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nonparametric statistical inference
Questions and Answers of
Nonparametric Statistical Inference
A die is weighted so that when it is tossed, one particular number will come up with a probability of 1/2 and each of the five other numbers will come up with a probability of 1/10. Suppose that each
Suppose we know that the probability that it will rain on any given day is p and is independent of the weather on all other days. As a meteorologist, what should your pattern of weather predictions
Suppose there are two curtains and behind each curtain is either gold or a goat.Suppose the prior probability that there is gold behind a given curtain is 1/2, is the same for each of the curtains,
You experiment by withdrawing a ball, checking its color, and returning it. Given that the ball is white, what is the probability that the container before you is container 1? If you repeat the
There are two containers of black and white balls. In container 1, 1/3 of the balls are black and 2/3 are white. In container 2, 2/3 of the balls are black and 1/3 are white. The container before you
Continuing with the cost assumptions of the previous problem.(a) In general, when you observe a feature x, what is the average cost if you decide 0? What is it if you decide 1? (These should be
Consider the situation in question 4 and assume now that it costs $12 if you decide 0 when it really was a 1 and $10 if you decide 1 when it really was a 0.(a) If you use the Bayes decision rule that
For the situation in question 4, consider two decision makers, Ivan and Derek, who make a decision solely on the basis of prior probabilities (i.e., without observing any features). Ivan thinks that
Consider two classes 0 and 1 with P (0) = 0.6. Suppose that the conditional probability density functions of the feature x are p(x|0) = 1 for 0 ≤ x ≤ 1 and p(x|1) = 2x for 0 ≤ x ≤ 1.(a) What
True or false? If the features, classifications, and probabilities are fixed, there can be more than one Bayes rule.
What is a Bayes rule?
What is Bayes theorem? Why is it true? How might Bayes theorem be useful in determining the probability of a hypothesis, given some evidence?
Are there uncountably many rational numbers?
Assuming that a decision rule for a pattern recognition problem is as follows:decide 1 (with cost $10 for a wrong answer) when 0 ≤ x ≤ 1, and decide 0(with cost $5 for a wrong answer) when 1 < x
In formulating the pattern recognition problem, we consider only the probability of error as our criterion for success. In some cases, this is not such an appropriate criterion since certain errors
If 0 and 1 are the subsets for which a decision rule c decides 0 and 1, respectively, write the expression for the average error rate of c?
Define what is meant by the probability of error of a decision rule. For any particular learning problem, what is the smallest probability of error of a decision rule? Is the best decision rule the
(a) As discussed in Chapter 1, a very common choice for object representation is a vector of real numbers. In some cases, the entries of the vector take on a discrete set (i.e., finite number) of
If the problem is to classify an email as spam or not spam, what features might you consider?
What are the advantages of choosing/using more features and what are the disadvantages?
What is a decision rule? If there are 10 possible feature vectors, how many possible decision rules are there?
If the probability that Alice hits any given point on the target is the same as the probability that she hits any other given point, does that imply that she is as likely to hit a point within 1 inch
Consider the function defined by p(x) =2 − 2x 0 ≤ x < 1 0 otherwise.(a) Sketch p(x).(b) Is p(x) a valid probability density function? Why or why not?(c) What is the probability that x ≥ 1/2?(d)
If p1(x) and p2(x) are two probability density functions, consider the function pλ(x) = λp1(x) + (1 − λ)p2(x).(a) Is it true that pλ(x) is a valid density function for any λ ∈ [0, 1]? Why or
Given the probability density function f (x), what is the expression for P(a ≤x ≤b) (that is, the probability that x takes a value in the range [a,b])?
True or false: A probability density can take on values less than 0.
True or false: A probability density can take on values greater than 1.
True or False: Depending on the circumstances, the subjective probability of an event can be less than, equal to, or greater than its statistical probability.
True or False: if a fair coin is tossed infinitely many times, each time with a probability of 0.5 for getting heads and 0.5 for getting tails, the probability that heads will come up every time
True or False: If we toss an unbiased coin (50% heads and 50% tails) 100 times, then by the Law of Large Numbers we will have exactly 50 heads and 50 tails.
Using the suppositions in question 3, suppose also that a ball is randomly drawn and then replaced, then again a ball is randomly drawn and replaced.What is the probability that the same ball is
What is the difference between drawing with replacement and drawing without replacement? Does this difference affect the probabilities in any way? Explain.
Given the same suppositions as in the previous question, suppose also that the balls come in two shapes, round and egg-shaped, where the number of round balls is r and the number of egg-shaped balls
Suppose that the probability of drawing a white ball from a bowl containing only black and white balls is w/(w + b), where w is the number of white balls and b is the number of black balls. What is
Why is it true that the number of ways of selecting r items from a set of n items is nCr = n!r!(n−r)!?
True or False: For any events A and B, P(A and B) < P(A)
True or False: For any events A and B, we have P(A or B) = P(A) + P(B).
If the events A and B are independent, what is P(A&B)?
True or False: For any event A, we have P(not A) = 1 − P(A)?
Given an opaque bowl containing six all blue balls, seven all red balls, and nothing else, if a ball is withdrawn at random, what is the probability that the ball is blue?
About the problem of induction (a) What is the problem of induction? (b) How does the reliability of induction compare with the reliability of deduction? (c) How might statistical learning theory say
Is classification a special case of estimation? What differences are there between typical cases of classification and typical cases of estimation?
For a 12 × 12 grayscale image (256 grayscale levels), how many dimensions are there for the feature vector? How many different possible feature vectors are there?
If we want to use the values of F different features, in order to classify objects, where each feature can have any of G different values, what is the dimension of the feature space?
What is a feature space? What do the dimensions of such a space represent?What is a vector? What is a feature vector?
Investigate recent developments in the debate over GM foods. Does the new information alter any of the major arguments in the debate? Explain.
Some people advocate giving the choice about GM foods to consumers by requiring labeling on all products that contain genetically modified ingredients. Do you think this is a good idea? Why or why
Ignoring any safety issues of GM foods, make a list of as many ways as you can think of in which GM foods might be beneficial to humanity. Then, ignoring any benefits of GM foods, make a list of as
Propose an experiment that could be used to test the safety of a GM food product, such as corn. Describe your experiment in detail, and discuss any practical difficulties that might be involved in
Can the GM crops cause any unforeseen environmental damage, such as transferring their genes into weeds (thereby making “superweeds”) or killing animals besides the insect pests?
Do the new proteins contained in GM foods cause allergic reactions in some people?
Do GM foods have any toxic effects in humans?
Education correlates with healthier lives as well as longer lives, which means that more educated people tend to have lower health care costs. Can you think of any ways in which this fact could be
Smoking is well known to be one of the biggest risk factors in disease and early death. As a result, the government and health care groups have often undertaken expensive advertising campaigns in
Why do you think education contributes to longer life?For example, do you think the mere fact of staying in school makes people live longer, or is it the extra learning associated with more years of
Figure 10 presents a wealth of data. Interpret it carefully.How does it support the hypothesis that more education leads to longer life? Explain.
If you incorrectly conclude that professional comedians have a mean IQ score greater than 110 when their actual mean IQ score is 100, have you made a type I error or a type II error?assume that we
What are the two possible conclusions that can be reached about the claim being tested?assume that we want to use a 0.05 significance level to test the claim that the mean IQ score of professional
What are the two possible conclusions that can be reached about the null hypothesis?assume that we want to use a 0.05 significance level to test the claim that the mean IQ score of professional
If the test results in a P-value of 0.007, what do you conclude about the given claim?assume that we want to use a 0.05 significance level to test the claim that the mean IQ score of professional
What is the alternative hypothesis?assume that we want to use a 0.05 significance level to test the claim that the mean IQ score of professional comedians is greater than 110.
What is the null hypothesis?assume that we want to use a 0.05 significance level to test the claim that the mean IQ score of professional comedians is greater than 110.
What is the alternative hypothesis that results from the claim that the proportion of convicted felons who serve time in prison is equal to 0.6?
What is the null hypothesis that is used for testing the claim that the proportion of college graduates who voted in the last election is greater than 0.5?
What is the alternative hypothesis that results from the claim that the mean brain volume of adults is larger than 1,126 cm3?
Finding a job Through networking. In a survey of 703 randomly selected workers, 429 got their jobs through networking (based on data from Taylor Nelson Sofres Research).a. Use the sample data with a
In a study of smokers who tried to quit smoking with nicotine patch therapy, 39 were smoking one year after the treatment, and 32 were not smoking one year after the treatment (based on data from
Randomly selected cans of Coke are measured for the amount of cola, in ounces. The sample values listed below have a mean of 12.19 ounces and a standard deviation of 0.11 ounce. Assume that we want
Hypothesis Testing in the news. Find a news article or research report that describes (perhaps not explicitly) a hypothesis test for a population proportion.Attach the article and summarize the
county data. The Statistical Abstract of the United States and the Current Population Survey provide an extensive supply of social, economic, and vital statistics at the county, state, and local
Women college students. Use the Internet or library references to find the proportion of college students in the United States who are women. Test the claim that the proportion of women students at
smoking. Use the Internet or library references to determine the proportion of Americans who smoke. Test the claim that the proportion of students at your college who smoke is different from the
left-Handedness. Given the claim that 10% of Americans are left-handed, randomly select at least 50 students at your college and determine whether they are left-handed. Test the claim with a formal
clinical Test. In clinical tests of the drug Lipitor, 863 patients were treated with the drug and 19 of them experienced flu symptoms (based on data from Parke-Davis).Test the claim that the
natural gas use. According to the Energy Information Administration, 53.0% of households nationwide used natural gas for heating in 1997. A recent survey of 3,600 randomly selected households showed
internet use. When 3011 adults were surveyed in a Pew Research Center poll, 2198 said that they use the Internet.Is it OK for a newspaper reporter to write that “3/4 of all adults use the
Abortion survey. An annual survey of first-year college students, conducted by the Higher Education Research Institute at UCLA, asks approximately 276,000 students about their attitudes on a variety
Poverty. According to recent estimates, 12.1% of the 4,342 people in Custer County, Idaho, live in poverty. Assume that the people in this county represent a random sample of all people in Idaho.
drug use. A Department of Health and Human Services study of illegal drug use among 12- to 17-year-olds reported a decrease in use (from 11.4% in 1997) to 9.9%now. Suppose a survey in a large high
Married Adults. In a recent year, 125.8 million adults, or 58.6% of the adult American population, were married. In a New England town, a simple random sample of 1,445 adults includes 56.0% who are
grade Pressure. A study commissioned by the U.S. Department of Education, based on responses from 1,015 randomly selected teenagers, concluded that 44% of teenagers cite grades as their greatest
smoking and college Education. A survey showed that among 785 randomly selected subjects who completed four years of college, 144 smoke and the others do not smoke (based on data from the American
Voter Poll. In a pre-election poll, a candidate for district attorney receives 205 of 400 votes. Assuming that the people polled represent a random sample of the voting population, test the claim
Alternative Hypothesis. The claim of p 6 0.25 can never be supported if the sample proportion pn is greater than 0.25.decide whether the statement makes sense (or is clearly true) or does not make
Alternative Hypothesis. In a two-tailed hypothesis test of a claim about a proportion, the P-value is the area to the right of the standard score, z.
null and Alternative Hypotheses. In a hypothesis test involving a claim made about a population proportion, if there is sufficient evidence to reject the null hypothesis, then there is sufficient
null Hypothesis. In a test of the claim that a majority of Americans favor registration of all handguns, the null hypothesis is p = 0.5 and the alternative hypothesis is p 7 0.5.decide whether the
P-Value. A P-value of 0.00001 is obtained when sample data are used to test the claim that the majority of car crashes occur within 5 miles of home. What does this Pvalue tell us?
scream. A survey of 61,647 people included several questions about office relationships. Of the respondents, 26%reported that bosses scream at employees. The survey is an Elle/MSNBC.COM survey in
distribution. In conducting a hypothesis test as described in this section, which term best describes the particular distribution that is used: uniform, normal, probability, weighted?
notation. What do p, pn, and P-value represent?
Power of a Test. Using a search engine such as Google, search for “power” of a hypothesis test. Describe what the power of a hypothesis test is.
Hypothesis Test Applet. Using a search engine such as Google, search for “hypothesis testing” and “applet.” Find an applet and run it. Describe how the applet works and what it illustrates.
county data. The Statistical Abstract of the United States and the Current Population Survey provide an extensive supply of social, economic, and vital statistics at the county, state, and local
comparisons with national Averages. Choose several variables that are relatively easy to measure in a class or sample of students. The variables should involve a quantity that can be averaged (for
H0: The mean length of a bolt in the suspension system of new Audi cars is 3.456 centimeters.Ha: The mean length of a bolt in the suspension system of new Audi cars is not equal to 3.456
H0: The lottery is fair.Ha: The lottery is biased.a null and alternative hypothesis are given. Without using the terms “null hypothesis” and “alternative hypothesis,” identify the type I
H0: The defendant is not guilty.Ha: The defendant is guilty.a null and alternative hypothesis are given. Without using the terms “null hypothesis” and “alternative hypothesis,” identify the
H0: The patient is free of a particular disease.Ha: The patient has the disease.a null and alternative hypothesis are given. Without using the terms “null hypothesis” and “alternative
ncAA Football coach salaries. A simple random sample of 40 salaries of NCAA football coaches in the NCAA has a mean of $415,953. The standard deviation of all salaries of NCAA football coaches is
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