Applications Of Statistical Sampling To Auditing 1st Edition Alvin A. Arens, James K. Loebbecke - Solutions

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In a study of graduate students who took the Graduate Record Exam (GRE), the Educational Testing Service recently reported that for the quantitative exam, U.S.citizens had a mean of 529 and standard deviation of 127, whereas the non-U.S. citizens had a mean of 649 and standard deviation of 129.(a)
A teacher summarizes grades on the midterm exam by Min = 26, Q1 = 67, Median = 80, Q3 = 87, Max = 100, Mean = 76, Mode = 100, Standard dev. = 76, IQR = 20.She incorrectly recorded one of these. Which one do you think it was? Why?
Ten people are randomly selected in Florida and another 10 people are randomly selected in Alabama. Table 3.20 provides summary information on mean income.The mean is higher in Alabama both in rural areas and in urban areas. Which state has the larger overall mean income?(The reason for this
For Table 3.2 (page 43), explain why the mean of these 50 observations is not necessarily the same as the violent crime rate for the entire U.S. population.
For a sample with mean ¯y, adding a constant c to each observation changes the mean to ¯y+c, and the standard deviation s is unchanged. Multiplying each observation by c changes the mean to c¯y and the standard deviation to |c|s.(a) Scores on a difficult exam have a mean of 57 and a standard
In a Child Health Survey conducted in a school, 1567 children answered “yes” and 433 children answered “no”in response to the question: “Do you drink milk?”(a) Estimate the probability that a randomly selected child drinks milk.(b) Estimate the probability that a child does not drink
Suppose a software for statistical inference methods often sets the default probability of a correct inference to be 0.95.Suppose we make an inference about the population proportion of people who like to go trekking, and we consider this separately for men and for women. LetAdenote the outcome
SaveYour Earth, anNGO, conducted a survey asking subjects whether they are a member of anyNGOworking for environmental protection and whether they would be willing to purchase a sapling. Table 4.4 shows the results.(a) Estimate the probability that a randomly selected adult is a member of an
Let y = number of sports a person can play. For residents of Country A, y has probability distribution P(0) = 0.05, P(1) = 0.85, and P(2) = 0.10, with negligible probability for higher values of y.(a) Is y a discrete or a continuous variable?Why?(b) Construct a table showing the probability
Let y denote the number of times a person was admitted to a hospital within the past 12 months. According to results from recent Region Health Surveys, for a randomly chosen person in Country A, the probability distribution of y is approximately P(0) = 0.85, P(1) = 0.13, P(2) = 0.01, and P(3) =
A ticket for a lottery costs $1. With probability 0.0000001, you win 2 million dollars ($2,000,000), and with probability 0.9999999 you win nothing. Let y denote the winnings from buying one ticket. Construct the probability distribution for y. Show that the mean of the distribution equals 0.20,
Let y be the outcome of selecting a single digit using a random number generator.(a) Construct the probability distribution for y. (This type of distribution is called a uniform distribution, because of the uniform spread of probabilities across the possible outcomes.)(b) Find the mean of this
For a normal distribution, find the probability that an observation falls (a) at least one standard deviation above the mean; (b) at least one standard deviation below the mean.
For a normal distribution, verify that the probability between(a) μ − σ and μ + σ equals 0.68.(b) μ − 1.96σ and μ + 1.96σ equals 0.95.(c) μ − 3σ and μ + 3σ equals 0.997.(d) μ − 0.67σ and μ + 0.67σ equals 0.50.
Find the z-value for which the probability that a normal variable exceeds μ + zσ equals (a) 0.01, (b) 0.025,(c) 0.05, (d) 0.10, (e) 0.25, (f) 0.50.
Find the z-value such that for a normal distribution the interval from μ − zσ to μ + zσ contains (a) 50%,(b) 90%, (c) 95%, (d) 99% of the probability.
Find the z-values corresponding to the (a) 90th, (b)95th, (c) 99th percentiles of a normal distribution.
If z is the number such that the interval from μ−zσto μ+zσ contains 90% of a normal distribution, then explain why μ + zσ is the 95th percentile.
If z is the positive number such that the interval fromμ − zσ to μ + zσ contains 50% of a normal distribution, then(a) Which percentile is (i) μ + zσ? (ii) μ − zσ?(b) Find this value of z. Using this result, explain why the upper quartile and lower quartile of a normal distribution are
What proportion of a normal distribution falls(a) above a z-score of 2.10?(b) below a z-score of −2.10?(c) between z-scores of −2.10 and 2.10?
Find the z-score for the number that is less than only 1% of the values of a normal distribution.
A teacher wants to know how many students have a test score at the 96th percentile or higher.(a) How many standard deviations above the mean is the 96th percentile?(b) For the normal distribution of scores with mean 75 and standard deviation 8, find the score for the 96th percentile.
According to a recent Population Survey, selfemployed individuals in a country work an average of 55 hours per week, with a standard deviation of 18. If this variable is approximately normally distributed, what proportion averaged more than 40 hours per week?
The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a standardized measure used in studies with high-risk infants. It has approximately a normal distribution with a mean of 100 and a standard deviation of 16.(a) What proportion of children have an MDI of at least
For a study in a State University Hospital, 3856 pregnant women who reported information on length of gestation until birth had mean = 284 days and standard deviation= 12 days. A baby is classified as premature if the gestation time is 258 days or less.(a) If mothers’ gestation times are normally
Suppose that the weekly use of ground water in a village is approximately normally distributed, with a mean of 20 gallons and a standard deviation of 5 gallons.(a) What proportion of villagers use more than 25 gallons per week?(b) Assuming that the standard deviation and the normal form would
On the midterm exam in mathematics, an instructor always gives a grade of A to students who score between 90 and 95. One year, the scores have approximately a normal distribution with mean 83 and standard deviation 7.About what proportion of the students get an A?
Student A’s scores in Country X have a distribution with μ = 400 and σ = 75, and Student B’s scores in Country Y have a distribution with μ = 25 and σ = 4.0.Which score is relatively higher, Student A = 500 or Student B = 36? Explain.
Suppose that consumption expenditure by households in a city has an approximately normal distribution with a mean of $5400 and a standard deviation of $1800.The consumption expenditure by one particular household is $8000.(a) Find the z-score corresponding to that value.(b) What proportion of the
An energy study found that household use of electricity had a mean of 583 and a standard deviation of 450 kWh (kilowatt-hours).(a) If the distribution was normal, what percentage of the households used above 800 kWh?(b) Do you think the distribution is truly normal?Why or why not?
Five students—the females Angeline and Belle and the males Charlie,David, and Edmund—are rated equally qualified for admission to medical school, ahead of other applicants. However, all but two positions have been filled for entering the class. The admissions committee can admit only two more
Construct the sampling distribution of the sample proportion of heads, for flipping a balanced coin(a) Once.(b) Twice. (Hint: The possible samples are (H, H), (H, T),(T, H), (T, T).)(c) Three times. (Hint: There are 8 possible samples.)(d) Four times. (Hint: There are 16 possible samples.)(e)
The probability distribution associated with the outcome of rolling a balanced die has probability 1/6 attached to each integer, {1, 2, 3, 4, 5, 6}. Let (y1, y2) denote the outcomes for rolling the die twice.(a) Enumerate the 36 possible (y1, y2) pairs (e.g., (2, 1)represents a 2 followed by a
An exit poll of 547 voters in the state election indicated that 55% voted for Candidate X, with most of the rest voting for Candidate Y.(a) If actually 50% of the population voted for Candidate X, find the standard error of the sample proportion voting for this candidate, for this exit poll.
According to Population Reports, population distribution of number of years of education for women in Country C has a mean of 10.6 and a standard deviation of 4.0.Find the mean and standard error of the sampling distribution of ¯y for a random sample of (a) 6 women,(b) 49 women, (c) 100 women.
Refer to Exercise 4.6.The mean and standard deviation of the probability distribution for the lottery winnings y are μ = 0.20 and σ = 632.45.Suppose you play the lottery 1 million times. Let ¯y denote your average winnings.(a) Find the mean and standard error of the sampling distribution of
According to a General Regional Survey, in South Asia, the distribution of y = pair of shoes has a mean of 6.5 and a standard deviation of 4.9.Suppose these are the population mean and standard deviation.(a) Does y have a normal distribution? Explain.(b) For a random sample of 700 adults (the size
The scores on the Psychomotor Development Index(PDI), a scale of infant development, are approximately normal with mean 100 and standard deviation 15.(a) An infant is selected at random. Find the probability that PDI is below 95.(b) A study uses a random sample of 36 infants. Specify the sampling
A study plans to sample randomly 100 government records of farms to estimate the mean acreage of farms in Place A. Results from an earlier study suggest that 400 acres is a reasonable guess for the population standard deviation of farm size.(a) Approximate the probability that the sample mean
According to the Census Report in Country A, the number of children in a household has a mean of 3.5 and a standard deviation of 2.0.Suppose the Census Report instead had estimated this mean using a random sample of 300 homes, and that sample had a mean of 4.2 and standard deviation of 1.5.(a)
At a university, 55% of the 6500 students are female.The student newspaper reports results of a survey of a random sample of 60 students about the activity clubs they wish to join. They report that their sample contained 32 females.(a) Explain how you can set up a variable y to represent gender.(b)
The distribution of family size in a particular society is skewed to the right, with μ = 10.4 and σ = 6.0.These values are unknown to an anthropologist, who samples families to estimate mean family size. For a random sample of 40 families, the anthropologist gets a mean of 8.6 and a standard
Use the applet for the Sampling Distribution for the Sample Proportion at www.pearsonglobaleditions.com/Agresti to illustrate this concept. Set the population proportion as 0.50 and sample size n = 100.(a) Simulate once (setting the number of samples to 1 and clicking on Draw Sample) and report the
Use the applet for the Sampling Distribution for the Sample Mean for continuous variables at www.pearsonglobaleditions.com/Agresti to investigate the sampling distribution of ¯y.(a) Select the skewed population distribution and set the skewness = 2. Take 10,000 samples of size 50 each. How does
Go to the applet for the Sampling Distribution for the Sample Mean for discrete variables at www.pearsonglobaleditions.com/Agresti.(a) Construct a population distribution that you think is plausible for y = number of cars in a house.(b) Draw a single sample of size n = 1000 to reflect results of a
For a single roll of a die, let y = 1 for an even number and y = 0 for an odd number, to simulate the vote in an election with two equally preferred candidates.(a) Construct the probability distribution for y, and find its mean.(b) The die is rolled 10 times, yielding an even number six times and
(Class Exercise) Refer to Exercises 1.11 and 1.12(pages 21 and 22). Using the population defined by your class or using the Students data file, the instructor will select a variable, such as weekly time watching television.(a) Construct a histogram or stem-and-leaf plot of the population
Rainbow City was designed to attract orphan children.Its current population of 30,000 children has a mean age of 6 years and a standard deviation of 10 years. The distribution of ages is skewed to the left, reflecting the predominance of older children. A random sample of 100 children of Rainbow
Refer to the previous exercise. Describe the sampling distribution of ¯y (a) for a random sample of size n = 1; (b) if you sample all 30,000 children.
(Class Exercise) Table 4.5 provides the ages of all 50 heads of households in a small Nova Scotian fishing village. The data are in the data file Ages at the text website. The distribution of these ages is characterized byμ = 47.18 and σ = 14.74.(a) Construct a stem-and-leaf plot or histogram of
(a) Which distribution does the sample data distribution tend to resemble more closely—the sampling distribution or the population distribution? Explain.(b) Explain carefully the difference between a sample data distribution and the sampling distribution of ¯y. Illustrate your answer for a
The Central Bureau of Statistics in Sunshine Village asked mothers aged between 22 and 26 about the ideal number of children. The probability distribution is approximately P(1) = 0.02, P(2) = 0.14, P(3) = 0.07, P(4) = 0.33, P(5) = 0.15, and P(6 or more) = 0.28.(a) Because the last category is open
For a normal distribution, show that(a) The upper quartile equals μ + 0.67σ.(b) According to the 1.5(IQR) criterion, an outlier is an observation falling more than 2.7 standard deviations below or above the mean, and this happens for only 0.7% of the data.
In an exit poll of 3646 voters in the 2017 election in a town, 50.8% said they voted for Candidate R and 49.2%said they voted for Candidate S. Based on this information, would you be willing to predict the winner of the election? Explain your reasoning.
For an election exit poll that uses random sampling, find the standard error of the sample proportion voting for a candidate for whom the population proportion is 0.50, when n = 100, when n = 1000, and when n = 10000.In each case, predict an interval within which the sample proportion is almost
The standard error of a statistic describes(a) The standard deviation of the sampling distribution of that statistic.(b) The standard deviation of the sample data.(c) How close that statistic is likely to fall to the parameter that it estimates.(d) The variability in the values of the statistic for
The Central Limit Theorem implies that(a) All variables have bell-shaped sample data distributions if a random sample contains at least about 30 observations.(b) Population distributions are normal whenever the population size is large.(c) For large random samples, the sampling distribution of ¯y
True or False: As the sample size increases, the standard error of the sampling distribution of ¯y increases. Explain your answer.4.54.* A junior college admits students only if they score above 500 on a standardized achievement test. Applicants from Region A have a mean of 550 and a standard
From the formula on page 84, the standard deviation of a discrete probability distribution is (a) When y can equal only 0 and 1, letting = P(y = 1) and 1 = P(y = 0), show that = and that - = (1-). (b) Show that the standard error of a sample proportion for a random sample of size n equals
Refer to the formula for the normal distribution curve shown in the footnote on page 85. Show that this curve is symmetric, by showing that for any constantc, the curve has the same value at y = μ+ c as at y = μ− c.(The integral of f (y) for y between μ + zσ and∞equals the tail probability
The standard error formula σ¯y = σ/√n treats the population size as infinitely large relative to the sample size n. The formula for σ¯y for a finite population size denoted by N is N-n Oy = VN-1 The term (N-n)/(N-1) is called the finite popula- tion correction. (a) When n 300 students are
A general rule states that for independent observations, the variance of Σyi is the sum of the variances, which is nσ2 for n observations.(a) Explain intuitively why Σyi would have a larger variance than a single observation y.(b) Since the variance of a probability distribution is σ2 = E(y −
Ellenberg (2014) noted that when you use sample data to rank states by brain cancer rates, the highest ranking state (South Dakota) and the nearly lowest ranking state (North Dakota) had relatively small sample sizes.Also, when schools in North Carolina were ranked by their average improvement in
Explain how it is possible to use different audit approaches to obtain a given level of assurance. Provide two examples of different audit approaches that would result in the same level of assurance.
Identify three circumstances in which it may be desirable to judgmentally select a sample size and sample items rather than use statistical sampling methods.
Identify five factors that are likely to affect judgmentally determined sample size for substantive tests of transactions and details of balances. Explain the relationship between the factors and the sample size.
Which two statistical methods are applicable to compliance tests of transactions? Identify the circumstances in which each of these two methods is likely to be favored.
Identify the major considerations affecting reliability level and upper precision limit for compliance tests. Explain how the auditor will decide when to use a reliability of 95% compared to 80%.
What four statistical methods are most commonly used for substantive tests of transactions? Identify the circumstances in which each of the methods is most likely to be used.
What are the factors affecting the reliability level of substantive tests of transactions? Under what circumstances is the auditor likely to use a high confidence level? When is a low confidence level or no substantive tests of transactions most likely?
Identify and distinguish among the eleven statistical methods that are available for direct tests of balances.
Explain why attributes sampling is unlikely to be used for direct tests of balances. Under what circumstances is attributes sampling likely to be used ?
There are limited circumstances in which unstratified mean-per-unit is likely to be used. Why? What are the circumstances?
Identify the circumstances in which difference and ratio estimation can be used in auditing.
Identify the circumstances in which dollar unit sampling is most likely to be used. Also identify the circumstances in which an alternative method is likely.
Explain why it is desirable to select all larger items for audit testing, regardless of the statistical method used. What decision rule was recommended in the book to identify those large items ?
What is meant by a fallback method? Describe the three fallback methods commonly used for statistical methods.
Identify the factors that affect the reliability level for direct tests of balances. What are the circumstances in which a high reliability level would be used ?
The following are 13 factors that would probably affect the sample size for compliance tests of transactions:(1) Upper precision limit(2) Quality of compliance evidence available(3) Planned reliance on internal control(4) Adequacy of the client’s documentation and records (5) Compliance
This is your first year in charge of the audit of the All-America Bank. All- America is a publicly held bank in excellent condition; it has been a client of the firm for several years. Two years ago, the bank installed a highly computerized system for processing transactions. The system and all
Steadfast Wholesaling Company purchases a line of approximately 42,000 different items and wholesales to a variety of industries. The company utilizes an elaborate EDP perpetual inventory system as a basis for its operations (promising deliveries, etc.) as well as financial reporting.You have been
Following are four situations in which you will use statistical sampling. Indicate what precision and reliability values you will use for each and justify your answer. Be sure to indicate whether reliability is one-sided or two-sided, and why.a. You are performing a detailed compliance test in a
If you were to perform hypothesis tests in the situations described in 11-19 c and d, what Beta risks would you use ?11-19 c and d:Following are four situations in which you will use statistical sampling. Indicate what precision and reliability values you will use for each and justify your answer.
Suppose you selected a dollar unit sample of 300 items, expecting to find zero errors, and evaluated the results using an upper precision limit of 1.0%. What actions would you take if you found:a. 3 errors?b. 10 errors?c. 30 errors?
What are the purposes of analytical tests from an auditor’s point of view?
Identify the major factors motivating auditors to seek more effective analytical testing techniques.
What is meant by regression analysis? Describe two audit situations in which regression analysis might be used.
Distinguish between inference and prediction as each is used in regression. Which is a more appropriate term when the auditor is verifying the reasonableness of an account balance with regression ?
What is meant by the term variables in regression? Distinguish between independent variables and dependent variables. Assume the auditor was using regression analysis to make inferences about total cost of sales. The variables being considered are commercial sales and wholesale sales. Identify the
What are the characteristics of simple linear regression? Provide an example in which this method might be appropriate.
Distinguish between linear and curvilinear regression. What are the advantages and disadvantages of each ?
Distinguish among the following three terms and define the standard formulation of independent and dependent variables for each term: simple linear regression, simple curvilinear regression, multiple linear regression, and multiple curvilinear regression.
Using simple linear regression, define each term for the formulation Y = a + bX. Use this formula to determine February’s expected value of repair and maintenance expense for a rock crusher if there is $$2,000$ of repairs and maintenance monthly regardless of volume. The expense is expected to
How would an auditor ascertain what information should be used to determine the values of a and b in question 10-9? Assuming the information is available, how would the values of a and b be determined ?Question 10-9:Using simple linear regression, define each term for the formulation Y = a + bX.
What is meant by a data set in regression analysis ? How many data sets would there be if 26 months of data were used to determine the value of a and b for each of four independent variables for purposes of estimating total sales ?
What is meant by the coefficient of correlation in regression analysis? What do the values .9 and — .82 for two different coefficients of correlation indicate ? How is the coefficient of correlation obtained in a particular situation?
What is meant by a confidence interval for a regression line? Compare in concept the confidence interval for regression analysis and for variables sampling.
Compare a t-distribution to a normal distribution. Which is used for regression analysis ? Assume there were 20 data sets used to determine values for a and b. How many degrees of freedom should be used to determine the t value ? What will be the t value for a two-sided interval at a 95% confidence
What is meant by an outlier in regression analysis? How would the auditor find which of the current year’s monthly recorded repairs and maintenance expense totals are outliers ? What would the auditor do with outliers ?
What is meant by a Markov chain process? What might be the purpose of using a Markov chain in the audit of inventory ? What information would be needed to make the calculations?

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Applications Of Statistical Sampling To Auditing 1st Edition Alvin A. Arens, James K. Loebbecke - Solutions

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