Concept explainers
The accompanying data on x = frequency (MHz) and y = output power (W) for a certain laser configuration was read from a graph in the article “Frequency Dependence in RF Discharge Excited Waveguide CO2 Lasers" (IEEE J. of Quantum Electronics, 1984: 509-514).
X | 60 | 63 | 77 | 100 | 125 | 157 | 186 | 222 |
V | 16 | 17 | 19 | 21 | 22 | 20 | 15 | 5 |
A computer analysis yielded the following information for a quadratic regression model:
- a. Does the quadratic model appear to be suitable for explaining observed variation in output power by relating it to frequency?
- b. Would the simple linear regression model be nearly as satisfactory as the quadratic model?
- c. Do you think it would be worth considering a cubic model?
- d. Compute a 95% CI for expected power output when frequency is 100.
- e. Use a 95% PI to predict the power from a single experimental run when frequency is 100.
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Probability and Statistics for Engineering and the Sciences
- Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?arrow_forwardAn attempt was made to construct a regression model explaining student scores in intermediate economics courses (Waldauer, Duggal, and Williams 1992). The population regression model assumed thatY = total student score in intermediate economics coursesX1 = mathematics score on Scholastic Aptitude TestX2 = verbal score on Scholastic Aptitude TestX3 = grade in college algebra (A = 4, B = 3, C = 2, D = 1)X4 = grade in college principles of economics courseX5 = dummy variable taking the value 1 if the student is female and 0 if maleX6 = dummy variable taking the value 1 if the instructor is male and 0 if femaleX7 = dummy variable taking the value 1 if the student and instructor are the same gender and 0 otherwiseThis model was fitted to data on 262 students. Next we report t-ratios, so that tj is the ratio of the estimate of bj to its associated estimated standard error. These ratios are as follows:t1 = 4.69, t2 = 2.89, t3 = 0.46, t4 = 4.90,t5 = 0.13, t6 = -1.08, t7 = 0.88The objective of…arrow_forward11) A simple linear regression model based on 20 observations. The F-stat for the model is 21.44 and the SSE is 1.41. The standard error for the coefficient of X is 0.2. a) Complete the ANOVA table. b) Find the t-stat of the co-efficient of X c) Find the co-efficient of X.arrow_forward
- )A county real estate appraiser wants to develop a statistical model to predict the appraised value of 3) houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(u) = Bo + Bix, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 73 houses in Fast Meadow, the following results were obtained: y = 73.80 + 19.72x What are the properties of the least squares line, y = 73.80 + 19.72x? A) Average error of prediction is 0, and SSE is minimum. B) It will always be a statistically useful predictor of y. C) It is normal, mean 0, constant variance, and independent. D) All 73 of the sample y-values fall on the line.arrow_forwardSuppose the simple linear regression model, Yi = β0 + β1 xi + Ei, is used to explain the relationship between x and y. A random sample of n = 12 values for the explanatory variable (x) was selected and the corresponding values of the response variable (y) were observed. A summary of the statistics is presented in the photo attached. Let b1 denote the least squares estimator of the slope coefficient, β1. What is the value of b1?arrow_forwardA forecaster used the regression equation Qt = a + bt+c₁D₁ + C2D2 + c3D3 and quarterly sales data for 2004/–2021/V (t = 1, ..., 64) for an appliance manufacturer to obtain the results shown below. Q is quarterly sales, and D₁, D2 and D3 are dummy variables for quarters /, //, and ///. DEPENDENT VARIABLE: QT OBSERVATIONS: VARIABLE INTERCEPT T D1 D2 D3 R-SQUARE 64 0.8768 PARAMETER ESTIMATE 30.0 1.5 10.0 25.0 40.0 F-RATIO P-VALUE ON F 107.982 0.0001 STANDARD ERROR 12.80 0.70 3.00 7.20 15.80 T-RATIO 2.34 2.14 3.33 3.47 2.53 P-VALUE 0.0224 0.0362 0.0015 0.0010 0.0140 At the 5 percent level of significance, is there a statistically significant trend in sales?arrow_forward
- 14.9 The article "Pulp Brightness Reversion: Influ- ence of Residual Lignin on the Brightness Reversion of Bleached Sulfite and Kraft Pulps" (TAPPI [1964]: 653-662) proposed a quadratic regression model to de- scribe the relationship between x = degree of delignifica- tion during the processing of wood pulp for paper and y = total chlorine content. Suppose that the population regression model is y =22075x-4x² + e a. Graph the regression function 220 +75x4x² over x values between 2 and 12. (Substitute x = 2, 4, 6, 8, 10, and 12 to find points on the graph, and connect them with a smooth curve.) b. Would mean chlorine content be higher for a degree of delignification value of 8 or 10? c. What is the change in mean chlorine content when the degree of delignification increases from 8 to 9? From 9 to 10?arrow_forwardCompute for the necessary linear regressions based on the given data. (can use Excel or Minitab for this) An article in the Tappi Journal (March, 1986) presented data on green liquor Na2S concentration (in grams per liter) and paper machine production (in tons per day). The data (read from a graph) are shown as follows: (a) Fit a simple linear regression model with y green liquor Na2S concentration and x production. Draw a scatter diagram of the data and the resulting least squares fitted model.(b) Find the fitted value of y corresponding to x = 910 and the associated residual.arrow_forwardConsider the multiple regression model Y₁ = Bo + B₁x1₁j + B₂x2j+B3 x 3j+ €j under the usual assumptions labelled A1, A2, A3, A4, A5, A6. Briefly explain which type of graphs are performed in the analysis of residuals.arrow_forward
- A study was conducted to see whether heart rate (y) on swimmers linearly related to their age (x1) and swimming time for 2000 meters (x2). A random sample of ten swimmers was selected and the result is shown in the following Microsoft Excel output. (a)Interpret the value of R2 from the output. (b)Conduct a hypothesis test to test whether the linear regression model is fit or not using a = 0.05. (c)Calculate the 95% confidence interval for the coefficient value for age.arrow_forwardThe measure of standard error can also be applied to the parameter estimates resulting from linear regressions. For example, consider the following linear regression equation that describes the relationship between education and wage: WAGEi=β0+β1EDUCi+εi where WAGEi is the hourly wage of person i (i.e., any specific person) and EDUCi is the number of years of education for that same person. The residual εi encompasses other factors that influence wage, and is assumed to be uncorrelated with education and have a mean of zero. Suppose that after collecting a cross-sectional data set, you run an OLS regression to obtain the following parameter estimates: WAGEi=−11.5+6.1 EDUCi If the standard error of the estimate of β1 is 1.32, then the true value of β1 lies between(4.78, 4.12, 3.46, 5.44) and (6.76, 7.42, 8.74) . As the number of observations in a data set grows, you would expect this range to (DECREASE , INCREASE) in size.arrow_forwardA study was conducted to see whether heart rate (y) on swimmers linearly related to their age (x1) and swimming time for 2000 meters (x2). A random sample of ten swimmers was selected and the result is shown in the following Microsoft Excel output. (a) Interpret the value of R2 from the output. (b) Conduct a hypothesis test to test whether the linear regression model is fit or not using a = 0.05. (c) Calculate the 95% confidence interval for the coefficient value for age.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning