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free/or 0.5$ |
| m6868 | A sensor is used to monitor the performance of a nuclear reactor. The sensor accurately reflects the state of the reactor with a probability of .97. But with a probability of .02, it gives a false alarm (by reporting excessive radiation even though the reactor is performing normally), and with a probability of .01, it misses excessive radiation (by failing to report excessive radiation even though the reactor is performing abnormally).
(a) What is the probability that a sensor will give an incorrect report, that is, either a false alarm or a miss?
(b) To reduce costly shutdowns caused by false alarms, management introduces a second completely independent sensor, and the reactor is shut down only when both sensors report excessive radiation. (According to this perspective, solitary reports of excessive radiation should be viewed as false alarms and ignored, since both sensors provide accurate information much of the time.) What is the new probability that the reactor will be shut down because of simultaneous false alarms by both the first and second sensors?
(c) Being more concerned about failures to detect excessive radiation, someone who lives near the nuclear reactor proposes an entirely different strategy: Shut down the reactor whenever either sensor reports excessive radiation. (According to this point of view, even a solitary report of excessive radiation should trigger a shutdown, since a failure to detect excessive radiation is potentially catastrophic.) If this |
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| m6869 | A set of data is mounded, with a mean of 450 and a variance of 625. Approximately what proportion of the observations is?
a. Greater than 425?
b. Less than 500?
c. Greater than 525? |
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| m6870 | a. Show that the centered s-point moving average series of Section 16.2 can be written as follows:
b. Show that
Discuss the computational advantages of this formula in the seasonal adjustment of monthly time series. |
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| m6871 | A simple random sample is to be taken of 527 business majors in a college to estimate the proportion favoring greater emphasis on business ethics in the curriculum. How many observations are necessary to ensure that a 95% confidence interval for the population proportion extends at most 0.06 on each side of the sample proportion? |
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| m6872 | A simple random sample of 400 from a total 1,395 colleges in the United States maintained that 39 colleges use the text Statistics Made Difficult and Boring. Find a 95% confidence interval for the proportion of all colleges using this text. |
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| m6873 | A small commuter airline flies planes that can seat up to 8 passengers. The airline has determined that the probability that a ticketed passenger will not show up for a flight is 0.2. For each flight the airline sells tickets to the first 10 people placing orders. The probability distribution for the number of tickets sold per flight is shown in the accompanying table. For what proportion of the airline s flights does the number of ticketed passengers showing up exceed the number of available seats? (Assume independence between the number of tickets sold and the probability that a ticketed passenger will show up.) |
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| m6874 | A small private university is planning to start a volunteer football program. A random sample of alumni is surveyed. It was found that 250 were in favor of this program, 75 were opposed, and 25 had no opinion.
a. Estimate the percent of alumni in favor of this program. Let α = 0.05.
b. Estimate the percent of alumni opposed to this volunteer football program with a 90% confidence level. |
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| m6875 | A small town contains a total of 1,800 households. The town is divided into three districts, containing 820, 540, and 440 households, respectively. A stratified random sample of 300 households contains 120, 90, and 90 households, respectively, from these three districts. Sample members were asked to estimate their total energy bills for the winter months. The respective sample means were $290, $352, and $427, and the respective sample standard deviations were $47, $61, and $93.
a. Use an unbiased estimation procedure to estimate the mean winter energy bill for all households in this town.
b. Use an unbiased estimation procedure to find an estimate of the variance of the estimator of part a.
c. Find a 95% confidence interval for the population mean winter energy bill for households in this town. |
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| m6876 | A social psychologist wishes to test the assertion that our attitude toward other people tends to reflect our perception of their attitude toward us. A randomly selected member of each of 12 couples who live together is told (in private) that his or her partner has rated that person at the high end of a 0 to 100 scale of trustworthiness. The other member is told (also in private) that his or her partner has rated that person at the low end of the trustworthiness scale. Each person is then asked to estimate, in turn, the trustworthiness of his or her partner, yielding the following results. (According to the original assertion, the people in the trustworthy condition should give higher ratings than should their partners in the untrustworthy condition.)
(a) Use T to test the null hypothesis at the .01 level.
(b) Specify the approximate p-value for this test result. |
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| m6877 | A social scientist cross-classifies the responses of 100 randomly selected people on the basis of gender and whether or not they favor strong gun control laws to obtain the following:
(a) Using the .05 level of significance, test the null hypothesis for gender and attitude toward gun control.
(b) Specify the approximate p - value for the test result.
(c) How might these results be reported in the literature? |
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| m6878 | A state has a law requiring motorists to carry insurance. It was estimated that, despite this law, 6.0% of all motorists in the state are uninsured. A random sample of 100 motorists was taken. Use the Poisson approximation to the binomial distribution to estimate the probability that at least 3 of the motorists in this sample are uninsured. Also indicate what calculations would be needed to find this probability exactly if the Poisson approximation was not used. |
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| m6879 | A state senator believes that 25% of all senators on the Finance Committee will strongly support the tax proposal she wishes to advance. Suppose that this belief is correct and that 5 senators are approached at random.
a. What is the probability that at least 1 of the 5 will strongly support the proposal?
b. What is the probability that a majority of the 5 will strongly support the proposal? |
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| m6880 | A statistics instructor is interested in the ability of students to assess the difficulty of a test they have taken. This test was taken by a large group of students, and the average score was 78.5. A random sample of eight students was asked to predict this average score. Their predictions were as follows:
72 83 78 65 69 77 81 71
Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a two-sided alternative and a 10% significance level. |
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| m6881 | A stock market analyst claims expertise in picking stocks that will outperform the corresponding industry norms. This analyst is presented with a list of 5 high-technology stocks and a list of 5 airline stocks, and she is invited to nominate, in order, the 3 stocks that will do best on each of these 2 lists over the next year. The analyst claims that success in just 1 of these 2 tasks would be a substantial accomplishment. If, in fact, the choices are made randomly and independently, what is the probability of success in at least 1 of the 2 tasks merely by chance? Given this result, what do you think of the analyst s claim? |
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| m6882 | A stock market analyst examined the prospects of the shares of a large number of corporations. When the performance of these stocks was investigated one year later, it turned out that 25% performed much better than the market average, 25%, much worse, and the remaining 50%, about the same as the average.
Forty percent of the stocks that turned out to do much better than the market were rated good buys by the analyst, as were 20% of those that did about as well as the market and 10% of those that did much worse. What is the probability that a stock rated a good buy by the analyst performed much better than the average? |
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| m6883 | A stockbroker is interested in the factors influencing the rate of return on the common stock of banks. For a sample of 30 banks, the following regression was estimated by least squares:
where
y = percentage rate of return on common stock of bank
x1 = percentage rate of growth of bank’s earnings
x2 = percentage rate of growth of bank’s assets
x3 = loan losses as percentage of bank’s assets
x4 = 1 if bank head office is in New York City and 0 otherwise
The numbers below the coefficients are the coefficient standard errors.
a. Interpret the estimated coefficient on x4.
b. Interpret the coefficient of determination, and use it to test the null hypothesis that, taken as a group, the four independent variables do not linearly influence the dependent variable.
c. Let ei denote the residuals from the fitted regression and y`i the in-sample predicted values of the dependent variable. The least squares regression of e`i on y`i yielded coefficient of determination 0.082. What can be concluded from this finding? |
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| m6884 | A store has determined that 30% of all lawn mower purchasers will also purchase a service agreement. In 1 month 280 lawn mowers are sold to customers, who can be regarded as a random sample of all purchasers.
a. What is the standard error of the sample proportion of those who will purchase a service agreement?
b. What is the probability that the sample proportion will be less than 0.32?
c. Without doing the calculations, state in which of the following ranges the sample proportion is most likely to be: 0.29 to 0.31, 0.30 to 0.32, 0.31 to 0.33, or 0.32 to 0.34. |
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| m6885 | A store owner stocks an out-of-town newspaper that is sometimes requested by a small number of customers. Each copy of this newspaper costs her 70 cents, and she sells them for 90 cents each. Any copies left over at the end of the day have no value and are destroyed.
Any requests for copies that cannot be met because stocks have been exhausted are considered by the store owner as a loss of 5 cents in goodwill. The probability distribution of the number of requests for the newspaper in a day is shown in the accompanying table. If the store owner defines total daily profit as total revenue from newspaper sales, less total cost of newspapers ordered, less goodwill loss from unsatisfied demand, what is the expected profit if four newspapers are order? |
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| m6886 | A student committee has 6 members-4 undergraduate and 2 graduate students. A subcommittee of 3 members is to be chosen randomly so that each possible combination of 3 of the 6 students is equally likely to be selected. What is the probability that there will be no graduate students on the subcommittee? |
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| m6887 | A student feels that 70% of her college courses have been enjoyable and the remainder have been boring. This student has access to student evaluations of professors and finds out that professors who had previously received strong positive evaluations from their students have taught 60% of his enjoyable courses and 25% of his boring courses. Next semester the student decides to take three courses, all from professors who have received strongly positive student evaluations. Assume that this student s reactions to the three courses are independent of one another.
a. What is the probability that this student will find all three courses enjoyable?
b. What is the probability that this student will find at least one of the courses enjoyable? |
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