10 The slant shear test is the most widely accepted procedure for assessing the quality

Example 7.10 The slant shear test is the most widely accepted procedure for assessing the quality

  of a bond between a repair material and its concrete substrate. The article “Testing the Bond Between Repair Materials and Concrete Substrate” (ACI Materials J., 1996: 553–558) reported that in one particular investigation, a sample of 48 shear

  strength observations gave a sample mean strength of 17.17 Nmm 2 and a sample standard deviation of 3.28 Nmm 2 . A lower confidence bound for true average shear

  strength m with confidence level 95 is (3.28)

  That is, with a confidence level of 95, we can say that m . 16.39 .

  ■

  EXERCISES Section 7.2 (12–27)

  12. A random sample of 110 lightning flashes in a certain

  the sample mean CO 2 level (ppm) was 654.16, and the sam-

  region resulted in a sample average radar echo duration

  ple standard deviation was 164.43.

  of .81 sec and a sample standard deviation of .34 sec

  a. Calculate and interpret a 95 (two-sided) confidence

  (“Lightning Strikes to an Airplane in a Thunderstorm,” J. of

  interval for true average CO 2 level in the population of

  Aircraft, 1984: 607–611). Calculate a 99 (two-sided) con-

  all homes from which the sample was selected.

  fidence interval for the true average echo duration m, and

  b. Suppose the investigators had made a rough guess of 175

  interpret the resulting interval.

  for the value of s before collecting data. What sample

  13. The article “Gas Cooking, Kitchen Ventilation, and

  size would be necessary to obtain an interval width of 50

  Exposure to Combustion Products” (Indoor Air, 2006:

  ppm for a confidence level of 95?

  65–73) reported that for a sample of 50 kitchens with gas

  14. The article “Evaluating Tunnel Kiln Performance” (Amer.

  cooking appliances monitored during a one-week period,

  Ceramic Soc. Bull., Aug. 1997: 59–63) gave the following

  CHAPTER 7 Statistical Intervals Based on a Single Sample

  summary information for fracture strengths (MPa) of

  19. The article “Limited Yield Estimation for Visual Defect

  n 5 169 ceramic bars fired in a particular kiln: x 5 89.10,

  Sources” (IEEE Trans. on Semiconductor Manuf., 1997:

  s 5 3.73 .

  17–23) reported that, in a study of a particular wafer inspec-

  a. Calculate a (two-sided) confidence interval for true aver-

  tion process, 356 dies were examined by an inspection

  age fracture strength using a confidence level of 95.

  probe and 201 of these passed the probe. Assuming a stable

  Does it appear that true average fracture strength has

  process, calculate a 95 (two-sided) confidence interval for

  been precisely estimated?

  the proportion of all dies that pass the probe.

  b. Suppose the investigators had believed a priori that the

  20. The Associated Press (October 9, 2002) reported that in a

  population standard deviation was about 4 MPa. Based

  survey of 4722 American youngsters aged 6 to 19, 15

  on this supposition, how large a sample would have been

  were seriously overweight (a body mass index of at least 30;

  required to estimate m to within .5 MPa with 95 confi-

  this index is a measure of weight relative to height).

  dence?

  Calculate and interpret a confidence interval using a 99

  15. Determine the confidence level for each of the following

  confidence level for the proportion of all American young-

  large-sample one-sided confidence bounds:

  sters who are seriously overweight.

  a. Upper bound: x 1 .84s 1n

  21. In a sample of 1000 randomly selected consumers who had

  b. Lower bound: x 2 2.05s 1n

  opportunities to send in a rebate claim form after purchas-

  c. Upper bound: x 1 .67s 1n

  ing a product, 250 of these people said they never did so

  16. The alternating current (AC) breakdown voltage of an insu-

  (“Rebates: Get What You Deserve,” Consumer Reports,

  lating liquid indicates its dielectric strength. The article

  May 2009: 7). Reasons cited for their behavior included too

  “Testing Practices for the AC Breakdown Voltage Testing of

  many steps in the process, amount too small, missed dead-

  Insulation Liquids” (IEEE Electrical Insulation Magazine,

  line, fear of being placed on a mailing list, lost receipt, and

  1995: 21–26) gave the accompanying sample observations on

  doubts about receiving the money. Calculate an upper con-

  breakdown voltage (kV) of a particular circuit under certain

  fidence bound at the 95 confidence level for the true pro-

  conditions.

  portion of such consumers who never apply for a rebate. Based on this bound, is there compelling evidence that the

  true proportion of such consumers is smaller than 13?

  Explain your reasoning.

  22. The technology underlying hip replacements has changed as

  a. Construct a boxplot of the data and comment on inter-

  these operations have become more popular (over 250,000

  esting features.

  in the United States in 2008). Starting in 2003, highly

  b. Calculate and interpret a 95 CI for true average break-

  durable ceramic hips were marketed. Unfortunately, for too

  down voltage m. Does it appear that m has been precisely

  many patients the increased durability has been counterbal-

  estimated? Explain.

  anced by an increased incidence of squeaking. The May 11,

  c. Suppose the investigator believes that virtually all values

  2008, issue of the New York Times reported that in one study

  of breakdown voltage are between 40 and 70. What sam-

  of 143 individuals who received ceramic hips between 2003

  ple size would be appropriate for the 95 CI to have a

  and 2005, 10 of the hips developed squeaking.

  width of 2 kV (so that m is estimated to within 1 kV with

  a. Calculate a lower confidence bound at the 95 confi-

  95 confidence)?

  dence level for the true proportion of such hips that develop squeaking.

  17. Exercise 1.13 gave a sample of ultimate tensile strength

  b. Interpret the 95 confidence level used in (a).

  observations (ksi). Use the accompanying descriptive statis- tics output from Minitab to calculate a 99 lower confi-

  23. The Pew Forum on Religion and Public Life reported on

  dence bound for true average ultimate tensile strength, and

  Dec. 9, 2009, that in a survey of 2003 American adults, 25

  interpret the result.

  said they believed in astrology.

  a. Calculate and interpret a confidence interval at the 99

  N

  Mean Median TrMean StDev SE Mean

  confidence level for the proportion of all adult

  4.59 0.37 Americans who believe in astrology.

  Minimum Maximum

  Q1

  Q3

  b. What sample size would be required for the width of a

  99 CI to be at most .05 irrespective of the value of ? pˆ 24. A sample of 56 research cotton samples resulted in a

  18. The article “Ultimate Load Capacities of Expansion Anchor

  sample average percentage elongation of 8.17 and a sam-

  Bolts” (J. of Energy Engr., 1993: 139–158) gave the follow-

  ple standard deviation of 1.42 (“An Apparent Relation

  ing summary data on shear strength (kip) for a sample of

  Between the Spiral Angle , the Percent Elongation E f 1 ,

  38-in. anchor bolts: n 5 78, x 5 4.25, s 5 1.30 . Calculate

  and the Dimensions of the Cotton Fiber,” Textile

  a lower confidence bound using a confidence level of 90

  Research J., 1978: 407–410). Calculate a 95 large-sam-

  for true average shear strength.

  ple CI for the true average percentage elongation m. What

  7.3 Intervals Based on a Normal Population Distribution

  assumptions are you making about the distribution of

  has approximately a standard normal distribution. Now pro-

  percentage elongation?

  ceed as in the derivation of the interval for p by making a

  25. A state legislator wishes to survey residents of her district to

  probability statement (with probability 12a ) and solving

  see what proportion of the electorate is aware of her position

  the resulting inequalities for m — see the argument just after

  on using state funds to pay for abortions.

  a. What sample size is necessary if the 95 CI for p is to have a width of at most .10 irrespective of p?

  Number of

  b. If the legislator has strong reason to believe that at least

  absences

  23 of the electorate know of her position, how large a

  Frequency

  sample size would you recommend? 26. The superintendent of a large school district, having once

  27. Reconsider the CI (7.10) for p, and focus on a confidence

  had a course in probability and statistics, believes that the

  level of 95. Show that the confidence limits agree quite

  number of teachers absent on any given day has a Poisson

  well with those of the traditional interval (7.11) once two

  distribution with parameter m. Use the accompanying data

  successes and two failures have been appended to the sam-

  on absences for 50 days to obtain a large-sample CI for m.

  ple [i.e., (7.11) based on x12 S’s in n14 trials]. [Hint:

  [Hint: The mean and variance of a Poisson variable both

  1.96 < 2 . Note: Agresti and Coull showed that this adjust-

  equal m, so

  ment of the traditional interval also has an actual confidence level close to the nominal level.]

  X2m Z5 1mn