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,

Population Mean: σ Known

In Chapter 8 we said that the σ known case corresponds to applications in which historical data and/or other information are available that enable us to obtain a good estimate of the population standard deviation prior to sampling. In such cases the population standard de­viation can, for all practical purposes, be considered known. In

Population Mean: s Unknown

In this section we describe how to conduct hypothesis tests about a population mean for the σ unknown case. Because the σ unknown case corresponds to situations in which an estimate of the population standard deviation cannot be developed prior to sampling, the sample must be used to develop an estimate of both μ

Population Proportion

In this section we show how to conduct a hypothesis test about a population proportion p. Using p0 to denote the hypothesized value for the population proportion, the three forms for a hypothesis test about a population proportion are as follows. The first form is called a lower tail test, the second form is

Hypothesis Testing and Decision Making

In the previous sections of this chapter we have illustrated hypothesis testing applications that are considered significance tests. After formulating the null and alternative hypotheses, we se­lected a sample and computed the value of a test statistic and the associated p-value. We then compared thep-value to a controlled probability of a Type I error,

Calculating the Probability of Type II Errors

In this section we show how to calculate the probability of making a Type II error for a hypothesis test about a population mean. We illustrate the procedure by using the lot- acceptance example described in Section 9.6. The null and alternative hypotheses about the mean number of hours of useful life for a

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Determining the Sample Size for a Hypothesis Test About a Population Mean

Assume that a hypothesis test is to be conducted about the value of a population mean. The level of significance specified by the user determines the probability of making a Type I error for the test. By controlling the sample size, the user can also control the probability of making a Type II error.

Big Data and Hypothesis Testing

We have seen that interval estimates of the population mean m and the population propor­tion p narrow as the sample size increases. This occurs because the standard error of the associated sampling distributions decrease as the sample size increases. Now consider the relationship between interval estimation and hypothesis testing that we discussed earlier in