3 Suppose the relationship between applied stress x and time-to-failure y is

Example 12.3 Suppose the relationship between applied stress x and time-to-failure y is

  described by the simple linear regression model with true regression line y 5 65 2 1.2x and s58 . Then for any fixed value x of stress, time-to-failure has a normal distribution with mean value

  65 2 1.2x and standard deviation 8.

  Roughly speaking, in the population consisting of all (x, y) points, the magnitude of a typical deviation from the true regression line is about 8. For x 5 20 , Y has

  mean value m Y 20 5 65 2 1.2(20) 5 41, so

  P(Y . 50 when x 5 20) 5 P aZ .

  8 b 5 1 2 (1.13) 5 .1292

  The probability that time-to-failure exceeds 50 when applied stress is 25 is, because

  m Y 25 5 35,

  P(Y . 50 when x 5 25) 5 P aZ .

  8 b 5 1 2 (1.88) 5 .0301

  These probabilities are illustrated as the shaded areas in Figure 12.5.

  y

  P(Y

  50 when x 20) .1292

  P(Y

  50 when x 25) .0301

  35 True regression line

  Figure 12.5 Probabilities based on the simple linear regression model

  Suppose that Y 1 denotes an observation on time-to-failure made with x 5 25

  and Y 2 denotes an independent observation made with x 5 24 . Then Y 1 2Y 2 is nor- mally distributed with mean value E(Y 1 2Y 2 )5b 1 5 21.2 , variance V(Y 1 2Y 2 )5s 2 1s 2 5 128 , and standard deviation 1128 5 11.314 . The prob- ability that Y 1 exceeds Y 2 is

  12.1 The Simple Linear Regression Model

  b 5 P(Z . .11) 5 .4562

  That is, even though we expected Y to decrease when x increases by 1 unit, it is not unlikely that the observed Y at x11 will be larger than the observed Y at x.

  ■

  EXERCISES Section 12.1 (1–11)

  1. The efficiency ratio for a steel specimen immersed in a phos-

  Construct scatter plots of NO x emissions versus age. What

  phating tank is the weight of the phosphate coating divided

  appears to be the nature of the relationship between these

  by the metal loss (both in mgft 2 ). The article “Statistical

  two variables? [Note: The authors of the cited article com-

  Process Control of a Phosphate Coating Line” (Wire J. Intl.,

  mented on the relationship.]

  May 1997: 78–81) gave the accompanying data on tank tem-

  3. Bivariate data often arises from the use of two different tech-

  perature (x) and efficiency ratio (y).

  niques to measure the same quantity. As an example, the

  accompanying observations on x5 hydrogen concentration

  Ratio

  .84 1.31 1.42 1.03 1.07 1.08 1.04 (ppm) using a gas chromatography method and y5 concen- tration using a new sensor method were read from a graph in

  the article “A New Method to Measure the Diffusible

  Ratio

  Hydrogen Content in Steel Weldments Using a Polymer Electrolyte-Based Hydrogen Sensor” (Welding Res., July

  1997: 251s–256s).

  a. Construct stem-and-leaf displays of both temperature and

  x

  efficiency ratio, and comment on interesting features.

  b. Is the value of efficiency ratio completely and uniquely

  y

  determined by tank temperature? Explain your reasoning.

  c. Construct a scatter plot of the data. Does it appear that

  Construct a scatter plot. Does there appear to be a very strong

  efficiency ratio could be very well predicted by the value

  relationship between the two types of concentration meas-

  of temperature? Explain your reasoning.

  urements? Do the two methods appear to be measuring

  2. The article “Exhaust Emissions from Four-Stroke Lawn

  roughly the same quantity? Explain your reasoning.

  Mower Engines” (J. of the Air and Water Mgmnt. Assoc.,

  4. A study to assess the capability of subsurface flow wetland sys-

  1997: 945–952) reported data from a study in which both a

  tems to remove biochemical oxygen demand (BOD) and vari-

  baseline gasoline mixture and a reformulated gasoline were

  ous other chemical constituents resulted in the accompanying

  used. Consider the following observations on age (yr) and

  data on x 5 BOD mass loading (kghad) and y 5 BOD mass

  NO x emissions (gkWh):

  removal (kghad) (“Subsurface Flow Wetlands—A Performance Evaluation,” Water Envir. Res., 1995: 244–247).

  6 7 8 9 10 a. Construct boxplots of both mass loading and mass

  Age

  16 9 0 12 4 removal, and comment on any interesting features.

  1.24 b. Construct a scatter plot of the data, and comment on any

  1.42 interesting features.

  CHAPTER 12 Simple Linear Regression and Correlation

  5. The article “Objective Measurement of the Stretchability of

  ered regressing y 5 28-day standard-cured strength (psi)

  Mozzarella Cheese” (J. of Texture Studies, 1992: 185–194)

  against x 5 accelerated strength (psi) . Suppose the equation

  reported on an experiment to investigate how the behavior of

  of the true regression line is y 5 1800 1 1.3x .

  mozzarella cheese varied with temperature. Consider the

  a. What is the expected value of 28-day strength when accel-

  accompanying data on x 5 temperature and y 5 elongation

  erated strength

  () at failure of the cheese. [Note: The researchers were

  b. By how much can we expect 28-day strength to change

  Italian and used real mozzarella cheese, not the poor cousin

  when accelerated strength increases by 1 psi?

  widely available in the United States.]

  c. Answer part (b) for an increase of 100 psi. d. Answer part (b) for a decrease of 100 psi.

  x

  59 63 68 72 74 78 83 8. Referring to Exercise 7, suppose that the standard deviation

  of the random deviation is 350 psi. P

  a. What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of acceler- ated strength is 2000?

  a. Construct a scatter plot in which the axes intersect at

  b. Repeat part (a) with 2500 in place of 2000.

  (0, 0). Mark 0, 20, 40, 60, 80, and 100 on the horizontal

  c. Consider making two independent observations on 28-day

  axis and 0, 50, 100, 150, 200, and 250 on the vertical

  strength, the first for an accelerated strength of 2000 and

  axis.

  the second for x 5 2500 . What is the probability that the

  b. Construct a scatter plot in which the axes intersect at

  second observation will exceed the first by more than

  (55, 100), as was done in the cited article. Does this

  1000 psi?

  plot seem preferable to the one in part (a)? Explain

  d. Let Y 1 and Y 2 denote observations on 28-day strength when

  your reasoning.

  x5x and x5x , respectively. By how much would x

  c. What do the plots of parts (a) and (b) suggest about the

  have to exceed x 1 in order that P(Y 2 .Y 1 ) 5 .95 ?

  nature of the relationship between the two variables?

  9. The flow rate y (m 3 min) in a device used for air-quality

  6. One factor in the development of tennis elbow, a malady that

  measurement depends on the pressure drop x (in. of water)

  strikes fear in the hearts of all serious tennis players, is the

  across the device’s filter. Suppose that for x values between

  impact-induced vibration of the racket-and-arm system at ball

  5 and 20, the two variables are related according to the simple

  contact. It is well known that the likelihood of getting tennis

  linear regression model with true regression line

  elbow depends on various properties of the racket used.

  y 5 2.12 1 .095x .

  Consider the scatter plot of x5 racket resonance frequency

  a. What is the expected change in flow rate associated with

  (Hz) and y 5 sum of peak-to-peak acceleration (a character-

  a 1-in. increase in pressure drop? Explain.

  istic of arm vibration, in msecsec) for n 5 23 different rack-

  b. What change in flow rate can be expected when pressure

  ets (“Transfer of Tennis Racket Vibrations into the Human

  drop decreases by 5 in.?

  Forearm,” Medicine and Science in Sports and Exercise,

  c. What is the expected flow rate for a pressure drop of

  1992: 1134–1140). Discuss interesting features of the data

  10 in.? A drop of 15 in.?

  and scatter plot.

  d. Suppose s 5 .025 and consider a pressure drop of 10 in. What is the probability that the observed value of flow rate

  y

  will exceed .835? That observed flow rate will exceed .840? e. What is the probability that an observation on flow rate

  when pressure drop is 10 in. will exceed an observation on

  flow rate made when pressure drop is 11 in.?

  10. Suppose the expected cost of a production run is related to the

  size of the run by the equation

  . Let Y denote an observation on the cost of a run. If the variables’ size and cost

  y 5 4000 1 10x

  are related according to the simple linear regression model,

  could it be the case that P(Y . 50 when x 5 100) 5 .05

  and ? Explain.

  P(Y . 6500 when x 5 200) 5 .10

  11. Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature (°F ) in the chamber in

  22 x

  which the reaction takes place according to the simple lin-

  ear regression model with equation y 5 5.00 2 .01x and s 5 .075 .

  7. The article “Some Field Experience in the Use of an

  a. What is the expected change in reaction time for a 1°F

  Accelerated Method in Estimating 28-Day Strength of

  increase in temperature? For a 10°F increase in

  Concrete” (J. of Amer. Concrete Institute, 1969: 895) consid-

  temperature?

  12.2 Estimating Model Parameters

  b. What is the expected reaction time when temperature is

  d. What is the probability that two independently observed

  200°F? When temperature is 250°F?

  reaction times for temperatures 1° apart are such that the

  c. Suppose five observations are made independently on reac-

  time at the higher temperature exceeds the time at the

  tion time, each one for a temperature of 250°F. What is the

  lower temperature?

  probability that all five times are between 2.4 and 2.6 hr?