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Basic Statistics Understanding Conventional Methods And Modern Insights 1st Edition Rand R. Wilcox - Solutions

For the data in table 2.2, suppose a histogram is to be created with bins defined as follows: −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80.Determine the frequencies for each bin and construct a histogram.
The heights of 30 male Egyptian skulls from 4000 BC were reported by Thomson and Randall-Maciver (1905) to be 121 124 129 129 130 130 131 131 132 132 132 133 133 134 134 134 134 135 135 136 136 136 136 137 137 138 138 138 140 143.Create a histogram with bins extending from 120–125, 125–130, and
For the data in the previous problem, does the boxplot rule (described in chapter 2) indicate that 143 is an outlier?
What do the last two problems suggest about using a histogram to detect outliers?
Table 3.5 shows the exam scores for 27 students. Create a stem-and-leaf display using the digit in the ones position as the stem.
If the leaf is the hundredths digit, what is the stem for the number 34.679?
Consider the values 5.134, 5.532, 5.869, 5.809, 5.268, 5.495, 5.142, 5.483, 5.329, 5.149, 5.240, 5.823. If the leaf is taken to be the tenths digit, why would this make an uninteresting stem-and-leaf display?
For the boxplot in figure 3.11, determine, approximately, the quartiles, the interquartile range, and the median. Approximately how large is the largest value not declared an outlier?
In figure 3.11, about how large must a value be to be declared an outlier? How small must it be?Table 3.5 Examination Scores 83 69 82 72 63 88 92 81 54 57 79 84 99 74 86 71 94 71 80 51 68 81 84 92 63 99 91
Create a boxplot for the data in table 3.1.
Create a boxplot for the data in table 3.2.
Describe a situation where the sample histogram is likely to give a good indication of the population histogram based on 100 observations.
Comment generally on how large a sample size is needed to ensure that the sample histogram will likely provide a good indication of the population histogram?
When trying to detect outliers, discuss the relative merits of using a histogram versus a boxplot.
A sample histogram indicates that the data are highly skewed to the right. Is this a reliable indication that if all individuals of interest could be measured, the resulting histogram would also be highly skewed?
If the possible values for x are 0, 1, 2, 3, 4, 5, and the corresponding values for p(x)are .2, .2, .15, .3, .35, .2, .1, respectively, does p(x) qualify as a probability function?
If the possible values for x are 2, 3, 4 and the corresponding values for p(x) are .2,−0.1, .9, respectively, does p(x) qualify as a probability function?
If the possible values for x are −1, 2, 3, 4, and the corresponding values for p(x) are.1, .15, .5, .25, respectively, does p(x) qualify as a probability function?
If the possible values for x are 2, 3, 4, 5, and the corresponding values for p(x) are.2, .3, .4, .1, respectively, what is the probability of observing a value less than or equal to 3.4?
In problem 4, what is the probability of observing a value less than or equal to 1?
In problem 4, what is the probability of observing a value greater than 3?
In problem 4, what is the probability of observing a value greater than or equal to 3?
If the probability of observing a value less than or equal to 6 is .3, what is the probability of observing a value greater than 6?
For the probability function x : 0,1 p(x) : .7,.3 verify that the mean and variance are .3 and .21, respectively. What is the probability of getting a value less than the mean?
Imagine that an auto manufacturer wants to evaluate how potential customers will rate handling for a new car being considered for production. Also suppose that if all potential customers were to rate handling on a 4-point scale, 1 being poor and 4 being excellent, the corresponding probabilities
If the possible values for x are 1, 2, 3, 4, 5 with probabilities .2, .1, .1, .5, .1, respectively, what are the population mean, variance and standard deviation?
In problem 11, determine the probability of getting a value within one standard deviation of the mean. That is, determine the probability of getting a value between μ−σ and μ+ σ.
If the possible values for x are 1, 2, 3 with probabilities .2, .6 and .2, respectively, what is the mean and standard deviation?
In problem 13, suppose the possible values for x are now 0, 2, 4 with the same probabilities as before. Will the standard deviation increase, decrease, or stay the same? Verify your answer.
For the probability function x : 1,2,3,4,5 p(x) : .15,.2,.3,.2,.15 determine the mean, the variance, and the probability that the a value is less than the mean.
For the probability function x : 1,2,3,4,5 p(x) : .1,.25,.3,.25,.1 would you expect the variance to be larger or smaller than the variance associated with the probability function used in the previous exercise? Verify your answer by computing the variance for the probability function given here.
For the probability function x : 1,2,3,4,5 p(x) : .2,.2,.2,.2,.2 would you expect the variance to be larger or smaller than the variance associated with the probability function used in the previous exercise? Verify your answer by computing the variance.
For the following probabilities Income Age High Medium Low< 30 .030 .180 .090 30–50 .052 .312 .156> 50 .018 .108 .054 determine (a) the probability someone is under 30, (b) the probability that someone has a high income given that they are under 30, (c) the probability of someone having a low
For the previous exercise, are income and age independent?
Coleman (1964) interviewed 3,398 schoolboys and asked them about their self-perceived membership in the “leading crowd.” Their response was either yes, they were a member, or no they were not. The same boys were also asked about their attitude concerning the leading crowd. In particular, they
Let Y be the cost of a home and let X be a measure of the crime rate. If the variance of the cost of a home changes with X, does this mean that cost of a home and the crime rate are dependent?
If the probability of Y < 6 is .4 given that X = 2, and if the probability of Y < 6 is.3 given that X = 4, does this mean that X and Y are dependent?
If the range of possible Y values varies with X, does this mean that X and Y are dependent?
For a binomial with n = 10 and p = .4, use table 2 in appendix B to determine(a) p(0), the probability of exactly 0 successes, (b) P(X ≤ 3), (c) P(X < 3),(d) P(X > 4), (e) P(2 ≤ X ≤ 5).
For a binomial with n = 15 and p = .3, use table 2 in appendix B to determine(a) p(0), the probability of exactly 0 successes, (b) P(X ≤ 3), P(X < 3), (c)P(X > 4), (d) P(2 ≤ X ≤ 5).
For a binomial with n = 15 and p = .6, use table 2 to determine the probability of exactly 10 successes.
For a binomial with n = 7 and p = .35, what is the probability of exactly 2 successes.
For a binomial with n = 18 and p = .6, determine the mean and variance of X, the total number of successes.
For a binomial with n = 22 and p = .2, determine the mean and variance of X, the total number of successes.
For a binomial with n = 20 and p = .7, determine the mean and variance of pˆ, the proportion of observed successes.
For a binomial with n = 30 and p = .3, determine the mean and variance of pˆ.
For a binomial with n = 10 and p = .8, determine (a) the probability that pˆ is less than or equal to .7, (b) the probability that pˆ is greater than or equal to .8, (c) the probability that pˆ is exactly equal to .8.
A coin is rigged so that when it is flipped, the probability of a head is .7. If the coin is flipped three times, which is the more likely outcome, exactly three heads, or two heads and a tail?
Imagine that the probability of head when flipping a coin is given by the binomial probability function with p = .5. (So the outcomes are independent.) If you flip the coin nine times and get nine heads, what is the probability of head on the tenth flip?
The Department of Agriculture of the United States reports that 75% of all people who invest in the futures market lose money. Based on the binomial probability function, with n = 5, determine(a) the probability that all 5 lose money.(b) the probability that all 5 make money.(c) the probability
If for a binomial, p = .4 and n = 25, determine (a) P(X < 11), (b) P(X ≤ 11),(c) P(X > 9) and (d) P(X ≥ 9)
In the previous problem, determine the mean of X, the variance of X, the mean of pˆ), and the variance of p
Given that Z has a standard normal distribution, use table 1 in appendix B to determine (a) P(Z ≥ 1.5), (b) P(Z ≤ −2.5), (c) P(Z < −2.5), (d) P(−1 ≤ Z ≤ 1).
If Z has a standard normal distribution, determine (a) P(Z ≤ .5),(b) P(Z > −1.25), (c) P(−1.2 < Z < 1.2), (d) P(−1.8 ≤ Z < 1.8).
If Z has a standard normal distribution, determine (a) P(Z < −.5),(b) P(Z < 1.2), (c) P(Z > 2.1), (d) P(−.28 < Z < .28).
If Z has a standard normal distribution, find c such that (a) P(Z ≤c) = .0099,(b) P(Z c) = .5691, (d) P(−c ≤ Z ≤c) = .2358.
If Z has a standard normal distribution, find c such that (a) P(Z >c) = .0764,(b) P(Z >c) = .5040, (c) P(−c ≤ Z
If X has a normal distribution with mean μ = 50 and standard deviation σ = 9, determine (a) P(X ≤ 40), (b) P(X < 55), (c) P(X > 60), (d) P(40 ≤ X ≤ 60).
If X has a normal distribution with mean μ = 20 and standard deviation σ = 9, determine (a) P(X < 22), (b) P(X > 17), (c) P(X > 15), (d) P(2 < X < 38).
If X has a normal distribution with mean μ = .75 and standard deviation σ = .5, determine c is (a) P(X < .25), (b) P(X > .9), (a)P(X c)=.382, (c) P(.5 < X < 1), (d) P(.25 < X < 1.25).
If X has a normal distribution, determine c such that P(μ −cσ < X < μ+cσ) = .95.Hint: Convert the above expression so that the middle term has a standard normal distribution.
If X has a normal distribution, determine c such that P(μ −cσ < X < μ+cσ) = .8.
Assuming that the scores on a math achievement test are normally distributed with mean μ = 68 and standard deviation σ = 10, what is the probability of getting a score greater than 78?
In the previous problem, how high must someone score to be in the top 5%? That is, determine c such that P(X >c) = .05.
A manufacturer of car batteries claims that the life of their batteries is normally distributed with mean μ = 58 months and standard deviation σ = 3.Determine the probability that a randomly selected battery will last at least 62 months.
Assume that the income of pediatricians is normally distributed with meanμ =$100,000 and standard deviation σ = 10,000. Determine the probability of observing an income between $85,000 and $115,000.
Suppose the winnings of gamblers at Las Vegas are normally distributed with mean μ = −300 (the typical person loses $300), and standard deviation σ = 100.Determine the probability that a gambler does not lose any money.
A large computer company claims that their salaries are normally distributed with mean $50,000 and standard deviation 10,000. What is the probability of observing an income between $40,000 and $60,000?
Suppose the daily amount of solar radiation in Los Angeles is normally distributed with mean 450 calories and standard deviation 50.Determine the probability that for a randomly chosen day, the amount of solar radiation is between 350 and 550.
If the cholesterol levels of adults are normally distributed with mean 230 and standard deviation 25, what is the probability that a randomly sampled adult has a cholesterol level greater than 260?
If after one year, the annual mileage of privately owned cars is normally distributed with mean 14,000 miles and standard deviation 3,500, what is the probability that a car has mileage greater than 20,000 miles?
Can small changes in the tails of a distribution result in large changes in the population mean, μ, relative to changes in the median?
Explain in what sense the population variance is sensitive to small changes in a distribution.
For normal random variables, the probability of being within one standard deviation of the mean is .68. That is, P(μ− σ ≤ X ≤ μ +σ) = .68 if X has a normal distribution. For nonnormal distributions, is it safe to assume that this probability is again .68? Explain your answer.
If a distribution appears to be bell-shaped and symmetric about its mean, can we assume that the probability of being within one standard deviation of the mean is .68?
Can two distributions differ by a large amount yet have equal means and variances?
If a distribution is skewed, is it possible that the mean exceeds the .85 quantile?
For a binomial with n = 25 and p = .5, determine (a) P(pˆ ≤ 15/25), (b)P(pˆ > 15/25), (c) P(10/25 ≤ pˆ ≤ 15/25).
Many research teams intend to conduct a study regarding the proportion of people who have colon cancer. If a random sample of ten individuals could be obtained, and if the probability probability of having colon cancer is .05, what is the probability that a research team will get pˆ = .1?
In the previous problem, what is the probability of pˆ = .05?
Someone claims that the probability of losing money, when using an investment strategy for buying and selling commodities, is .1. If this claim is correct, what is the probability of getting pˆ ≤ .05 based on a random sample of 25 investors?
You interview a married couple and ask the wife whether she supports the current leader of their country. Her husband is asked the same question. Describe why it might be unreasonable to view these two responses as a random sample.
Imagine that a thousand research teams draw a random sample from a binomial distribution with p = .4, with each study based on a sample size of 30.So this would result in 1,000 pˆ values. If these 1,000 values were averaged, what, approximately, would be the result?
In the previous problem, if you computed the sample variance of the pˆ values, what, approximately, would be the result?
Suppose n = 16, σ = 2, and μ = 30.Assume normality and determine (a)P(X¯ ≤ 29), (b) P(X¯ > 30.5), (c) P(29 ≤ X¯ ≤ 31)
Suppose n = 25, σ = 5, and μ = 5.Assume normality and determine (a)P(X¯ ≤ 4), (b) P(X¯ > 7), (c) P(3 ≤ X¯ ≤ 7).
Someone claims that within a certain neighborhood, the average cost of a house isμ =$100,000 with a standard deviation of σ = $10,000. Suppose that based on n = 16 homes, you find that the average cost of a house is X¯ = $95,000.Assuming normality, what is the probability of getting a sample
In the previous problem, what is the probability of getting a sample mean between$97,500 and $102,500?
A company claims that the premiums paid by its clients for auto insurance has a normal distribution with mean μ = 750 dollars and standard deviation σ = 100 dollars. Assuming normality, what is the probability that for n = 9 randomly sampled clients, the sample mean will a value between 700 and
Imagine you are a health professional interested in the effects of medication on the diastolic blood pressure of adult women. For a particular drug being investigated, you find that for n = 9 women, the sample mean is X¯ = 85 and the sample variance is s 2 = 160.78. Estimate the standard error of
You randomly sample 16 observations from a discrete distribution with meanμ = 36 and variance σ2 = 25.Use the central limit theorem to determine(a) P(X¯ < 34), (b) P(X¯ < 37), (c) P(X¯ > 33), (d) P(34 < X¯ < 37).
You sample 25 observations from a non-normal distribution with mean μ = 25 and variance σ2 = 9.Use the central limit theorem to determine (a) P(X¯ < 24),(b) P(X¯ < 26), (c) P(X¯ > 24), (d) P(24 < X¯ < 26).
Referring to the previous problem, describe a situation where reliance on the central limit theorem to determine P(X¯ < 24) might be unsatisfactory.
Describe situations where a normal distribution provides a good approximation of the sampling distribution of the mean.
For the values 4, 8, 23, 43, 12, 11, 32, 15, 6, 29, verify that the McKean–Schrader estimate of the standard error of the median is 7.57.
In the previous example, how would you argue that the method used to estimate the standard error of the median is a reasonable approach?
For the values 5, 7, 2, 3, 4, 5, 2, 6, 7, 3, 4, 6, 1, 7, 4, verify that the McKean–Schrader estimate of the standard error of the median is .97.
In the previous example, how would you argue that the method used to estimate the standard error of the median might be highly inaccurate?
In problem 20, would it be advisable to approximate P(M ≤ 4) using equation (5.6)?
For the values 2, 3, 5, 6, 8, 12, 14, 18, 19, 22, 201, why would you suspect that the McKean–Schrader estimate of the standard error of the median will be smaller than the standard error of the mean? (Hint: Consider features of data that can have a large impact on s, the sample standard

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