Uniform Probability Distribution

Consider the random variable x representing the flight time of an airplane traveling from Chicago to New York. Suppose the flight time can be any value in the interval from 120 minutes to 140 minutes. Because the random variable x can assume any value in that interval, x is a continuous rather than a

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Normal Probability Distribution

The most commonly used probability distribution for describing a continuous random variable is the normal probability distribution. The normal distribution has been used in a wide variety of practical applications in which the random variables are heights and weights of people, test scores, scientific measurements, amounts of rainfall, and other similar values. It is

Normal Approximation of Binomial Probabilities

In Section 5.5 we presented the discrete binomial distribution. Recall that a binomial experi­ment consists of a sequence of n identical independent trials with each trial having two possible outcomes, a success or a failure. The probability of a success on a trial is the same for all trials and is denoted by p.

Exponential Probability Distribution

The exponential probability distribution may be used for random variables such as the time between arrivals at a hospital emergency room, the time required to load a truck, the distance between major defects in a highway, and so on. The exponential probability density function follows. As an example of the exponential distribution, suppose that

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The Electronics Associates Sampling Problem

The director of personnel for Electronics Associates, Inc. (EAI), has been assigned the task of developing a profile of the company’s 2500 managers. The characteristics to be identified include the mean annual salary for the managers and the proportion of managers having completed the company’s management training program. Using the 2500 managers as the

Selecting a Sample

In this section we describe how to select a sample. We first describe how to sample from a finite population and then describe how to select a sample from an infinite population. 1. Sampling from a Finite Population Statisticians recommend selecting a probability sample when sampling from a finite popu­lation because a probability sample

Point Estimation

Now that we have described how to select a simple random sample, let us return to the EAI problem. A simple random sample of 30 managers and the corresponding data on annual salary and management training program participation are as shown in Table 7.2. The notation x1, x2, and so on is used to

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Introduction to Sampling Distributions

In the preceding section we said that the sample mean X is the point estimator of the population mean m, and the sample proportion p is the point estimator of the population proportion p. For the simple random sample of 30 EAI managers shown in Table 7.2, the point estimate of m is X

Sampling Distribution of x

In the previous section we said that the sample mean X is a random variable and its probability distribution is called the sampling distribution of X. This section describes the properties of the sampling distribution of X. Just as with other probability distributions we studied, the sampling distribution of X has an expected value

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Sampling Distribution of p

The sample proportion p is the point estimator of the population proportion p. The formula for computing the sample proportion is where x = the number of elements in the sample that possess the characteristic of interest n = sample size As noted in Section 7.4, the sample proportion p is a random variable

Properties of Point Estimators

In this chapter we showed how sample statistics such as a sample mean X, a sample standard deviation 5, and a sample proportion p can be used as point estimators of their corresponding population parameters m, s, and p. It is intuitively appealing that each of these sample statistics is the point estimator of

Other Sampling Methods

We described simple random sampling as a procedure for sampling from a finite population and discussed the properties of the sampling distributions of X and p when simple random sampling is used. Other methods such as stratified random sampling, cluster sampling, and systematic sampling provide advantages over simple random sampling in some of these

Big Data and Standard Errors of Sampling Distributions

The purpose of statistical inference is to use sample data to quickly and inexpensively gain insight into some characteristic of a population. Therefore, it is important that we can expect the sample to look like, or be representative of, the population that is being investigated. In practice, individual samples always, to varying degrees, fail

Population Mean: σ Known

In order to develop an interval estimate of a population mean, either the population stand­ard deviation s or the sample standard deviation 5 must be used to compute the margin of error. In most applications s is not known, and 5 is used to compute the margin of error. In some applications, large amounts

Population Mean: s Unknown

When developing an interval estimate of a population mean we usually do not have a good estimate of the population standard deviation either. In these cases, we must use the same sample to estimate both m and s. This situation represents the s unknown case. When 5 is used to estimate s, the margin

Determining the Sample Size

In providing practical advice in the two preceding sections, we commented on the role of the sample size in providing good approximate confidence intervals when the population is not normally distributed. In this section, we focus on another aspect of the sample size issue. We describe how to choose a sample size large enough

Population Proportion

In the introduction to this chapter we said that the general form of an interval estimate of a population proportion p is The sampling distribution of p plays a key role in computing the margin of error for this interval estimate. In Chapter 7 we said that the sampling distribution of p can be

Big Data and Confidence Intervals

We have seen that confidence intervals are powerful tools for making inferences about pop­ulation parameters. We now consider the ramifications of big data on confidences intervals for means and proportions, and we return to the data-collection problem of online news service PenningtonDailyTimes.com (PDT). Recall that PDT’s primary source of revenue is the sale of

Developing Null and Alternative Hypotheses

It is not always obvious how the null and alternative hypotheses should be formulated. Care must be taken to structure the hypotheses appropriately so that the hypothesis testing conclusion provides the information the researcher or decision maker wants. The context of the situation is very important in determining how the hypotheses should be stated.

Type I and Type II Errors

The null and alternative hypotheses are competing statements about the population. Either the null hypothesis H0 is true or the alternative hypothesis Ha is true, but not both. Ideally the hypothesis testing procedure should lead to the acceptance of H0 when H0 is true and the rejection of H0 when Ha is true. Unfortunately,