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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

Let a be a number between 0 and 1, and define a sequence by and for . Substituting for its representation in terms of and , then substituting for , and so on, results in The fact that Wt depends not
Resistance observations (ohms) for subgroups of a certain type of register gave the following summary quantities:i ni si i ni si 1 4 430.0 22.5 11 4 445.2 27.3 2 4 418.2 20.6 12 4 430.1 22.2 3 3
An alternative to the p chart for the fraction defective is the np chart for number defective. This chart has, and the number of defectives from each sample is plotted on the chart.Construct such a
The following numbers are observations on tensile strength of synthetic fabric specimens selected from a production process at equally spaced time intervals. Construct appropriate control charts, and
The number of scratches on the surface of each of 24 rectangular metal plates is determined, yielding the following data: 8, 1, 7, 5, 2, 0, 2, 3, 4, 3, 1, 2, 5, 7, 3, 4, 6, 5, 2, gr gsi 5 402 i 5
Observations on shear strength for 26 subgroups of test spot welds, each consisting of six welds, yield, and . Calculate control limits for any relevant control charts.
Consider the single-sample plan that utilizes and when . Determine the values of AOQ and ATI for selected values of p, and graph each of these against p. Also determine the value of AOQL.
Refer to Example 16.11, in which a single-sample plan with and was employed.a. Calculate AOQ for . What does this suggest about the value of p for which AOQ is a maximum and the corresponding AOQL?b.
Some sources advocate a somewhat more restrictive type of doubling-sampling plan in which ; that is, the lot is rejected if at either stage the (total) number of defectives is at least r1 (see the
Consider the double-sampling plan for which both sample sizes are 50.The lot is accepted after the first sample if the number of defectives is at most 1, rejected if the number of LTPD 5 .07 a 5 .05,
Develop a single-sample plan for which and in the case . Once values of n and c have been determined, calculate the achieved values of a and b for the plan.
Refer to Exercise 34 and consider the plan with and . Calculate P(A) for , and sketch the two OC curves on the same set of axes. Which of the two plans is preferable (leaving aside the cost of
A sample of 50 items is to be selected from a batch consisting of 5000 items. The batch will be accepted if the sample contains at most one defective item. Calculate the probability of lot acceptance
Consider the single-sample plan with and , as discussed in Example 16.11, but now suppose that the lot size is . Calculate P(A), the probability of accepting the lot, for using the hypergeometric
The target value for the diameter of a certain type of driveshaft is .75 in. The size of the shift in the average diameter considered important to detect is .002 in. Sample average diameters for
Construct a control chart for the data of Exercise 25 by using the transformation suggested in the text
In some situations, the sizes of sampled specimens vary, and larger specimens are expected to have more defects than smaller ones. For example, sizes of fabric samples inspected for flaws might vary
For what values will the LCL in a c chart be negative?
Refer to the data of Exercise 22, and construct a control chart using the sin transformation as suggested in the text.
When , what is the smallest value of for which the LCL in a p chart is positive?
When S2 is the sample variance of a normal random sample, has a chi-squared distribution with df, so from which This suggests that an alternative chart for controlling process variation involves
Calculate control limits for an S chart from the refractive index data of Exercise 11.Does the process appear to be in control with respect to variability? Why or why not?
The following data on the deviation from target in the parallel orientation is taken from Table 1 of the article cited in Example 16.5. Sometimes a transformation of the data is appropriate, either
Calculate control limits for the data of Exercise 8 using the robust procedure presented in this section.
Apply the supplemental rules suggested in the text to the data of Exercise 8.Are there any out-of-control signals?
Consider the control chart based on control limits m . 0 6 2.81 s/ 1na. What is the ARL when the process is in control?b. What is the ARL when and the process mean has shifted to ?c. How do the
The accompanying table gives sample means and standard deviations, each based on observations of the refractiv index of fiber-optic cable. Construct a control chart, and comment on its appearance.
Refer to Exercises 8 and 9, and now employ control limits based on using the sample ranges to estimate s. Does the process appear to be in control?
The table below gives data on moisture content for specimens of a certain type of fabric. Determine control limits for a chart with center line at height 13.00 based on , construct the control chart,
Consider a 3-sigma control chart with a center line at m0 and based on . Assuming normality, calculate the probability that a single point will fall outside the control limits when the actual process
If a process variable is normally distributed, in the long run virtually all observed values should be between and, giving a process spread of 6s.a. With LSL and USL denoting the lower and upper
A cork intended for use in a wine bottle is considered acceptable if its diameter is between 2.9 cm and 3.1 cm (so the lower specification limit is and the upper specification limit is ).a. If cork
The ranking procedure described in Exercise 35 is somewhat asymmetric, because the smallest observation receives rank 1, whereas the largest receives rank 2, and so on. Suppose both the smallest and
Refer to Exercise 33, and consider a confidence interval associated with the sign test: the sign interval. The relevant hypotheses are now versus .Let’s use the following rejection region: either
The sign test is a very simple procedure for testing hypotheses about a population median assuming only that the underlying distribution is continuous. To illustrate, consider the following sample of
The given data on phosphorus concentration in topsoil for four different soil treatments appeared in the article “Fertilisers for Lotus and Clover Establishment on a Sequence of Acid Soils on the
High-pressure sales tactics or door-to-door salespeople can be quite offensive. Many people succumb to such tactics, sign a purchase agreement, and later regret their actions. In the mid-1970s, the
In an experiment to study the way in which different anesthetics affect plasma epinephrine concentration, ten dogs were selected and concentration was measured while they were under the influence of
In a test to determine whether soil pretreated with small amounts of Basic-H makes the soil more permeable to water, soil samples were divided into blocks, and each block received each of the four
The accompanying data on cortisol level was reported in the article “Cortisol, Cortisone, and 11-Deoxycortisol Levels in Human Umbilical and Maternal Plasma in Relation to the Onset of Labor” (J.
The article “Production of Gaseous Nitrogen in Human Steady-State Conditions” (J. of Applied Physiology, 1972:155–159) reports the following observations on the amount of nitrogen expired (in
The accompanying data refers to concentration of the radioactive isotope strontium-90 in milk samples obtained from five randomly selected dairies in each of four different regions.1 6.4 5.8 6.5 7.7
Compute a 99% CI for using the data in Exercise 11.
Compute the 90% rank-sum CI for using the data in Exercise 10.
An experiment was carried out to compare the abilities of two different solvents to extract creosote impregnated in test logs. Each of eight logs was divided into two segments, and then one segment
Reconsider the situation described in Exercise 81 of Chapter 9 and the accompanying Minitab output (the Greek letter eta is used to denote a median).Mann-Whitney Confidence Interval and Test good
The accompanying data resulted from an experiment to compare the effects of vitamin C in orange juice and in synthetic ascorbic acid on the length of odontoblasts in guinea pigs over a 6-week period
A modification has been made to the process for producing a certain type of “time-zero” film (film that begins to develop as soon as a picture is taken). Because the modification involves extra
The article “A Study of Wood Stove Particulate Emissions”(J. of the Air Pollution Control Assoc., 1979: 724–728)reports the following data on burn time (hours) for samples of oak and pine. Test
In an experiment to compare the bond strength of two different adhesives, each adhesive was used in five bondings of two surfaces, and the force necessary to separate the surfaces was determined for
Suppose that observations are made on a process at times . On the basis of this data, we wish to test 1, 2,c, n X1, X2, c,H0: the Xi ’s constitute an independent and identically distributed
Reconsider the port alcohol content data from Exercise 14.22. A normal probability plot casts some doubt on the assumption of population normality. However, a dotplot shows a reasonable amount of
Reconsider the situation described in Exercise 39 of Section 9.3, and use the Wilcoxon test to test the appropriate hypotheses.
Here again is the data on expense ratio (%) for a sample of 20 large-cap blended mutual funds introduced in Exercise 1.53:1.03 1.23 1.10 1.64 1.30 1.27 1.25.78 1.05 .64 .94 .86 1.05 .75.09 0.79 1.61
Reconsider the situation described in Exercise 34 of Section 8.2, and use the Wilcoxon test with to test the relevant hypotheses.
The Fibonacci sequence of numbers occurs in various scientific contexts. The first two numbers in the sequence are 1,1. Then every succeeding number is the sum of the two previous numbers: 1, 1, 1
Do the successive digits in the decimal expansion of p behave as though they were selected from a random number table (or came from a computer’s random number generator)?a. Let p0 denote the long
Have you ever wondered whether soccer players suffer adverse effects from hitting “headers”? The authors of the article “No Evidence of Impaired Neurocognitive Performance in Collegiate Soccer
The NCAA basketball tournament begins with 64 teams that are apportioned into four regional tournaments, each involving 16 teams. The 16 teams in each region are then ranked (seeded) from 1 to
Let p1 denote the proportion of successes in a particular population. The test statistic value in Chapter 8 for testing was , where. Show that for the case , the chisquared test statistic value of
In a study to investigate the extent to which individuals are aware of industrial odors in a certain region(“Annoyance and Health Reactions to Odor from Refineries and Other Industries in Carson,
Much attention has recently focused on the incidence of concussions among athletes. Separate samples of soccer players, non-soccer athletes, and non-athletes were selected. The accompanying table
The accompanying two-way frequency table appears in the article “Marijuana Use in College” (Youth and Society, 1979: 323–334). Each of 445 college students was classified according to both
The authors of the article “Predicting Professional Sports Game Outcomes from Intermediate Game Scores” (Chance, 1992: 18–22) used a chi-squared test to determine whether there was any merit to
Qualifications of male and female head and assistant college athletic coaches were compared in the article “Sex Bias and the Validity of Believed Differences Between Male and Female Interscholastic
Does the phase of the moon have any bearing on birthrate?Each of 222,784 births that occurred during a period encompassing 24 full lunar cycles was classified according to lunar phase. The following
Consider the accompanying table displaying the sample proportions that fell in the various combinations of categories (e.g., 13% of those in the sample were in the first category of both factors).123
Suppose that in a particular state consisting of four distinct regions, a random sample of nk voters is obtained from the k th region for . Each voter is then classified according to which candidate
Suppose that in Exercise 32 each student had been categorized with respect to political views, marijuana usage, and religious preference, with the categories of this latter factor being Protestant,
Each individual in a random sample of high school and college students was cross-classified with respect to both political views and marijuana usage, resulting in the data displayed in the
A random sample of smokers was obtained, and each individual was classified both with respect to gender and with respect to the age at which he/she first started smoking. The data in the accompanying
Three different design configurations are being considered for a particular component. There are four possible failure modes for the component. An engineer obtained the following data on number of
The accompanying data on degree of spirituality for samples of natural and social scientists at research universities as well as for a sample of non-academics with graduate degrees appeared in the
A random sample of 175 Cal Poly State University students was selected, and both the email service provider and cell phone provider were determined for each one, resulting in the accompanying data.
The article “Human Lateralization from Head to Foot: SexRelated Factors” (Science, 1978: 1291–1292) reports for both a sample of right-handed men and a sample of right-handed women the number
The accompanying data refers to leaf marks found on white clover samples selected from both long-grass areas and short-grass areas (“The Biology of the Leaf Mark Polymorphism in Trifolium repens
The accompanying two-way table was constructed using data in the article “Television Viewing and Physical Fitness in Adults” (Research Quarterly for Exercise and Sport, 1990: 315–320). The
The article from which the data in Exercise 20 was obtained also gave the accompanying data on the composite mass/outer fabric mass ratio for high-quality fabric specimens.
The article “Nonbloated Burned Clay Aggregate Concrete”(J. of Materials, 1972: 555–563) reports the following data on 7-day flexural strength of nonbloated burned clay aggregate concrete
The article “A Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baumé and Refractometer Brix” (Amer. J. of Enol. and Vitic., 2006:486–490) gave duplicate measurements
An article in Annals of Mathematical Statistics reports the following data on the number of borers in each of 120 groups of borers. Does the Poisson pmf provide a rigidity for medium-quality fabric
In a genetics experiment, investigators looked at 300 chromosomes of a particular type and counted the number of sister-chromatid exchanges on each (“On the Nature of Sister-Chromatid Exchanges in
The article “Compatibility of Outer and Fusible Interlining Fabrics in Tailored Garments (Textile Res. J., 1997:137–142) gave the following observations on bendingˆu1, ˆu2'/'u ' 2 /'u1 pi(u1,
Each headlight on an automobile undergoing an annual vehicle inspection can be focused either too high (H), too low (L), or properly (N). Checking the two headlights simultaneously (and not
The article “A Probabilistic Analysis of Dissolved Oxygen–Biochemical Oxygen Demand Relationship in Streams” (J. Water Resources Control Fed., 1969: 73–90)reports data on the rate of
A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the
The article “Feeding Ecology of the Red-Eyed Vireo and Associated Foliage-Gleaning Birds” (Ecological Monographs, 1971: 129–152) presents the accompanying data on the variable the number of
A study of sterility in the fruit fly (“Hybrid Dysgenesis in Drosophila melanogaster: The Biology of Female and Male Sterility,” Genetics, 1979: 161–174) reports the following data on the
Consider a large population of families in which each family has exactly three children. If the genders of the three children in any family are independent of one another, the number of male children
a. Show that another expression for the chi-squared statistic is Why is it more efficient to compute x2 using this formula?b. When the null hypothesis is(i.e., for all i), how does the formula of
The response time of a computer system to a request for a certain type of information is hypothesized to have an exponential distribution with parameter sec (so if, the pdf of X under H0 is for).a.
The article “The Gap Between Wine Expert Ratings and Consumer Preferences” (Intl. J. of Wine Business Res., 2008: 335–351) studied differences between expert and consumer ratings by considering
It is hypothesized that when homing pigeons are disoriented in a certain manner, they will exhibit no preference for any direction of flight after takeoff (so that the direction X should be uniformly
The article “Racial Stereotypes in Children’s Television Commercials” (J. of Adver. Res., 2008: 80–93) reported the following frequencies with which ethnic characters appeared in recorded
Say as much as you can about the P-value for an upper-tailed chi-squared test in each of the following situations:a. b.c. d.e.
What conclusion would be appropriate for an upper-tailed chi-squared test in each of the following situations?a.b.c.d.
Given that for the model containing predictors x1, x4, x5, and x8 and for the model with predictors x1, x3, x5, and x6, what can you say about R2 for the model containing predictorsa. x1, x3, x4, x5,
Does exposure to air pollution result in decreased life expectancy? This question was examined in the article“Does Air Pollution Shorten Lives?” (Statistics and Public Policy, Reading, MA,
A sample of companies was selected, and the values of price and variables (such as quarterly dividend, previous year’s earnings, and debt ratio) were determined. When the multiple regression model
Coir fiber, derived from coconut, is an eco-friendly material with great potential for use in construction. The article“Seepage Velocity and Piping Resistance of Coir Fiber Mixed Soils” (J. of

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