=\) wind speed \((\mathrm{m} / \mathrm{sec})\) and \(y=\) residence half time were given and used to produce the following information. $$ \begin{array}{ll} a=0.0119 \quad b=3.4307 \quad n=13 \\ \text { SSTo }=73.937 \quad \text { SSResid }=27.890 \end{array} $$ a. What percentage of observed variation in residence half time can be attributed to the simple linear regression model? b. Give a point estimate of \(\sigma\) and interpret the estimate. c. Estimate the mean change in residence half time associated with a \(1-\mathrm{m} / \mathrm{sec}\) increase in wind speed. d. Calculate a point estimate of true average residence half time when wind speed is \(1 \mathrm{~m} / \mathrm{sec}\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete 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.