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Chapter 15: Problem 5

The flow rate in a device used for air quality measurement depends on the pressure drop \(x\) (inches of water) across the device's filter. Suppose that for \(x\) values between 5 and \(20,\) these two variables are related according to the simple linear regression model with population regression line \(y=-0.12+0.095 x\) a. What is the mean flow rate for a pressure drop of 10 inches? A drop of 15 inches? b. What is the average change in flow rate associated with a 1 inch increase in pressure drop? Explain.

Short Answer

Expert verified
The mean flow rate for a pressure drop of 10 inches is 0.83, and for a drop of 15 inches, it is 1.305. The average change in flow rate associated with a 1-inch increase in pressure drop is 0.095. This means that for every 1-inch increase in pressure drop, the flow rate increases by 0.095 on average.

Step by step solution

01

a. Calculate Mean Flow Rate for Pressure Drop of 10 inches

To calculate the mean flow rate when the pressure drop is 10 inches, plug in the value of x = 10 into the equation: \(y = -0.12 + 0.095 * x\) \(y = -0.12 + 0.095 * 10\) \(y = -0.12 + 0.95\) The mean flow rate when the pressure drop is 10 inches is: \(y = 0.83\)
02

a. Calculate Mean Flow Rate for Pressure Drop of 15 inches

To calculate the mean flow rate when the pressure drop is 15 inches, plug in the value of x = 15 into the equation: \(y = -0.12 + 0.095 * x\) \(y = -0.12 + 0.095 * 15\) \(y = -0.12 + 1.425\) The mean flow rate when the pressure drop is 15 inches is: \(y = 1.305\)
03

b. Calculate the Average Change in Flow Rate

To understand the average change in flow rate for a 1-inch increase in pressure drop, we examine the slope of the regression line. The slope, in this case, is 0.095. This means that for every 1-inch increase in pressure drop, the flow rate increases by 0.095 on average. In other words, an increase of 1 inch in pressure drop is associated with an average increase of 0.095 in the flow rate.

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Key Concepts

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

Mean Flow Rate Calculation
The mean flow rate is a measure used to describe the average rate at which a fluid travels through a given system. In exercises like the one provided, where the flow rate in an air quality measurement device is examined, the calculation becomes integral to understanding system behavior.

To determine the mean flow rate at specific values of pressure drop, we use a simple linear regression model. For instance, given the equation \( y = -0.12 + 0.095x \), we can calculate the mean flow rate for a pressure drop of 10 inches by substituting \( x = 10 \) into the equation, resulting in a flow rate of \( 0.83 \) units. Similarly, with a pressure drop of 15 inches, replacing \( x \) with 15 gives us a flow rate of \( 1.305 \) units.

Understanding these calculations is critical for engineers and scientists as they often use mean flow rates to predict performance, design systems, and evaluate the efficiency of different components within a system.
Regression Analysis
Regression analysis is a powerful statistical tool used to examine the relationship between two or more variables. In the context of our provided exercise, simple linear regression analysis helps to determine how the flow rate responds to changes in the pressure drop across a filter.

In simple linear regression, the relationship between the independent variable \( x \) and the dependent variable \( y \) is represented by a straight line, also known as the regression line, which is described by the formula \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Significance of Regression Analysis

Regression analysis is not only about drawing a line through the data. It鈥檚 about quantifying the strength of the relationship, making predictions, and identifying trends. By using regression analysis, one can make informed decisions based on the patterns observed in the data, which is indispensable for scientific research, economics, engineering, and many other fields.
Slope Interpretation
The slope of a regression line is a critical value that conveys a lot of information about the relationship between the variables under study. In simple linear regression, the slope \( m \) represents the estimated change in the dependent variable \( y \) for each one-unit change in the independent variable \( x \).

From our exercise's regression equation \( y = -0.12 + 0.095x \), the slope is \( 0.095 \), which means that for every inch increase in the pressure drop, the mean flow rate is predicted to increase by \( 0.095 \) units. This insight enables the user to anticipate how changes in one variable will affect the other, and it is instrumental in planning and optimization processes in various scientific and engineering applications.

Practical Implications

Correctly interpreting the slope is essential, as it impacts decision-making and predictions. For example, if a company is testing the efficiency of a filter, understanding how a small increase in pressure can significantly increase flow rate might influence design choices or operational strategies.

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Most popular questions from this chapter

The authors of the article "Age, Spacing and Growth Rate of Tamarix as an Indication of Lake Boundary Fluctuations at Sebkhet Kelbia, Tunisia" (Journal of Arid Environments [1982]: 43-51) used a simple linear regression model to describe the relationship between \(y=\) vigor (average width in centimeters of the last two annual rings) and \(x=\) stem density (stems/m \(^{2}\) ). The estimated model was based on the following data. Also given are the standardized residuals. \(\begin{array}{lrrrrr}x & 4 & 5 & 6 & 9 & 14 \\ \boldsymbol{y} & 0.75 & 1.20 & 0.55 & 0.60 & 0.65 \\ \text { Std resid } & -0.28 & 1.92 & -0.90 & -0.28 & 0.54 \\ \boldsymbol{x} & 15 & 15 & 19 & 21 & 22 \\ \boldsymbol{y} & 0.55 & 0.00 & 0.35 & 0.45 & 0.40 \\ \text { Std resid } & 0.24 & -2.05 & -0.12 & 0.60 & 0.52\end{array}\) a. What assumptions are required for the simple linear regression model to be appropriate? b. Construct a normal probability plot of the standardized residuals. Does the assumption that the random deviation distribution is normal appear to be reasonable? Explain. c. Construct a standardized residual plot. Are there any unusually large residuals? d. Is there anything about the standardized residual plot that would cause you to question the use of the simple linear regression model to describe the relationship between \(x\) and \(y ?\)

The standard deviation of the errors, \(\sigma_{e},\) is an important part of the linear regression model. a. What is the relationship between the value of \(\sigma_{e}\) and the value of the test statistic in a test of a hypotheses about \(\beta ?\) b. What is the relationship between the value of \(\sigma_{e}\) and the width of a confidence interval for \(\beta\) ?

Let \(x\) be the size of a house (in square feet) and \(y\) be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) Houses in this community range in size from 1000 to 3000 square feet. a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq. ft. of space? d. What is the average change in usage associated with a 1 sq. ft. increase in size? e. What is the average change in usage associated with a 100 sq. ft. increase in size? f. Would you use the model to predict mean usage for a 500 sq. ft. house? Why or why not?

The paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: \(85-88\) ) suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length (mm) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation 0.005 . a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m} ?\) c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) have a shell thickness of greater than \(0.175 ?\) Less than \(0.178 ?\)

Consider a test of hypotheses about, \(\beta\) the population slope in a linear regression model. a. If you reject the null hypothesis, \(\beta=0\), what does this mean in terms of a linear relationship between \(x\) and \(y ?\) b. If you fail to reject the null hypothesis, \(\beta=0,\) what does this mean in terms of a linear relationship between \(x\) and \(y ?\)
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