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Chapter 2: Problem 6

Describe each of the following correlation coefficients using the terms strong, moderate, or weak and positive or negative. a. \(r = 0.21 \qquad\) b. \(r = 鈥� 0.87\) c. \(r = 0.55 \qquad\) d. \(r = 鈥� 0.099\) e. \(r = 0.99 \qquad\) f. \(r = 鈥� 0.49\)

Short Answer

Expert verified
a. weak positive, b. strong negative, c. moderate to strong positive, d. weak negative, e. very strong positive, f. moderate negative

Step by step solution

01

Determine Correlation Strength and Direction for \(r = 0.21\)

The value of the correlation coefficient is 0.21, which falls in the range of 0.1 to 0.5. Hence it indicates weak positive correlation.
02

Determine Correlation Strength and Direction for \(r = -0.87\)

The value of the correlation coefficient is -0.87, which falls in the range of -0.9 to -0.5. Hence it indicates a strong negative correlation.
03

Determine Correlation Strength and Direction for \(r = 0.55\)

The value of the correlation coefficient is 0.55, which falls in the range of 0.5 to 0.9. Hence it indicates moderate to strong positive correlation.
04

Determine Correlation Strength and Direction for \(r = -0.099\)

The value of the correlation coefficient is -0.099, which falls in the range of -0.1 to -0.5. Hence it indicates weak negative correlation.
05

Determine Correlation Strength and Direction for \(r = 0.99\)

The value of the correlation coefficient is 0.99, which is close to 1. Hence it indicates a very strong positive correlation.
06

Determine Correlation Strength and Direction for \(r = -0.49\)

The value of the correlation coefficient is -0.49, which falls in the range of -0.5 to -0.1. Hence it indicates a moderate negative correlation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Analysis
Correlation analysis is a method used in statistics to measure the strength and direction of a linear relationship between two variables. It gives us a correlation coefficient, usually denoted as \( r \), which can range from \( -1 \) to \( 1 \).

This coefficient tells us two things about the relationship:
  • The direction: Whether the relationship is positive or negative.
  • The strength: How strong or weak the relationship is.

An \( r \) value closer to \( 1 \) or \( -1 \) indicates a stronger relationship, while a value closer to \( 0 \) suggests a weaker relationship. Understanding these nuances can help you analyze data more effectively.
Positive Correlation
Positive correlation occurs when two variables move in the same direction. This means if one variable increases, the other tends to increase as well, and vice versa.

The correlation coefficient \( r \) will be greater than \( 0 \). The closer \( r \) is to \( 1 \), the stronger the positive relationship.
  • Example: \( r = 0.21 \) is a weak positive correlation.
  • Example: \( r = 0.99 \) indicates a very strong positive correlation.

Understanding positive correlation can be useful in fields like finance, psychology, and research where predicting variables in tandem can offer insights.
Negative Correlation
Negative correlation happens when two variables move in opposite directions. If one variable increases, the other decreases.

A negative correlation coefficient \( r \) will be less than \( 0 \). The closer \( r \) is to \( -1 \), the stronger the negative relationship.
  • Example: \( r = -0.87 \) indicates a strong negative correlation.
  • Example: \( r = -0.099 \) shows a weak negative correlation.

Negative correlations are significant in understanding processes where factors inversely affect one another, like supply and demand in economics.
Statistical Strength
Statistical strength refers to how strong or weak the relationship is between two variables. It helps determine the confidence with which we can interpret the relationship.

Here's how it works:
  • Strong correlation: \( |r| \) closer to \( 1 \) or \( -1 \). Reliable indication of linear relationship.
  • Moderate correlation: \( |r| \) in the range of \( 0.5 \) to \( 0.7 \). Shows a reasonable degree of association.
  • Weak correlation: \( |r| \) closer to \( 0 \). Indicates a weaker association between variables.

Recognizing statistical strength is crucial for interpreting data accurately, guiding decision-making, and predicting outcomes reliably.

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Most popular questions from this chapter

An electronics store is selling car chargers for cell phones. The expense function is \(E=-300 p+13,000\) and the revenue function is \(R=-32 p^{2}+1,200 p .\) a. At what price would the maximum revenue be reached? b. What would that maximum revenue be? Round to the nearest cent. c. Graph the expense and revenue functions. Circle the breakeven points. d. Determine the prices at the breakeven points. Round to the nearest cent. e. Determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent.

The fixed costs of producing a Wild Widget are \(\$ 34,000 .\) The variable costs are \(\$ 5.00\) per widget. What is the average cost per widget of producing \(7,000\) Wild Widgets? Round to the nearest cent.

Rich and Betsy Cuik started a small business. They manufacture a microwavable coffee-to-go cup called Cuik Cuppa Coffee. It contains spring water and ground coffee beans in a tea-bag-like pouch. Each cup costs the company \(\$ 1.00\) to manufacture. The fixed costs for this product line are \(\$ 1,500\) . Rich and Betsy have determined the demand function to be \(q=-1,000 p+8,500,\) where \(p\) is the price for each cup. a. Write the expense equation in terms of the demand, q. b. Express the expense equation found in part a in terms of the price, p. c. Determine a viewing window on a graphing calculator for the expense function. Justify your answer. d. Draw and label the graph of the expense function. e. Write the revenue function in terms of the price. f. Graph the revenue function in a suitable viewing window. What price will yield the maximum revenue? What is the revenue at that price? Round both answers to the nearest cent. g. Graph the revenue and expense functions on the same coordinate plane. Identify the points of intersection using a graphing calculator. Round your answers to the nearest cent. Identify the price at the breakeven points.

Variable costs of producing widgets account for the cost of gas required to deliver the widgets to retailers. A widget producer finds the average cost of gas per widget. The expense equation was recently adjusted from \(E=4.55 q+69,000\) to \(E=4.98 q+69,000\) in response to the increase in gas prices. a. Find the increase in the average cost per widget. b. If the widgets are sold to retailers for \(\$ 8.00\) each, find the break even point prior to the adjustment in the expense function. c. After the gas increase, the company raised its wholesale cost from \(\$ 8\) to \(\$ 8.50\) . Find the breaker point after the adjustment in the expense function. Round to the nearest integer.

A CD storage rack is sold to stores at a wholesale price of \(\$ 18\) . a. If a store has a \(\$ 13\) markup, what is the retail price of the CD rack? b. Find the percent increase of the markup to the nearest percent.
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