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Chapter 12: Problem 119

The article "Determination of Biological Maturity and Effect of Harvesting and Drying Conditions on Milling Quality of Paddy" (J. Agric. Engr. Res., 1975: 353-361) reported the following data on date of harvesting ( \(x\), the number of days after flowering) and yield of paddy, a grain farmed in India ( \(y\), in \(\mathrm{kg} / \mathrm{ha}\) ). $$ \begin{aligned} &\begin{array}{l|cccccccc} x & 16 & 18 & 20 & 22 & 24 & 26 & 28 & 30 \\ \hline y & 2508 & 2518 & 3304 & 3423 & 3057 & 3190 & 3500 & 3883 \end{array}\\\ &\begin{array}{c|cccccccc} x & 32 & 34 & 36 & 38 & 40 & 42 & 44 & 46 \\ \hline y & 3823 & 3646 & 3708 & 3333 & 3517 & 3241 & 3103 & 2776 \end{array} \end{aligned} $$ a. Construct a scatter plot of the data. What model is suggested by the plot? b. Use a statistical software package to fit the model suggested in (a) and test its utility. c. Use the software package to obtain a prediction interval for yield when the crop is harvested 25 days after flowering, and also a confidence interval for expected yield in situations where the crop is harvested 25 days after flowering. How do these two intervals compare to each other? Is this result consistent with what you learned in simple linear regression? Explain. d. Use the software package to obtain a PI and CI when \(x=40\). How do these intervals compare to the corresponding intervals obtained in (c)? Is this result consistent with what you learned in simple linear regression? Explain. e. Carry out a test of hypotheses to decide whether the quadratic predictor in the model fit in (b) provides useful information about yield (presuming that the linear predictor remains in the model).

Short Answer

Expert verified
The plot suggests a quadratic model. Intervals confirm regression principles; quadratic term is significant. Outcomes vary based on \(x\) position.

Step by step solution

01

Plot the Data

First, we need to plot the provided data points, with the number of days after flowering, \(x\), on the x-axis and yield, \(y\), on the y-axis. Upon plotting, the trend suggests a quadratic relationship due to the parabolic shape observed in the data distribution.
02

Fit a Quadratic Model

Since the scatter plot suggested a quadratic relationship, we fit a quadratic model (\(y = ax^2 + bx + c\)) to the data using a statistical software package. We then evaluate the utility of the model using statistical tests, such as the F-test, to determine its suitability.
03

Prediction Interval for Harvesting at 25 Days

Using the statistical software, calculate the prediction interval (PI) for \(x = 25\). This interval estimates the range for individual yields that could occur when harvesting 25 days after flowering.
04

Confidence Interval for Expected Yield at 25 Days

Calculate the confidence interval (CI) for the expected yield when \(x = 25\). This CI reflects the range within which the average yield is expected to fall when the crop is harvested at this time. Compare this CI with the PI to note that the PI is usually wider, confirming knowledge from simple linear regression.
05

Prediction and Confidence Intervals for Harvesting at 40 Days

Repeat the calculations for \(x = 40\) to obtain both the prediction and confidence intervals. Compare these results to those obtained for \(x = 25\), noting that intervals can vary based on the position relative to the model's center, consistent with regression analyses.
06

Hypothesis Test for Quadratic Predictor

Perform a hypothesis test on the quadratic term (\(x^2\)) in the model to determine if it provides significant information about yield. This involves checking if the coefficient of the quadratic term is significantly different from zero, often using a t-test.

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Key Concepts

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

Milling Quality
Milling quality refers to how successfully grains, like paddy, can be processed into the final product without significant loss or damage. Here, researchers are interested in determining how the date of harvesting affects the yield and quality of the paddy. Factors such as moisture content, grain texture, and hardness influence milling quality. Understanding these factors can help optimize harvesting times for better quality and yield.
  • Moisture content: It plays a critical role in determining the quality of the grains, influencing the ease of milling.
  • Grain texture: Softer grains might break more easily, affecting overall quality.
Given the data, quadratic regression analysis is useful to establish the relationship between harvest timing (in days after flowering) and the yield, potentially impacting the milling quality. If grains are picked too early or too late, the milling quality might decrease, leading to lower yields and potential economic losses.
Prediction Interval
A prediction interval (PI) provides a range where individual observations are expected to fall, with a certain level of confidence. In the exercise, the PI was used to estimate yield when the crop is harvested at 25 and 40 days after flowering.
  • PI for 25 days: This interval offers a possible range of yields that might be observed for a single harvest at this time.
  • PI for 40 days: Similarly, this interval predicts the potential yield outcomes for harvests occurring later, at 40 days.
It is important to remember that the PI is broader than the confidence interval, as it takes into account the variability among individual observations. This means we might witness more fluctuation in potential yields when considering individual harvests at these times. Understanding PIs helps farmers and researchers anticipate variations in outcomes, aiding in better planning and management.
Confidence Interval
A confidence interval (CI) offers a range where the true mean of a population parameter, such as average yield, is expected to fall. In this study, the CI is utilized to estimate the average yield for crops harvested 25 and 40 days after flowering.
  • CI at 25 days: This interval estimates the true average yield if the crop is harvested at this stage.
  • Comparison with PI: The CI is usually narrower than the PI because it focuses on the average rather than individual data points.
Understanding the CI helps to make more informed decisions regarding agricultural practices by considering the expected average yields, rather than just individual outcomes. CIs are crucial in verifying results are consistent and reliable, especially in comparisons like those presented between 25 and 40 days in the exercise.
Hypothesis Testing
Hypothesis testing in the context of this exercise involves determining whether the quadratic predictor 鈥� part of the quadratic regression model 鈥� significantly aids in predicting paddy yield. This is achieved by assessing whether the inclusion of the quadratic term truly provides meaningful information.
  • Quadratic term test: Verify if the coefficient of the quadratic term is statistically different from zero, often using a t-test.
  • F-test: Sometimes used to test the overall fit of the model, checking how well it represents the data.
Successfully conducting hypothesis tests allows us to confirm or reject assumptions made during model building. If the quadratic term is significant, it validates its inclusion in the model as being impactful for estimating yields. Through hypothesis testing, one ensures that model refinements genuinely contribute to understanding the agricultural phenomena in question.

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Most popular questions from this chapter

Fit the model \(Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon\) to the data $$ \begin{array}{rrr} x_{1} & x_{2} & y \\ -1 & -1 & 1 \\ -1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & 4 \end{array} $$ a. Determine \(\boldsymbol{X}\) and \(\boldsymbol{y}\) and express the normal equations in terms of matrices. b. Determine the \(\hat{\boldsymbol{\beta}}\) vector, which contains the estimates for the three coefficients in the model. c. Determine \(\hat{\boldsymbol{y}}\), the predictions for the four observations, and also the four residuals. Find SSE by summing the four squared residuals. Use this to get the estimated variance MSE. d. Use the MSE and \(c_{11}\) to get a \(95 \%\) confidence interval for \(\beta_{1}\). e. Carry out a \(t\) test for the hypothesis \(H_{0}\) : \(\beta_{1}=0\) against a two-tailed alternative, and interpret the result. f. Form the analysis of variance table and carry out the \(F\) test for the hypothesis \(H_{0}: \beta_{1}=\beta_{2}\) \(=0\). Find \(R^{2}\) and interpret.

A regression of \(y=\) calcium content \((\mathrm{g} / \mathrm{L})\) on \(x=\) dissolved material \(\left(\mathrm{mg} / \mathrm{cm}^{2}\right)\) was reported in the article "Use of Fly Ash or Silica Fume to Increase the Resistance of Concrete to Feed Acids" (Mag. Concrete Res., 1997: 337-344). The equation of the estimated regression line was \(y=3.678+.144 x\), with \(r^{2}=.860\), based on \(n=23\). a. Interpret the estimated slope \(.144\) and the coefficient of determination .860. b. Calculate a point estimate of the true average calcium content when the amount of dissolved material is \(50 \mathrm{mg} / \mathrm{cm}^{2}\). c. The value of total sum of squares was SST \(=320.398\). Calculate an estimate of the error standard deviation \(\sigma\) in the simple linear regression model.

Torsion during hip external rotation and extension may explain why acetabular labral tears occur in professional athletes. The article "Hip Rotational Velocities During the Full Golf Swing" \((J\). Sport Sci. Med., 2009: 296 - 299) reported on an investigation in which lead hip internal peak rotational velocity \((x)\) and trailing hip peak external rotational velocity \((y)\) were determined for a sample of 15 golfers. Data provided by the article's authors was used to calculate the following summary quantities: $$ \begin{aligned} &S_{x x}=64,732.83, \quad S_{y y}=130,566.96, \\ &S_{x y}=44,185.87 \end{aligned} $$ Separate normal probability plots showed very substantial linear patterns. a. Calculate a point estimate for the population correlation coefficient. b. If the simple linear regression model were fit to the data, what proportion of variation in external velocity could be attributed to the model relationship? What would happen to this proportion if the roles of \(x\) and \(y\) were reversed? Explain. c. Carry out a test at significance level .01 to decide whether there is a linear relationship between the two velocities in the sampled population; your conclusion should be based on a \(P\)-value. d. Would the conclusion of (c) have changed if you had tested appropriate hypotheses to decide whether there is a positive linear association in the population? What if a significance level of \(.05\) rather than \(.01\) had been used?

If there is at least one \(x\) value at which more than one observation has been made, there is a formal test procedure for testing \(H_{0}: \mu_{Y \cdot x}=\beta_{0}+\beta_{1} x\) for some values \(\beta_{0}, \beta_{1}\) (the true regression function is linear) versus \(H_{\mathrm{a}}: H_{0}\) is not true (the true regression function is not linear) Suppose observations are made at \(x_{1}, x_{2}, \ldots, x_{c}\). Let \(Y_{11}, Y_{12}, \ldots, Y_{1 n_{1}}\) denote the \(n_{1}\) observations when \(x=x_{1} ; \ldots ; Y_{c 1}, Y_{c 2}, \ldots, Y_{c n_{c}}\) denote the \(n_{c}\) observations when \(x=x_{c}\). With \(n=\Sigma n_{i}\) (the total number of observations), SSE has \(n-2\) df. We break SSE into two pieces, SSPE (pure error) and SSLF (lack of fit), as follows: $$ \begin{aligned} \mathrm{SSPE} &=\sum_{i} \sum_{j}\left(Y_{i j}-\bar{Y}_{i} .\right)^{2} \\ &=\sum_{i} \sum_{j} Y_{i j}^{2}-\sum_{i} n_{i}\left(\bar{Y}_{i} .\right)^{2} \end{aligned} $$ $$ \text { SSLF }=\text { SSE }-\text { SSPE } $$ The \(n_{i}\) observations at \(x_{i}\) contribute \(n_{i}-1\) df to SSPE, so the number of degrees of freedom for SSPE is \(\Sigma_{i}\left(n_{i}-1\right)=n-c\) and the degrees of freedom for SSLF is \(n-2-(n-c)=c-2\). Let MSPE \(=\operatorname{SSPE} /(n-c), \operatorname{MSLF}=\operatorname{SSLF} /(c-2) .\) Then it can be shown that whereas \(E(\) MSPE \()=\sigma^{2}\) whether or not \(H_{0}\) is true, \(E\) (MSLF) \(=\sigma^{2}\) if \(H_{0}\) is true and \(E(\) MSLF \()>\sigma^{2}\) if \(H_{0}\) is false. Test statistic: \(F=\) MSLF/MSPE Rejection region: \(f \geq F_{\alpha, c-2, n-c}\) The following data comes from the article "Changes in Growth Hormone Status Related to Body Weight of Growing Cattle" (Growth, 1977: 241-247), with \(x=\) body weight and \(y=\) metabolic clearance rate/ body weight. $$ \begin{aligned} &\begin{array}{l|lllllll} x & 110 & 110 & 110 & 230 & 230 & 230 & 360 \\ \hline y & 235 & 198 & 173 & 174 & 149 & 124 & 115 \end{array}\\\ &\begin{array}{r|rrrrrrr} x & 360 & 360 & 360 & 505 & 505 & 505 & 505 \\ \hline y & 130 & 102 & 95 & 122 & 112 & 98 & 96 \end{array} \end{aligned} $$ (So \(c=4, n_{1}=n_{2}=3, n_{3}=n_{4}=4\).) a. Test \(H_{0}\) versus \(H_{\mathrm{a}}\) at level \(.05\) using the lackof-fit test just described. b. Does a scatter plot of the data suggest that the relationship between \(x\) and \(y\) is linear? How does this compare with the result of part (a)? (A nonlinear regression function was used in the article.)

Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article "Ion Beam-Assisted Etching of Aluminum with Chlorine" (J. Electrochem. Soc., 1985: 2010-2012) gives the accompanying data (read from a graph) on chlorine flow \((x\), in SCCM) through a nozzle used in the etching mechanism and etch rate \((y\), in \(100 \mathrm{~A} / \mathrm{min})\). $$ \begin{array}{l|lrrrrrrrr} x & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \\ \hline y & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0 \end{array} $$ a. Does the simple linear regression model specify a useful relationship between chlorine flow and etch rate? b. Estimate the true average change in etch rate associated with a 1-SCCM increase in flow rate using a \(95 \%\) confidence interval, and interpret the interval. c. Calculate a \(95 \%\) CI for \(\mu_{Y \cdot 3.0}\), the true average etch rate when flow \(=3.0\). Has this average been precisely estimated? d. Calculate a \(95 \%\) PI for a single future observation on etch rate to be made when flow \(=3.0 .\) Is the prediction likely to be accurate? e. Would the \(95 \%\) CI and PI when flow \(=2.5\) be wider or narrower than the corresponding intervals of parts (c) and (d)? Answer without actually computing the intervals. f. Would you recommend calculating a \(95 \%\) PI for a flow of 6.0? Explain. g. Calculate simultaneous CI's for true average etch rate when chlorine flow is \(2.0,2.5\), and \(3.0\), respectively. Your simultaneous confidence level should be at least \(97 \%\).
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