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Questions and Answers of
Statistics
A random sample of 15 automobile mechanics certified to work on a certain type of car was selected, and the time (in minutes) necessary for each one to diagnose a particular problem was determined,
Both a gravimetric and a spectrophotometric method are under consideration for determining phosphate content of a particular material. Twelve samples of the material are obtained, each is split in
Reconsider the situation described in Exercise 39 of Section 9.3, and use the Wilcoxon test to test the appropriate hypotheses.In Exercise 39
Use the large-sample version of the Wilcoxon test at significance level .05 on the data of Exercise 37 in Section 9.3 to decide whether the true mean difference between outdoor and indoor
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
Suppose that observations X1, X2,..., Xn are made on a process at times 1, 2,..., n. On the basis of this data, we wish to testH0: the Xi's constitute an independent and identically distributed
When installing a bath faucet, it is important to properly fasten the threaded end of the faucet stem to the watersupply line. The threaded stem dimensions must meet product specifications, otherwise
The accompanying table gives sample means and standard deviations, each based on n = 6 observations of the refractive index of fiber-optic cable. Construct a control chart, and comment on its
Refer to Exercise 11. An assignable cause was found for the unusually high sample average refractive index on day 22. Recompute control limits after deleting the data from this day. What do you
Consider the control chart based on control limits µ0±2.81 σ/√n. a. What is the ARL when the process is in control? b. What is the ARL when n = 4 and the process mean has shifted to µ = µ0 +
Three-dimensional (3D) printing is a manufacturing technology that allows the production of three-dimensional solid objects through a meticulous layering process performed by a 3D printer. 3D
Calculate control limits for the data of Exercise 8 using the robust procedure presented in this section.Data of Exercise 8
A manufacturer of dustless chalk instituted a quality control program to monitor chalk density. The sample standard deviations of densities for 24 different subgroups, each consisting of n = 8 chalk
Subgroups of power supply units are selected once each hour from an assembly line, and the high-voltage output of each unit is determined. a. Suppose the sum of the resulting sample ranges for 30
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 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?Data From Exercise 11
Refer to Exercise 1 and suppose the ten most recent values of the quality statistic are .0493, .0485, .0490, .0503, .0492, .0486, .0495, .0494, .0493, and .0488. Construct the relevant portion of the
When S2 is the sample variance of a normal random sample, (n - 1)S2/σ2 has a chi-squared distribution with n - 1 df, sofrom whichThis suggests that an alternative chart for controlling process
On each of the previous 25 days, 100 electronic devices of a certain type were randomly selected and subjected to a severe heat stress test. The total number of items that failed to pass the test was
A sample of 200 ROM computer chips was selected on each of 30 consecutive days, and the number of nonconforming chips on each day was as follows: 10, 18, 24, 17, 37, 19, 7, 25, 11, 24, 29, 15, 16,
When n = 150, what is the smallest value of p for which the LCL in a p chart is positive?
Refer to the data of Exercise 22, and construct a control chart using the sin21 transformation as suggested in the text. Exercise 22 A sample of 200 ROM computer chips was selected on each of 30
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
Construct a control chart for the data of Exercise 25 by using the transformation suggested in the text. Exercise 25 The accompanying observations are numbers of defects in 25 1-square-yard specimens
Containers of a certain treatment for septic tanks are supposed to contain 16 oz of liquid. A sample of five containers is selected from the production line once each hour, and the sample average
Suppose a control chart is constructed so that the probability of a point falling outside the control limits when the process is actually in control is .002. What is the probability that ten
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
The standard deviation of a certain dimension on an aircraft part is .005 cm. What CUSUM procedure will give an in-control ARL of 600 and an out-of-control ARL of 4 when the mean value of the
When the out-of-control ARL corresponds to a shift of 1 standard deviation in the process mean, what are the characteristics of the CUSUM procedure that has ARLs of 250 and 4.8, respectively, for the
Consider the single-sample plan with c = 2 and n 5 50, as discussed in Example 16.11, but now suppose that the lot size is N = 500. Calculate P(A), the probability of accepting the lot, for p = .01,
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
Refer to Exercise 34 and consider the plan with n = 100 and c = 2. Calculate P(A) for p = .01, .02,..., .05, and sketch the two OC curves on the same set of axes. Which of the two plans is preferable
Develop a single-sample plan for which AQL = .02 and LTPD = .07 in the case a = .05, β = .10. Once values of n and c have been determined, calculate the achieved values of α and β for the plan.
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 defectives is at
Some sources advocate a somewhat more restrictive type of doubling-sampling plan in which r1 = c2 + 1; that is, the lot is rejected if at either stage the (total) number of defectives is at least r1
Refer to Example 16.11, in which a single-sample plan with n = 50 and c = 2 was employed. a. Calculate AOQ for p = .01, .02,..., .10. What does this suggest about the value of p for which AOQ is a
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 LSL = 2.9 and the upper specification limit is USL
Consider the single-sample plan that utilizes n = 50 and c = 1 when N = 2000. Determine the values of AOQ and ATI for selected values of p, and graph each of these against p. Also determine the value
Observations on shear strength for 26 subgroups of test spot welds, each consisting of six welds, yield ∑x̅i = 10,980, ∑si = 402, and ∑ri 5 1074. Calculate control limits for any relevant
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, 4, 0, 10, 2, 6.
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
An alternative to the p chart for the fraction defective is the np chart for number defective. This chart has UCL = np̅ + 3 √np̅(1-p̅) , LCL = np̅ - 3√np̅(1-p̅), and the number of
Resistance observations (ohms) for subgroups of a certain type of register gave the following summary quantities:Construct appropriate control limits. Use xÌ =
Let a be a number between 0 and 1, and define a sequence W1, W2, W3,... by W0 = µ Wt = αXÌ t +(1-α)Wt-1 for t = 1,2 , . . . Substituting for Wt-1 its
If a process variable is normally distributed, in the long run virtually all observed values should be between µ - 3σ and µ + 3σ, giving a process spread of 6σ. a. With LSL and USL denoting the
In the case of known µ and σ, what control limits are necessary for the probability of a single point being outside the limits for an in-control process to be .005?
Consider a 3-sigma control chart with a center line at µ0 and based on n = 5. Assuming normality, calculate the probability that a single point will fall outside the control limits when the actual
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 Ï = .600, construct
Refer to the data given in Exercise 8, and construct a control chart with an estimated center line and limits based on using the sample standard deviations to estimate Ï. Is there any
An airline analyst wishes to estimate the proportion of all American adults who are afraid to fly because of potential terrorist attacks. To estimate this percentage, the analyst decides to survey
Suppose that you collect a random sample of 250 salaries for the salespersons employed by a large PC manufacturer. Furthermore, assume that you find that two of these salaries are considerably higher
Does it make sense to construct a histogram for the state of residence of randomly selected individuals in a sample? Explain why or why not.
Characterize the likely shape of a histogram of the distribution of scores on a midterm exam in a graduate statistics course?
Suppose that the histogram of a given income distribution is positively skewed. What does this fact imply about the relationship between the mean and median of this distribution?
"The midpoint of the line segment joining the first quartile and third quartile of any distribution is the median." Is this statement true or false? Explain your answer.
Explain why the standard deviation would likely not be a reliable measure of variability for a distribution of data that includes at least one extreme outlier?
Explain how a box plot can be used to determine whether the associated distribution of values is essentially symmetric?
When you are trying to discover whether there is a relationship between two categorical variables, why is it useful to transform the counts in a crosstabs to percentages of row or column totals? Once
In checking whether several times series, such as monthly exchange rates of various currencies, move together, why do most analysts look at correlations between their differences rather than
Suppose you have a crosstabs of two "Yes/No" categorical variables, with the counts shown as percentages of row totals. What will these percentages look like if there is absolutely no relationship
If you suspect that a company's advertising expenditures in a given month affect its sales in future months, what correlations would you look at to confirm your suspicions? How would you find them?
Suppose you have customer data on whether they have bought your product in a given time period, along with various demographics on the customers. Explain how you could use pivot tables to see which
Suppose you have data on student achievement in high school for each of many school districts. In spreadsheet format, the school district is in column A, and various student achievement measures are
If two variables are highly correlated, does this imply that changes in one cause changes in the other? If not, give at least one example from the real world that illustrates what else could cause a
Suppose there are two commodities A and B with strongly negatively correlated daily returns, such as a stock and gold. Is it possible to find another commodity with daily returns that are strongly
Suppose that you want to find the probability that event A or event B will occur. If these two events are not mutually exclusive, explain how you would proceed?
Consider an event that will either occur or not. For example, the event might be that California will experience a major earthquake in the next five years. You let p be the probability that the event
Suppose a couple is planning to have two children. Let B1 be the event that the first child is a boy, and let B2 be the event that the second child is a boy. You and your friend get into an argument
"If two events are mutually exclusive, they must not be independent events." Is this statement true or false? Explain your choice.
Is the number of passengers who show up for a particular commercial airline flight a discrete or a continuous random variable? Is the time between flight arrivals at a major airport a discrete or a
Suppose that officials in the federal government are trying to determine the likelihood of a major smallpox epidemic in the United States within the next 12 months. Is this an example of an objective
Consider the statement, "When there are a finite number of outcomes, then all probability is just a matter of counting. Specifically, if n of the outcomes are favorable to some event E, and there are
If there is uncertainty about some monetary outcome and you are concerned about return and risk, then all you need to see are the mean and standard deviation. The entire distribution provides no
Choose at least one uncertain quantity of interest to you. For example, you might choose the highest price of gas between now and the end of the year, the highest point the Dow Jones Industrial
Historically, the most popular measure of variability has been the standard deviation, the square root of the weighted sum of squared deviations from the mean, weighted by their probabilities.
Suppose a person flips a coin, but before you can see the result, the person puts her hand over the coin. At this point, does it make sense to talk about the probability that the result is heads? Is
For each of the following uncertain quantities, discuss whether it is reasonable to assume that the probability distribution of the quantity is normal.If the answer isn't obvious, discuss how you
Many basketball players and fans believe strongly in the "hot hand." That is, they believe that players tend to shoot in streaks, either makes or misses. If this is the case, why does the binomial
Suppose the demands in successive weeks for your product are normally distributed with mean 100 and standard deviation 20, and suppose your lead time for receiving a placed order is three weeks. A
For each of the following uncertain quantities, discuss whether it is reasonable to assume that the probability distribution of the quantity is binomial. If you think it is, what are the parameters
The Poisson distribution is often appropriate in the "binomial" situation of n independent and identical trials, where each trial has probability p of success, but n is very large and p is very
One disadvantage of a normal distribution is that there is always some probability that a quantity is negative, even when this makes no sense for the uncertain quantity. For example, the time a light
Explain why probabilities such as P(X < x) and P(X ˂ x) are equal for a continuous random variable?
State the major similarities and differences between the binomial distribution and the Poisson distribution?
A distribution we didn't discuss is the Bernoulli distribution. It is a binomial distribution with n = 1. In other words, it is the number of successes (0 or 1) in a single trial when the probability
For real applications, the normal distribution has two potential drawbacks: (1) It can be negative, and (2) It isn't symmetric. Choose some continuous random numeric outcomes of interest to you. Are
Several decision criteria besides EMV are suggested in the section. For each of the following criteria, rank all three decisions in Figure 6.1 from best to worst.a. Look only at the worst possible
In the file Bayes Rule for Disease.xlsx, explain why the probabilities in cells B9 and B10 (or those in cells C9 and C10) do not necessarily sum to 1, but why the probabilities in cells B9 and C9 (or
In using Bayes' rule for the presence of a disease (see Figure 6.17 and the file Bayes Rule for Disease.xlsx), we assumed that there are only two test results, positive or negative. Suppose there is
For the decision problem in Figure 6.1, use data tables to perform the following sensitivity analyses. The goal in each is to see whether decision 1 continues to have the largest EMV. In each part,
Some decision makers prefer decisions with low risk, but this depends on how risk is measured. As we mentioned in this section, variance (see the definition in problem 1) is one measure of risk, but
The fixed cost of $6 million in the Acme problem is evidently not large enough to make Acme abandon the product at the current time. How large would the fixed cost need to be to make the abandon
Perform a sensitivity analysis on the probability of a great market. To do this, enter formulas in cells B9 and B10 (see Figure 6.4) to ensure that the probabilities of "fair" and "awful" remain in
Sometimes it is possible for a company to influence the uncertain outcomes in a favorable direction. Suppose Acme could, by an early marketing blitz, change the probabilities of "great," "fair," and
Sometimes a "single-stage" decision can be broken down into a sequence of decisions, with no uncertainty resolved between these decisions. Similarly, uncertainty can sometimes be broken down into a
Can you ever use the material in this chapter to help you make your own real-life decisions? Consider the following. You are about to take an important and difficult exam in one of your MBA
If you examine the decision tree in Figure 6.12 (or any other decision trees from PrecisionTree), you will see two numbers (in blue font) to the right of each end node. The bottom number is the
Explain what it means in general when we say a risk-averse decision maker is willing to give up some EMV to avoid risk? How is this apparent in certainty equivalents of gambles?
Your company needs to make an important decision that involves large monetary consequences. You have listed all of the possible outcomes and the monetary payoffs and costs from all outcomes and all
You often hear about the trade-off between risk and reward. Is this trade-off part of decision making under uncertainty when the decision maker uses the EMV criterion? For example, how does this work
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