## Ixxr

Is called the sample variance.7 Its square root, a*, is called the sample standard deviation. At first sight, the use of (n 1 ) rather than n as the divisor in our definition of the sample variance may be rather surprising. The motivation for this formulation is that if the sample variance is defined in this way, it can be shown that the mean of its sampling distribution is the true population variance 8 that is The conclusion that the expected value of the sample variance is the population...

## Confidence Intervals For The Mean Of A Normal Distribution Population Variance Known

Let us assume that a random sample is taken from a normal distribution with an unknown mean and a known variance and that the objective is to find a confidence interval for the population mean. This problem is somewhat unrealistic, because rarely (if ever) will a population variance be precisely known and yet the mean be unknown. It does sometimes happen, however, that similar populations have been sampled so often in the past that the variance of the population of interest can be assumed known...

## The Hypergeometric Distribution

In Example 4.13, we considered a situation in which a sample of items from a very large consignment was to be checked for defectives. By assuming that the number sampled was extremely small relative to the total number of items in the consignment, we were able to approach the problem through use of the binomial distribution. However, in cases where the number of sample members is not a very small proportion of the total number of items in the population, the binomial distribution is...

## Tests Of The Mean Of A Normal Distribution Population Variance Known

We introduce the methodology of classical hypothesis testing by considering the case where a random sample of n observations, XUX2, , X, from a normal distribution with mean x and variance cr2, is available. The objective is to test hypotheses about the unknown population mean. Initially it will be assumed that the population variance is known. Later we will see that this assumption and that of normality can be relaxed when the number of sample observations is large. We begin with the problem...

## Grouped Data And Histograms

When a data set of interest contains only a few observations, the presentation of numerical measures of central location and dispersion, together with a plot such as Figure 2.1, typically provides an adequate summary. The purpose of including the plot is to give a visual impression of the distribution of the observations. However, most data sets met in practice contain many observations, and it is generally desirable to obtain a clearer picture of the distribution of such data. To illustrate...

## Choice Of Point Estimator

The problem that often arises in practice of how to choose an appropriate point estimator for a population parameter is by no means straightforward and, moreover, involves considerable mathematical intricacy beyond the scope of this text. We are able to make only a few brief comments on this important question. In Section 7.2, we saw that an attractive possibility is to choose the most efficient of all unbiased estimators, or perhaps the most efficient of a broad group of unbiased estimators....

## Jointly Distributed Continuous Random Variables

Section 4.4 introduced joint distributions for discrete random variables. Many of the concepts and results discussed there extend quite naturally to the case of continuous random variables. Let Xi, X , , XK be continuous random variables. (i) Their joint cumulative distribution function, FXl, X2, , . Uu x2, , xK) expresses the probability that simultaneously Xt is less than xu X2 is less than x2, and so on that is Fx Xl,XK(xux2, , xK) P(xl < v, n x < .v n n xK < xK) (ii) The cumulative...

## Tests Based On Independent Samples

Suppose now that we have a random sample of nx observations from a normal population with mean fix and variance ax and an independent random sample of ny observations from a normal population with mean jly and variance crY2. In Section 8.7, we saw that if the sample means are denoted X and Y, then the random variable has a standard normal distribution. If the two population variances are known, tests for the difference between the population means can be based on this result, using the same...

## Info

For a random sample of forty accounting students in a class using group learning techniques, the mean examination score was 322.12, and the sample standard deviation was 54.53. For an independent random sample of sixty-one students in the same course but in a class not using group learning techniques, the sample mean and standard deviation of the scores were 304.61 and 62.61, respectively.2 Find a 95 confidence interval for the difference between the two population mean scores. 37. In a...

## Plot Variety A Variety B

Cox, H. M. Nix, and H. Wichmann, Responsibility accounting and operational control for government units, Accounting Horizons, 3, no. 2 1989 , 38 48. 42 S. H. Akhter and G. R. Laczniak, The future U.S. business environment with strategic marketing implications for European exporters, European Journal of Marketing, 23, no. 5 1989 , 58-74. 43 H. Cooper, K. M. DeNeve, and F. Mosteller, Predicting professional sports game outcomes from intermediate game scores,...

## Expectations For Continuous Random Variables

In Section 4.3, we introduced the ideas of the expectation of a discrete random variable X and the expectation of a function of that random variable. These concepts extend to the case where the random variable is continuous, though the fact that here the probability of any specific value is 0 necessitates some modification in the method of evaluating expectations, as indicated in the box. Expectations for Continuous Random Variables Suppose that a random experiment leads to an outcome that can...

## Review Exercises

Suppose that you have an intelligent friend who has not studied probability. How would you explain Jo your friend the distinction between mutually exclusive events and independent events Illustrate your answer with suitable examples. 49. State, with reasons, whether each of the following statements is true or false a The complement of the union of two events is the intersection of their complements. b The sum of the probabilities of collectively exhaustive events must equal 1. c The number...

## Numerical Summary Of Grouped Data

A histogram provides a very convenient visual summary of a large set of numerical observations. However, an investigator will frequently want, in addition to this picture, some numerical summary measures of central tendency and dispersion. When the original data are available, this can be accomplished using the procedures discussed in Sections 2.2 and 2.3. Given modern computing resources, this typically provides only a modest computational burden, even for very large data sets. However, it...

## Example

We can now state that the estimates of the population mean, variance, and proportion of stocks for which the price-earnings ratio exceeded 8.5 are obtained through unbiased estimation procedures. However, the estimate of the population standard deviation, sx 3.97, is not obtained through an unbiased estimation procedure. An estimator that is not unbiased is said to be biased. The extent of the bias is the difference between the mean of the estimator and the...

## Worse

About the same 106 153 75 About the same 106 153 75 a Find the probability that if the forecast is for a worse performance in earnings, this outcome will result. b If the forecast is for an improvement in earnings, find the probability that this outcome fails to result. 71. A dean has found that 62 of entering freshmen and 78 of junior college transfers eventually graduate. Of all entering students, 73 are freshmen, and the remainder are junior college transfers. a What is the probability that...

## Normal Approximation To The Poisson Distribution

Let the random variable X denote the number of occurrences of an event in a particular interval of time and denote by A the expected, or mean, number of occurrences in that time interval. Then X obeys the Poisson distribution discussed in Section 4.7, with mean and variance Consider now the situation in which the mean number of occurrences, A, is large. Suppose that the time interval of interest is broken down into subintervals of equal width, as in Figure 5.19. Then the total number of...

## Suppose That A New Windmill Can Generate An Average Of 800

a Without assuming that the population variance is known, test the null hypothesis that the population mean weight of active ingredient per tablet is 5 grams. Use a two-sided alternative and a 5 significance level. State any assumptions that you make. b Stating any assumptions that you make, test the null hypothesis that the population standard deviation is .025 gram against the alternative hypothesis that the population standard deviation exceeds .025 gram. Use a 5 ' significance level. 63....

## Exercises

It is hypothesized that the more expert a group of people examining a corporation's financial report, the more variable will be their predictions about its future. Independent random samples, each of thirty individuals, from groups of different levels of expertise were chosen.39 The low-expertise group consisted of people who had just completed their first intermediate accounting course. Members of the high-expertise group had completed undergraduate studies and were employed by reputable...

## Example 417

An analyst predicted that 3.5 of all small corporations would file for bankruptcy in the coming year. For a random sample of 100 small corporations, estimate the probability that at least three will file for bankruptcy in the next year, assuming that the analyst's prediction is correct. The distribution of the number X of filings for bankruptcy is binomial, with n 100 and p .035, so the mean of the distribution is We will use the Poisson distribution, with mean A 3.5, to approximate our...

## Confidence Intervals For The Mean Of A Normal Population Population Variance Unknown

We can now use the Student's t distribution to derive confidence intervals for the mean of a normal population when the variance is unknown, using an argument similar to that of Section 8.2. Assume that a random sample of n observations is available from a normal population with mean fx and unknown variance and that confidence intervals for the population mean are required. Let X and s 2 denote the sample mean and variance. Then, from Section 8.3, we know that the random variable follows a...

## Manufacturer Of Detergent Claims That The Contents Of Boxes Soldweigh On Average At Least 16 Ounces

A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weights is known to be normal, with standard deviation .4 ounce. A random sample of sixteen boxes yielded a sample mean weight of 15.84 ounces. Test at the 10 significance level the null hypothesis that the population mean weight is at least 16 ounces. 2. A company which receives shipments of batteries tests a random sample of nine of them before agreeing to take a...

## Making Sense Of Numerical Information

Any manager operating in the business environment requires as much information as possible about the characteristics of that environment. In the modern era, thanks in part to the massive information storage capacities of computer systems, much of the available information is quantitative. For example, it may be necessary to assimilate movements in interest rates, stock market prices, money supply, or unemployment. Market research surveys are carried out to determine the strength of product...

## Economictrics Null Hypothesis Exercises

When a production process is operating correctly, the resistance in ohms of electrical components produced has a normal distribution with mean 92.0 and standard deviation 3.6. A random sample of four components was taken. a Find the mean of the sampling distribution of the sample mean resistance. b Find the variance of the sample mean. c Find the standard error of the sample mean. d What is the probability that the sample mean exceeds 93.0 ohms 2. The lifetimes of lightbulbs produced by a...

## Abc Acb Bac Bca Cab

This example is illustrated in the tree diagram of Figure 3.9. We begin at the intersection on the left-hand side of the figure by choosing one of the three letters to fill the first position. Following each of the emerging branches, we then have two possibilities for filling the second position. For example, if the letter A is in the first position, either B or C must be placed in the second position. Finally, once the first two positions have FIGURE 3.9 Tree diagram for Example 3.1 First...

## Fixed Cost 2000 Plus 2000for Each Day Taken To Complete Project Find Mean And Standard Deviation

a What is the probability that a randomly chosen project will take less than 3 days to complete b Find the expected time to complete a project. c Find the standard deviation of time required to complete a project. d The contractor's project cost is made up of two parts a fixed cost of 20,000, plus 2,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost. e If three projects are undertaken, what is the probability that at least two of them...

## Bags Of Chemical Produced By A Company Have Impurity Weights That Can Be Represented By A Normal Distribution With Mean

A car rental company has determined that the probability a car will need service work in any given month is .2. The company has 900 cars. a What is the probability that more than 200 cars will require service work in a particular month b What is the probability that fewer than 175 cars will need service work in a given month Use the normal approximation to the binomial distribution, without the continuity correction. 36. It is known that 10 of all the items produced by a particular...

## Random Sample Of 10 Stock Market Mutual Funds Was Taken

A process produces batches of a chemical whose impurity concentrations follow a normal distribution with variance 1.75. A random sample of twenty of these batches is chosen. Find the probability that the sample variance exceeds 3.10. 36. Monthly rates of return on the shares of a particular common stock are independent of one another and normally distributed with a standard deviation of 1.7 A sample of 12 months is taken. a Find the probability that the sample standard deviation is less...

## Find The Probability That A Randomly Chosen Student Has Not Visited A Museum

a Find the probability that a randomly chosen student has not visited a museum in the last year. b Find the means of the random variables X and Y. c Find and interpret the covariance between the random variables X and Y. 65. A basketball team's star 3-point shooter takes six 3-point shots in a game. Historically, he makes 40 of all 3-point shots attempted. Answer the following questions about the outcome of the six 3-point shots taken in this game, stating at the outset what assumptions you...

## Long-distance Taxi Service Owns 4 Vehicles.these Are Of Different Ages And Have Different Repair Records. The

a Find the expected number of cars that will be sold in the week. b Find the standard deviation of the number of cars that will be sold in the week. c The salesman receives for the week a salary of 250, plus an additional 300 for each car sold. Find the mean and standard deviation of his total salary for the week. d What is the probability that the salesman's salary for the week will be more than 1,000 61. A multiple-choice test has nine questions. For each question, there are four possible...

## Emotive And Loaded Statements

Numbers, in and of themselves, contain no value judgments. Data simply provide factual material, which could, of course, be useful on one side or another of a particular 6 The title of this section is inspired by D. Huff and I. Geis, How to Lie with Statistics New York Norton, 1954 . This delightful little book is essential reading for anyone with a serious interest in the presentation of statistical information. Also see H. Wainer, How to display data badly, American Statistician, 38 1984 ,...