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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with and . We already know that in this case, for any n, is normal
Referring to Exercise 82, suppose an investigator wishes to test versus based on a sample of 21 observations. The computed value of 20s2/.04 is 8.58.Place bounds on the P-value and then reach a
Chapter 7 presented a CI for the variance s2 of a normal population distribution. The key result there was that the rv has a chi-squared distribution with df. Consider the null hypothesis
A hot-tub manufacturer advertises that with its heating equipment, a temperature of 100°F can be achieved in at most 15 min. A random sample of 42 tubs is selected, and the time necessary to achieve
When are independent Poisson variables, each with parameter m, and n is large, the sample mean has approximately a normal distribution with and. This implies that has approximately a standard normal
The Dec. 30, 2009, the NewYork Times reported that in a survey of 948 American adults who said they were at least somewhat interested in college football, 597 said the current Bowl Championship
Annual holdings turnover for a mutual fund is the percentage of a fund’s assets that are sold during a particular year.Generally speaking, a fund with a low value of turnover is more stable and
The article “Caffeine Knowledge, Attitudes, and Consumption in Adult Women” (J. of Nutrition Educ., 1992:179–184) reports the following summary data on daily caffeine consumption for a sample
Contamination of mine soils in China is a serious environmental problem. The article “Heavy Metal Contamination in Soils and Phytoaccumulation in a Manganese Mine Wasteland, South China” (Air,
In Exercise 65, suppose the experimenter had believed before collecting the data that the value of s was approximately .30. If the experimenter wished the probability of a type II error to be .05
The relative conductivity of a semiconductor device is determined by the amount of impurity “doped” into the device during its manufacture. A silicon diode to be used for a specific purpose
A spectrophotometer used for measuring CO concentration[ppm (parts per million) by volume] is checked for accuracy by taking readings on a manufactured gas (called span gas)in which the CO
A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is
The accompanying data on cube compressive strength(MPa) of concrete specimens appeared in the article“Experimental Study of Recycled Rubber-Filled HighStrength Concrete” (Magazine of Concrete
The article “Heavy Drinking and Polydrug Use Among College Students” (J. of Drug Issues, 2008: 445–466) stated that 51 of the 462 college students in a sample had a lifetime abstinence from
Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p
The article “Analysis of Reserve and Regular Bottlings: Why Pay for a Difference Only the Critics Claim to Notice?”(Chance, Summer 2005, pp. 9–15) reported on an experiment to investigate
Let m denote true average serum receptor concentration for all pregnant women. The average for all women is known to be 5.63. The article “Serum Transferrin Receptor for the Detection of Iron
The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let m denote the true average reflectometer reading for a new type of paint under consideration. A test
Give as much information as you can about the P-value of a t test in each of the following situations:a. Upper-tailed test,b. Lower-tailed test,c. Two-tailed test,d. Upper-tailed test,e. Upper-tailed
Newly purchased tires of a certain type are supposed to be filled to a pressure of 30 lb/in2. Let m denote the true average pressure. Find the P-value associated with each given z statistic value for
Let m denote the mean reaction time to a certain stimulus.For a large-sample z test of versus , find the P-value associated with each of the given values of the z test statistic.a. 1.42b. .90c.
Pairs of P-values and significance levels,a, are given. For each pair, state whether the observed P-value would lead to rejection of H0 at the given significance level.a.b.c.d.e.f.
A manufacturer of plumbing fixtures has developed a new type of washerless faucet. Let (a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer
The article referenced in Example 8.11 also reported that in a sample of 106 wine consumers, 22 (20.8%) thought that screw tops were an acceptable substitute for natural corks.Suppose a particular
A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered.a. Does
A common characterization of obese individuals is that their body mass index is at least 30 [ , where height is in meters and weight is in kilograms]. The article “The Impact of Obesity on Illness
For a fixed alternative value m, show that as for either a one-tailed or a two-tailed z test in the case of a normal population distribution with known s.
Show that for any , when the population distribution is normal and s is known, the two-tailed test satisfies, so that is symmetric about m0.
A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows:a. Does this data suggest that the population mean
Reconsider the accompanying sample data on expense ratio(%) for large-cap growth mutual funds first introduced in Exercise 1.53 A normal probability plot shows a reasonably linear pattern.a. Is there
The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. The article“Nutrient Intakes and Dietary Patterns of Older Americans:A National Study” (J. of
A well-designed and safe workplace can contribute greatly to increased productivity. It is especially important that workers not be asked to perform tasks, such as lifting, that exceed their
The article “Uncertainty Estimation in Railway Track LifeCycle Cost” (J. of Rail and Rapid Transit, 2009) presented the following data on time to repair (min) a rail break in the high rail on a
The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples
The article “The Foreman’s View of Quality Control”(Quality Engr., 1990: 257–280) described an investigation into the coating weights for large pipes resulting from a galvanized coating
The true average diameter of ball bearings of a certain type is supposed to be .5 in. A one-sample t test will be carried a 5 .01 b(94) 5 .1 m 5 94 Ha H0: m 5 95 : m 2 95 s 5 1.20 x 5 94.32 m 5 76 n
The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in. Assume that the distribution of the melting point is normal with .a. Test versus
Reconsider the paint-drying situation of Example 8.2, in which drying time for a test specimen is normally distributed with . The hypotheses versus are to be tested using a random sample of
Answer the following questions for the tire problem in Example 8.7.a. If and a level test is used, what is the decision?b. If a level .01 test is used, what is b(30,500)?c. If a level .01 test is
Let the test statistic T have a t distribution when H0 is true. Give the significance level for each of the following situations:a. , rejection regionb. , rejection regionc. , rejection region or
Let the test statistic Z have a standard normal distribution when H0 is true. Give the significance level for each of the following situations:a. , rejection regionb. , rejection regionc. H ,
Reconsider the situation of Exercise 11 and suppose the rejection region is.a. What is a for this procedure?b. What is b when ? When ? Is this desirable?m 5 10.1 m 5 9.9 5z: z $ 2.51 or z # 22.656
Let denote a random sample from a normal population distribution with a known value of s.a. For testing the hypotheses versus(where m0 is a fixed number), show that the test with test statistic and
A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft.
The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each
A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m2. The mixture will not be used unless experimental evidence
Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second.
Let m denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing versus or versus
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample.Weld strength is measured as
For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different
For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:a. b.c. d.e.f. , where l is the parameter of an exponential distribution used to model component
a. Use the results of Example 7.5 to obtain a 95% lower confidence bound for the parameter l of an exponential distribution, and calculate the bound based on the data given in the example.b. If
Suppose are observed values resulting from a random sample from a symmetric but possibly heavy-tailed distribution. Let and fs denote the sample median and fourth spread, respectively. Chapter 11 of
Let . Then a % CI for m when n is large is The choice yields the usual interval derived in Section 7.2; if , this interval is not symmetric about. The width of this interval is . Show that w is
Let be a random sample from a uniform distribution on the interval [0, u], so that Then if , it can be shown that the rv has density functiona. Use to verify that and use this to derive a % CI for
Let be a random sample from a continuous probability distribution having median (so that).a. Show that so that (min(xi), max(xi)) is a confidence interval for with . [Hint: The complement of the
In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and the experiment terminates as soon as r of the components have failed. Suppose component
Chronic exposure to asbestos fiber is a well-known health hazard. The article “The Acute Effects of Chrysotile Asbestos Exposure on Lung Function” (Environ. Research, 1978: 360–372) reports
A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull
It is important that face masks used by firefighters be able to withstand high temperatures because firefighters commonly work in temperatures of 200–500°F. In a test of one type of mask, 11 of 55
Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees are selected for
High concentration of the toxic element arsenic is all too common in groundwater. The article “Evaluation of Treatment Systems for the Removal of Arsenic from Groundwater” (Practice Periodical of
An April 2009 survey of 2253 American adults conducted by the Pew Research Center’s Internet & American Life Project revealed that 1262 of the respondents had at some point used wireless means for
A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting
For each of 18 preserved cores from oil-wet carbonate reservoirs, the amount of residual gas saturation after a solvent injection was measured at water flood-out. Observations, in percentage of pore
A triathlon consisting of swimming, cycling, and running is one of the more strenuous amateur sporting events. The article “Cardiovascular and Thermal Response of Triathlon Performance” (Medicine
Example 1.11 introduced the accompanying observations on bond strength.a. Estimate true average bond strength in a way that conveys information about precision and reliability. [Hint:and .]b.
The article “Concrete Pressure on Formwork” (Mag. of Concrete Res., 2009: 407–417) gave the following observations on maximum concrete pressure (kN/m2):33.2 41.8 37.3 40.2 36.7 39.1 36.2 41.8
The following observations were made on fracture toughness of a base plate of 18% nickel maraging steel [“Fracture s 5 2.81 n 5 9 n 5 25 P(x2 , 14.611 or x2 . 37.652)n 5 22 P(10.98 # x2 # 36.78)n 5
The amount of lateral expansion (mils) was determined for a sample of pulsed-power gas metal arc welds used in LNG ship containment tanks. The resulting sample standard deviation was mils. Assuming
Determine the following:a. The 95th percentile of the chi-squared distribution withb. The 5th percentile of the chi-squared distribution withc. , where x2 is a chi-squared rv withd. , where x2 is a
Determine the values of the following quantities:a. b.c. d.e. f.
A more extensive tabulation of t critical values than what appears in this book shows that for the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20,
Exercise 13 of Chapter 1 presented a sample of observations on ultimate tensile strength, and Exercise 17 of the previous section gave summary quantities and requested a large-sample confidence
Exercise 72 of Chapter 1 gave the following observations on a receptor binding measure (adjusted distribution volume)for a sample of 13 healthy individuals: 23, 39, 40, 41, 43, 47, 51, 58, 63, 66,
A sample of 25 pieces of laminate used in the manufacture of circuit boards was selected, and the amount of warpage(in.) under particular conditions was determined for each piece, resulting in a
A study of the ability of individuals to walk in a straight line(“Can We Really Walk Straight?” Amer. J. of Physical Anthro., 1992: 19–27) reported the accompanying data on cadence (strides per
The observations on escape time given in Exercise 36 of Chapter 1 give a sample mean and sample standard deviation of 370.69 and 24.36, respectively.a. Calculate an upper confidence bound for
Silicone implant augmentation rhinoplasty is used to correct congenital nose deformities. The success of the procedure depends on various biomechanical properties of the human nasal periosteum and
A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.48 MPa and a sample standard deviation of .79 MPa (“Characterization of Bearing Strength
The article “Measuring and Understanding the Aging of Kraft Insulating Paper in Power Transformers” (IEEE Electrical Insul. Mag., 1996: 28–34) contained the following observations on degree of
According to the article “Fatigue Testing of Condoms”(Polymer Testing, 2009: 567–571), “tests currently used for t.005,40 t.05,40 t.05,25 t.05,15 t.1,15 condoms are surrogates for the
Determine the values of the following quantities:a.b. c.d. e.
Reconsider the CI (7.10) for p, and focus on a confidence level of 95%. Show that the confidence limits agree quite well with those of the traditional interval (7.11) once two successes and two
The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poisson distribution with
A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions.a. What sample size is
A sample of 56 research cotton samples resulted in a sample average percentage elongation of 8.17 and a sample standard deviation of 1.42 (“An Apparent Relation Between the Spiral Angle , the
The Pew Forum on Religion and Public Life reported on Dec. 9, 2009, that in a survey of 2003 American adults, 25%said they believed in astrology.a. Calculate and interpret a confidence interval at
The technology underlying hip replacements has changed as these operations have become more popular (over 250,000 in the United States in 2008). Starting in 2003, highly durable ceramic hips were
In a sample of 1000 randomly selected consumers who had opportunities to send in a rebate claim form after purchasing a product, 250 of these people said they never did so(“Rebates: Get What You
The Associated Press (October 9, 2002) reported that in a survey of 4722 American youngsters aged 6 to 19, 15%were seriously overweight (a body mass index of at least 30;this index is a measure of
The article “Ultimate Load Capacities of Expansion Anchor Bolts” (J. of Energy Engr., 1993: 139–158) gave the following summary data on shear strength (kip) for a sample of 3/8-in. anchor
The alternating current (AC) breakdown voltage of an insulating liquid indicates its dielectric strength. The article“Testing Practices for the AC Breakdown Voltage Testing of Insulation Liquids”
Determine the confidence level for each of the following large-sample one-sided confidence bounds:a. Upper bound:b. Lower bound:c. Upper bound:
The article “Evaluating Tunnel Kiln Performance” (Amer.Ceramic Soc. Bull., Aug. 1997: 59–63) gave the following summary information for fracture strengths (MPa) of ceramic bars fired in a
The article “Gas Cooking, Kitchen Ventilation, and Exposure to Combustion Products” (Indoor Air, 2006:65–73) reported that for a sample of 50 kitchens with gas cooking appliances monitored
A random sample of 110 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec(“Lightning Strikes to an Airplane
Consider the next 1000 95% CIs for m that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another.
A random sample of heat pumps of a certain type yielded the following observations on lifetime (in years):2.0 1.3 6.0 1.9 5.1 .4 1.0 5.3 15.7 .7 4.8 .9 12.2 5.3 .6a. Assume that the lifetime

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