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Statistics Probability Inference And Decision 1st Edition Robert L. Winkler, William L. Hays - Solutions
Using Equations (6.4.3) and (6.4.4), derive formula (6.4.5).
What is meant by the statement that “consistency, unlike the other three properties of estimators considered in the text, is a large-sample property”?
Show that for a sample of size N from a Poisson process, the sample mean is an unbiased estimator of both the mean and the variance of the process.
A sample of N independent observations is taken from a normally distributed population, with N > 2. Denote the N observations by Xh X2, • • •, XN.You want to estimate the mean of the population, ji. For each of the three estimators G, H, and M, answer the following questions.G = Xi, H = (Yi +
Suppose that the random variable X is normally distributed with mean n and variance a2. Consider the following two estimators of n, based on a sample of four independent observations, Xi, X2, X3, and X4:G = (Xi + X2 + X3 + X4)/4 H = (4Ah 3X2 + 2X3 + X4)/10.(a) Is G unbiased?(b) Is H unbiased?(c)
If a sample of size Ni is taken from a normal population with mean pi and variance cr\, and a sample of size N2 is taken from a normal population with mean n2 and variance a2, what is the distribution of the difference between the two sample means, Mi — M2? What if the populations were not normal
Suppose that samples are taken from two normal populations. The first population has mean 80 and variance 81, and the second population has mean 100 and variance 100. If Ni = 9 is the size of the sample from the first population and N2 — 10 is the size of the sample from the second population,
Answer (a)-(c) in Exercise 36 if the sample size is 400, and compare your answers to these obtained when N = 16. In each case, how would your answers be affected if the assumption of normality was not applicable?
Suppose that X is normally distributed with mean 150 and variance 400. If a sample of 16 independent trials is randomly chosen from the X population, find(a) P(M> 150)(b) P(145 < Af < 149)(c) the mean and variance of M.
To demonstrate the central limit theorem under a skewed population dis¬tribution, follow the same procedure as in Exercise 34 but perform the follow¬ing transformation before computing sample means:Random number X 0,1, 2,3 0 4,5,6 1 7,8 2 9 3
To demonstrate the central limit theorem, draw 100 samples of size five from a random number table and calculate the sample mean for each of the 100 samples. Construct a frequency distribution of sample means. Do the same for 100 samples of size ten and compare the two frequency distri¬butions.
The argument for the normal theory of error, presented in Section 4.23 of the text, was based on the normal approximation to the binomial, assuming the action of a vast number of “influences” that might affect any given nu¬merical observation. However, can one make a rationale for the normal
If X is normally distributed with mean 50 and variance 100, Y is normally distributed with mean 80 and variance 25, Z is normally distributed with mean 100 and variance 144, and X, Y, and Z are independent, find the dis¬tribution of:(a) X — Y (c) 6Z + 4X-30F(b) X+Y-Z (d) (X+Y + Z)/3.
For the population distribution given in Exercise 28, find the sampling dis¬tribution of the sample variance for a sample of three independent observations by enumerating the possible combinations of events for samples of size three and translating these into values of the sample variance.
For the population distribution given in Exercise 28, find the expectation and the variance of the sample mean from a sample of size three(a) directly from the sampling distribution of the sample mean(b) from the mean and variance of the population distribution.
From the population distribution given in Exercise 28, find the sampling distribution of the sample median based on three independent observations from the population. Compare the standard error of the sample median with the standard error of the sample mean.
Given the following very simple population distribution, construct the theo¬retical sampling distribution for the sample mean based on three independent observations from the population.x P{X = x)9 1/6 6 1/2 3 1/3[Hint-. Enumerate the possible combinations of events for samples of size 3, and then
Starting with Equation (5.17.9), derive the computational formulas (5.17.10)and (5.17.11).
For the data in Exercise 15, find the mean income and the median income.Which do you think is a better measure of central tendency? Why? Also, find the sample variance and standard deviation [Hint: To make this easier, first transform the data so that you will not have to work with such large
In a group of 50 boys and 50 girls of high-school age, the number of calories consumed on a particular day by each person can be summarized by the following two frequency distributions:Class interval Males Females 5000-5499 4500-4999 4000-4499 3500-3999 3000-3499 2500-2999 2000-2499 1500-1999
.Construct the standardized scores for the twelve students whose raw scores are given in Exercise 22, and then show that the mean of these standardized scores is 0 and the standard deviation of the standardized scores is 1.
The scores for twelve students on a particular examination were as follows:18 15 19 27 13 30 24 11 5 16 17 20 Compute the mean, median, standard deviation, and average absolute deviation.
How does this assumption show up in the cumulative frequency polygon?How does the value of the sample mean computed from a grouped distribu¬tion depend on the particular choice of intervals? Does the error introduced by treating each score as equal in value to the midpoint of its interval have any
When one finds the median of a grouped distribution by interpolation, what is he assuming about the distribution of values within a given class interval?
For the data in Exercise 11, compute the sample median(a) directly from the data(b) from the grouped frequency distribution with i — .0625(c) from the grouped frequency distribution with i — .1250.Explain any differences in your three answers. Also, compute the mode according to (b) and (c) and
For the data in Exercise 11, compute the sample mean(a) directly from the data(b) from the grouped frequency distribution with i = .0625(c) from the grouped frequency distribution with i = .1250.Explain any differences in your three answers.
Suppose that a group of 1000 college students were classified by sex (700 males, 300 females) and by class in school (400 freshmen, 300 sophomores, 200 juniors, and 100 seniors). If sex is independent of class in school, con¬struct a contingency table showing all possible joint frequencies (such
A box of loose change is known to contain 100 coins of the following denom¬inations: 35 pennies, 50 nickels, and 15 quarters. You draw a random sample of ten coins, with replacement. What is the frequency distribution of coins which you should expect to get? What is the probability of obtaining
The incomes of the families in a particular neighborhood are shown below.Construct a histogram with i = 1000, another histogram with i = 5000, and another histogram with i —three histograms and discuss.333 to represent this data. Compare the 5,500 12,500 8,600 11,000 11,000 8,600 31,000 6,500
Suppose that thirty-two persons participated in a series of gambles and that the net gain for each person is shown below. Construct a histogram, a fre¬quency polygon, and a cumulative frequency polygon for this set of data.$2.74 -$1.05 $ .25 -$1.39 4.09 - 1.56 .37 - 2.08.95 - .36 .09 - .48 2.29 -
In a learning experiment, cats were given successive trials at learning a complicated maze. A given cat was trained until he could traverse the maze without error five times in a row. If the cat could not go through the maze five times successfully after 30 trials, training was stopped. The numbers
For the two frequency distributions obtained in Exercise 11, construct the corresponding frequency polygons and cumulative frequency polygons.
The diameters of parts produced by a certain production process are given below. Starting at 0 and using i = .0625, construct a frequency distribution for these data. Repeat the procedure with i = .1250 and comment on any differences in the two distributions.581 .630 .460 .511 .351 .180 .450 .240
In one week, thirty patients were admitted to a large state hospital. A list of the patients classified by sex and by tentative diagnosis follows. For each sex form a frequency distribution by type of diagnosis and display this infor¬mation via a pair of histograms superimposed on the same
Make up two examples from your area of interest in which a frequency distribution would very likely require the employment of unequal or open class intervals.
Comment on the following statement: “The process of data-gathering and reporting often involves the loss or deliberate sacrifice of some potential information.”
What is one assuming about a set of data grouped into a frequency distribu¬tion when he: (a) considers the real, rather than the apparent, limits to be the class boundaries; and (b) uses the midpoint of any interval to represent all of the scores grouped into that interval.
In your own words, carefully distinguish between a probability distribution, a frequency distribution, and a sampling distribution.
What level of measurement would you expect to attain when measuring the following variables?(a) temperature (e) brand preference(b) income (f) academic achievement(c) sex (g) color of hair(d) height (h) occupation.
In many institutions of higher education, a student’s record is evaluated by a grade point average. For example, 4 points might be given for each semester hour of A, 3 for each hour of B, 2 for each hour of C, 1 for each hour of D, and 0 for each hour of F. These numerical values are multiplied
Distinguish between independent random sampling and nonindependent random sampling. By considering a simple experiment such as the drawing of a series of cards from a well-shufRed deck of cards, give examples of both independent and dependent random sampling. How does this distinction relate to the
If one has a fair coin, he can generate his own table of random numbers in the following manner: let Xi = 1 if the coin comes up heads on the first toss, and x\ = 0 if it comes up tails. Similarly, let X2 = 1 if it comes up heads on the second toss and let x2 = 0 if it comes up tails. Define X3 and
Why are random samples of such importance in statistics?
The random variable X has the following probability distribution:
graph the PMF and the CDF of X explain the relationship between the PMF and the CDF find P(X> 2.5)find P(-l < X < 4)find P(-1 < X< 4)find P(X< -3)find P(X = 1).The random variable X has the probability distribution given by the rule?
Suppose that the face value of a playing card is regarded as a random variable, with an ace counting as 1 and any face card (Jack, Queen, King) counting as 10. You draw one card at random from a well-shuffled deck. Construct a PMF showing the probability distribution for this random variable and
Why is it necessary to deal with probability densities rather than probabilities such as P(X =a) when the variable under consideration is continuous?
The density function of X is given by the rule x for 0 < x < 1, f(x) = 2- x for 1 < x < 2, 0 elsewhere.
Does the function fix) =2x/3 0for — 1 < x < 2, elsewhere, satisfy the two requirements for a density function?
Suppose that a person agrees to pay you $10 if you throw at least one six in four tosses of a fair die. How much would you have to pay him for this opportunity in order to make it a “fair” gamble?
Explain what is meant by the statement, “on any single trial, we do not expect the expectation.”
For the distribution of Exercise 1, find the mean, the mode, and the median.Is the distribution skewed positively or negatively?
For the distribution of Exercise 2, find the mean, the mode, the median, and the mid-range and compare these measures of location, or central tendency.Is this distribution symmetric or skewed?
Find the following fractiles of the distribution in Exercise 1:(a) .25 (b) .75 (c) .01(d) .33 (e) .90.
Explain why it is possible for a discrete random variable to have more than one median.
For the distribution of Exercise 5, find the mean, the mode, and the median, and comment on the relative value of these measures in this example as measures of the “typical value” of the random variable.
Can you determine the mode of a distribution by examining the CDF?Explain for both discrete and continuous distributions.21.Suppose that X represents the daily sales of a particular product, and that the probability distribution of X is as follows:
For the distribution of Exercise 1, find the variance, the standard deviation, the expected absolute deviation, and the range. Also, find these same values under the assumption that the largest value of X is 12 rather than 6. In light of your results, comment on the relative merit of the different
For the distribution of Exercise 3, find(a) E(X) (b) E{X2) (c) F(4X + 12)(d) var (X) (e) var (4X + 12).
Find the mean and variance of the random variables X and I in Exercise 8.
What do we mean when we say that the expectation of a random variable can be thought of as a center of gravity?
If a random variable has a mean of ten and a variance of zero, graph its distribution.
Prove that in general, E(X2) is not equal to [E(X)]2. [Hint: If they are equal, what can be said about the variance, o-2?]
Criticize the following statement: If the mean of a distribution is 50, and the standard deviation is 10, then the “best bet” about any case drawn at random is 50, and, on the average, one can expect to be in error by 10 points?
If the density function of X is given by a + bx2 for 0 < x < 1, f(x) =0 elsewhere, and E(X) = §, find a and b.
Find the following fractiles of the distribution in Exercise 5:(a) .01 (b) .05 (c) .25(d) .40 (e) .70 (f) .85.
From your own main area of interest, think of two variables that would be expected to have reasonably symmetric distributions, two variables that would be expected to have positively skewed distributions, and two variables that would be expected to have negatively skewed distributions.
The range is the easiest to compute of all of the measures of dispersion which we discussed. In view of this, why is the range not preferred to the standard deviation, which is much more difficult to compute?
What advantage does the standard deviation have over the variance as a measure of dispersion?
Suppose that the joint distribution of X and F is represented by the following table:1 X 2 3graph the joint PMF of X and Y (you will need a three-dimensional graph)are X and Y independent? Explain your answer, determine the marginal distributions of X and Y determine the conditional distribution of
Complete the following table, given that X and F are independent.
Complete the following table, given that P(X =P(Y = 3 \X = 2) = i
From your major field of interest, make up two examples of variables that you might reasonably expect to be independent. Also, make up four examples of variables that should be associated to some degree, two examples involving variables with a positive relationship and two examples involving
There is a real danger in confusing the idea of causation with that of statis¬tical association. Why do you think these two concepts are so often confused?For example, for centuries it was thought that swampy air when breathed“caused” malaria (hence the name, literally “bad air”). Comment
Prove that if X and F are independent, then their covariance and their cor¬relation coefficient are equal to zero \_Note\ The reverse implication is not true in general.]
Prove that cov (aX, bY) = ab cov (X, F).
If var (X) = 50, var (X -f- F) = 80, and var (X — F) = 40, find var (F)and cov (X, Y).44.Suppose that X and F are continuous random variables with joint density function given by the rule k(x + y)fix, y) = •for 0 < x < 2, 0 < y < 2, 0 elsewhere.
For the distribution in Exercise 1, find the first three moments about the origin and the first three moments about the mean.
Suppose that on a final exam in statistics the mean was 50 and the standard deviation was 10. Find the following:(i) the standardized (z) scores of students receiving the following grades:50, 25, 0, 100, 64.(ii) the raw grades corresponding to standardized scores of-2, 2, 1.95, -2.58, 1.65, .33.
For the distribution in Exercise 2, find the distribution of the corresponding standardized random variable, z. In what ways are the distributions of X and z similar and in what ways are they dissimilar?
The covariance is not a good measure of the strength of a relationship between two random variables because it depends so much on the units of measure¬ment of the two variables. However, if X and Y are standardized random variables, this problem should be eliminated. Is it? [Hint: Compare cov (X,
For the distribution of Exercise 1, calculate P(| X — p | > 2) and compare this wfith the upper bound for this probability obtained from Tchebycheff’s inequality.
For the distribution of Exercise 2, calculate P(| X — p | > 1.5) and compare this witha) the upper bound obtained from Tchebycheff’s inequality and withb) the upper bound obtained from the stronger form of Tchebycheff’s inequality (3.27.3).
A rough computational check on the accuracy of a standard deviation is that around six times the standard deviation should, in general, include almost the entire range of values for the distribution on which it is based.Do you see any reason why this rule should work? Must it be true?
Explain the terms “stationary” and “independent” as applied to a series of Bernoulli trials.
A multiple choice examination is so constructed that, in principle, the prob¬ability of a correct choice for any item by guessing alone isf. If the test consists of 10 items, what is the probability that a student will have exactly 5 correct answers if he is just guessing?
The probability that an item produced by a certain production process is defective is .05. (a) Assuming stationarity and independence, what is the probability that a lot of 10 items from the process contains no defectives?(b) What is the probability that the first defective item is the fifth item
Suppose that the probability that a duck hunter will successfully hit a duck is .25 on any given shot. He decides to hunt until he has hit 4 birds, (a) What is the expected number of shots that he will have to take? (b) What is the probability that he will take more than 10 but fewer than 15 shots?
Suppose that the probability that a particular football team wins a certain game is p, and that this probability remains constant from game to game.(a) If the outcomes of the games played by the team are independent of each other, for what value of p would the team have at least an even chance of
If the expected number of successes in N Bernoulli trials is 3 and the variance of the number of successes is 2.1, find N and p.
The probability that a person who is exposed to a certain contagious disease will catch it is .20.(a) Find the probability that the 20th person exposed to this disease is the 5th one to catch it.(b) If 20 persons are exposed to the disease, what is the expected number of persons who will catch the
Compare the binomial, Pascal, and geometric distributions. Is it possible to determine binomial probabilities by using the Pascal and geometric distri¬butions? The event “no successes in ten trials,” which is a binomial event, is related to what event involving the geometric distribution?
A manufacturer of parts for automobiles knows that the proportion of defec¬tive parts in the past has been about 10 percent. However, today he has taken a random sample of 10 parts and found that 2 were defective. What is the probability of his getting two or more defective parts by chance in such
In the primary elections of a certain state, any candidate for governor must carry at least 55 percent of the counties in order to win the nomination out¬right, without the necessity for a run-off election. A public opinion poll run on a sample of ten counties indicates that the candidate will
Compute the binomial probabilities, the Poisson approximation, and the normal approximation for the following values of N and p:(a) N = 10, p = .5 (b) N = 10, p = .05(c) N = 20, p = .5 (d) N = 20, p = .05.Comment on the relative merit of the two approximations in each case.
Under what conditions can we approximate a binomial distribution by using a Poisson distribution, and under what conditions can we approximate a binomial distribution by using a normal distribution? Justify your answers with a brief intuitive discussion.
For any given match, the probabilities that a soccer team will win, tie, or lose are .3, .4, and .3. If the team plays ten matches and the outcomes are assumed to be independent, what is the probability that they will win 5, tie 3, and lose 2?
What is the probability that in 24 tosses of a fair die, each face will occur exactly 4 times? What is the probability that in 6 tosses, each face will occur exactly once?
Just as the binomial distribution is a special case of a much more general principle in algebra (the binomial expansion), so the multinomial distribu¬tion is a special case of another quite general mathematical principle. See if you can state the general rule, and then use this rule to prove that
Explain the differences among the binomial, multinomial, and hypergeometric distributions. When is the binomial a good approximation to the hyper¬geometric?
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