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Chapter 13: Problem 22

The article "Effects of Enhanced UV-B Radiation on Ribulose-1,5-Biphosphate, Carboxylase in Pea and Soybean" (Environmental and Experimental Botany [1984]: 131-143) included the accompanying data on pea plants, with \(y=\) sunburn index and \(x=\) distance \((\mathrm{cm})\) from an ultraviolet light source. \(\begin{array}{lllllllll}x & 18 & 21 & 25 & 26 & 30 & 32 & 36 & 40 \\ y & 4.0 & 3.7 & 3.0 & 2.9 & 2.6 & 2.5 & 2.2 & 2.0 \\ x & 40 & 50 & 51 & 54 & 61 & 62 & 63 & \\ y & 2.1 & 1.5 & 1.5 & 1.5 & 1.3 & 1.2 & 1.1 & \end{array}\) $$ \begin{array}{lc} \sum x=609 & \sum y=33.1 \quad \sum x^{2}=28,037 \\ \sum y^{2}=84.45 & \sum x y=1156.8 \end{array} $$ Estimate the mean change in the sunburn index associated with an increase of \(1 \mathrm{~cm}\) in distance in a way that includes information about the precision of estimation.

Short Answer

Expert verified
On average, for each increase of 1 cm in distance from the light source, the sunburn index decreases by 0.035.

Step by step solution

01

Understand the linear regression model

The mathematical model for this linear regression analysis is given by \(y = b_0 + b_1 * x + \epsilon\), where \(y\) is the dependent variable (sunburn index), \(x\) is the independent variable (distance from light source), \(b_0\) is the y-intercept, \(b_1\) is the slope of the line (mean change in \(y\) per unit change in \(x\)), and \(\epsilon\) is the error term.
02

Recall the formulas for the regression coefficients

The formula for the slope \(b_1\) of the regression line is:\(b_1 = (n*\sum(xy) - \sum(x)*\sum(y)) / (n*\sum(x^2) - (\sum(x))^2)\),where 'n' is the number of observations, and the summations are taken over all observations. The y-intercept \(b_0\) can then be calculated as:\(b_0 = \bar{y} - b_1 * \bar{x}\),where \(\bar{y}\) is the mean of \(y\) values and \(\bar{x}\) is the mean of \(x\) values.
03

Calculate the regression coefficients using the given summations

First, calculate the number of observations 'n' by counting the given pairs of (x, y). From the data, there are 14 pairs, so n = 14.Second, calculate the means \(\bar{x}\) and \(\bar{y}\) as \(\bar{x} = \sum(x)/n = 609 / 14 = 43.5\) and \(\bar{y} = \sum(y)/n = 33.1 / 14 = 2.364\).Next, plug these, and the given summation values into the equations:\(b_1 = (14*1156.8 - 609*33.1) / (14*28037 - 609^2) = -0.035\),\(b_0 = 2.364 - -0.035*43.5 = 3.88\).
04

Answer the problem statement

The problem asks for the mean change in the sunburn index associated with an increase of 1 cm in distance. This is represented by the slope of the regression line \(b_1\), which is -0.035. This implies that for each increase of 1 cm in distance from the light source, the sunburn index decreases by 0.035, on average.

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Key Concepts

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

Regression Coefficients
In the exploration of relationships between two numerical variables, regression coefficients play a pivotal role in linear regression analysis. These coefficients, denoted by b0 and b1, are the pillars that quantify the linkage between the independent and dependent variables.

The y-intercept (or b0) specifies where the estimated regression line crosses the y-axis when all independent variables are at zero. It can be perceived as the starting point of the predicted value when no influence from the independent variables is exerted.

On the other hand, the slope (or b1) represents the mean change in the dependent variable for each one-unit shift in the independent variable. It's the heart of our quest to estimate the expected change with precision; it tells us how steeply the line slopes upward or downward, revealing the nature of the relationship. Is it positive, indicating that as the independent variable increases, so does the dependent variable? Or is it negative, suggesting an inverse relationship?

Interpreting the Coefficients

By decoding these coefficients, we shed light on the causal connections within our data. In the given example, the slope b1 was calculated to be -0.035. This indicates for every additional centimeter in distance from the UV light source, the index of sunburn decreases. This value is our compass; it guides us in predicting how one variable will behave as another changes.

However, it is essential to remember that coefficients alone do not tell the entire story. They give direction and magnitude but must be considered within the larger context of the model's assumptions and the collected data's reliability.
Dependent Variable
The dependent variable is the centerpiece of any regression analysis; the outcome we aim to predict or understand. Acting much like the subject of a sentence, it's the primary focus of our investigation that we expect to change or vary in response to different factors.

In the context of our study, the dependent variable is the sunburn index of plants, denoted by y. It depends on one or more independent variables, which in our case, is the distance from a UV light source. Think of the dependent variable as a kite dancing in the wind鈥攖he wind being the independent variables. It moves in tune with the forces acting upon it.

The Role in Regression

The whole purpose of linear regression is to create an equation that predicts the dependent variable's value based on the independent variables. The accuracy and usability of our model rest upon how well we understand and measure this variable.

For students and researchers alike, it's critical to identify and measure the dependent variable accurately. Any errors here can lead to mistaken conclusions or misguided predictions. Hence, one should be meticulous in defining what the dependent variable is and how it's captured within the dataset.
Independent Variable
The independent variable is the experimenter's lever; by moving this, we observe its influence on the dependent variable. This is the variable that we believe will cause change and that we either manipulate (as in an experiment) or observe changes in (as in observational studies).

In our plant study, the independent variable is the distance from the UV light source, symbolized as x. Unlike the dependent variable, which is our subject, the independent variable is akin to the verb鈥攊t's what we think does the acting or influencing.

Choosing the Right Variables

Selecting the appropriate independent variable is a cornerstone in constructing a valid regression model. It should be something that is likely to be the cause of changes in the dependent variable. Otherwise, we could end up with misleading results due to confounding factors or reverse causality.

It's also essential to measure the independent variable with precision. Inaccuracies here could dilute the observed effect on the dependent variable or, worse, suggest a relationship where none exists.
Mean Change Estimation
When we delve into statistics, particularly regression analysis, mean change estimation is a central concept that allows us to predict the average change in the dependent variable for a given change in the independent variable. It brings to life the practical significance of our mathematical explorations.

In the context of the given problem, estimating the mean change in the sunburn index per centimeter increase in distance showcases the use of linear regression in making quantitative predictions. We are looking for a tangible figure to represent an abstract relationship鈥攁 figure that measures the impact.

Estimating with Accuracy

The slope of our regression line, b1, is this estimate of mean change. In plain language, it answers the question, 'If I move one unit along the X axis, how much will the Y variable change on average?'. For our plant study, a slope of -0.035 translates to a reduction in the sunburn index by 0.035 for every extra centimeter of distance from the UV light source.

The precision of this estimation is paramount. It is influenced by the data's quality and the appropriateness of the regression model. Uncertainties in mean change estimation must always be acknowledged as they highlight the confidence we can place in our predictive abilities. Properly communicating this precision鈥攐r the lack thereof鈥攊s as crucial as the estimate itself, for it guides decision-making and shapes expectations of future outcomes.

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Most popular questions from this chapter

The article "Performance Test Conducted for a Gas Air-Conditioning System" (American Society of Heating, Refrigerating, and Air Conditioning Engineering [1969]: 54 ) reported the following data on maximum outdoor temperature \((x)\) and hours of chiller operation per day \((y)\) for a 3 -ton residential gas air- conditioning system: \(\begin{array}{rrrrrrr}x & 72 & 78 & 80 & 86 & 88 & 92 \\ y & 4.8 & 7.2 & 9.5 & 14.5 & 15.7 & 17.9\end{array}\) Suppose that the system is actually a prototype model, and the manufacturer does not wish to produce this model unless the data strongly indicate that when maximum outdoor temperature is \(82^{\circ} \mathrm{F}\), the true average number of hours of chiller operation is less than \(12 .\) The appropriate hypothesis is then $$ H_{0}: \alpha+\beta(82)=12 \text { versus } H_{a}: \alpha+\beta(82)<12 $$

In anthropological studies, an important characteristic of fossils is cranial capacity. Frequently skulls are at least partially decomposed, so it is necessary to use other characteristics to obtain information about capacity. One such measure that has been used is the length of the lambda-opisthion chord. The article "Vertesszollos and the Presapiens Theory" (American Journal of Physical Anthropology \([1971]\) ) reported the accompanying data for \(n=7\) Homo erectus fossils. \(x\) (chord \(\begin{aligned}&\text { length in } \mathrm{mm} \text { ) } & 78 & 75 & 78 & 81 & 84 & 86 & 87\end{aligned}\) \(y\) (capacity in \(\mathrm{cm}^{3}\) ) \(\begin{array}{lllllll}850 & 775 & 750 & 975 & 915 & 1015 & 1030\end{array}\) Suppose that from previous evidence, anthropologists had believed that for each 1 -mm increase in chord length, cranial capacity would be expected to increase by \(20 \mathrm{~cm}^{3}\). Do these new experimental data strongly contradict prior belief?

'The article "Photocharge Effects in Dye Sensitized \(\mathrm{Ag}[\mathrm{Br}, \mathrm{I}]\) Emulsions at Millisecond Range Exposures" (Photographic Science and Engineering [1981]: \(138-144\) ) gave the accompanying data on \(x=\%\) light absorption and \(y=\) peak photovoltage. \(\begin{array}{rrrrrrrrrr}x & 4.0 & 8.7 & 12.7 & 19.1 & 21.4 & 24.6 & 28.9 & 29.8 & 30.5 \\ y & 0.12 & 0.28 & 0.55 & 0.68 & 0.85 & 1.02 & 1.15 & 1.34 & 1.29\end{array}\) \(\sum x=179.7 \quad \sum x^{2}=4334.41\) \(\sum y=7.28 \quad \sum y^{2}=7.4028 \quad \sum x y=178.683\) a. Construct a scatterplot of the data. What does it suggest? b. Assuming that the simple linear regression model is appropriate, obtain the equation of the estimated regression line. c. How much of the observed variation in peak photovoltage can be explained by the model relationship? d. Predict peak photovoltage when percent absorption is 19.1, and compute the value of the corresponding residual. e. The authors claimed that there is a useful linear relationship between the two variables. Do you agree? Carry out a formal test. f. Give an estimate of the average change in peak photovoltage associated with a \(1 \%\) increase in light absorption. Your estimate should convey information about the precision of estimation. g. Give an estimate of true average peak photovoltage when percentage of light absorption is 20 , and do so in a way that conveys information about precision.

It seems plausible that higher rent for retail space could be justified only by a higher level of sales. A random sample of \(n=53\) specialty stores in a chain was selected, and the values of \(x=\) annual dollar rent per square foot and \(y=\) annual dollar sales per square foot were determined, resulting in \(r=.37\) ("Association of Shopping Center Anchors with Performance of a Nonanchor Specialty Chain Store,"Journal of Retailing [1985]: \(61-74\) ). Carry out a test at significance level \(.05\) to see whether there is in fact a positive linear association between \(x\) and \(y\) in the population of all such stores.

An}\( experiment to study the relationship between \)x=\( time spent exercising (min) and \)y=\( amount of oxygen consumed during the exercise period resulted in the following summary statistics. $$ \begin{aligned} &n=20 \quad \sum x-50 \quad \sum y-16,705 \quad \sum x^{2}-150 \\ &\sum y^{2}=14,194,231 \quad \sum x y=44,194 \end{aligned} $$ a. Estimate the slope and \)y\( intercept of the population regression line. b. One sample observation on oxygen usage was 757 for a 2 -min exercise period. What amount of oxygen consumption would you predict for this exercise period, and what is the corresponding residual? c. Compute a \)99 \%$ confidence interval for the true average change in oxygen consumption associated with a 1 -min increase in exercise time.
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