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

Inferences About the Difference Between Two Population Means: σ1 and σ2 Known

Letting μ1 denote the mean of population 1 and μ2 denote the mean of population 2, we will focus on inferences about the difference between the means: μ1 – μ2. To make an inference about this difference, we select a simple random sample of n1 units from popula­tion 1 and a second simple random

Inferences About the Difference Between Two Population Means: s1 and s2 Unknown

In this section we extend the discussion of inferences about the difference between two population means to the case when the two population standard deviations, s1 and s2, are unknown. In this case, we will use the sample standard deviations, s1 and s2, to estimate the unknown population standard deviations. When we use the

Inferences About the Difference Between Two Population Means: Matched Samples

Suppose employees at a manufacturing company can use two different methods to per­form a production task. To maximize production output, the company wants to iden­tify the method with the smaller population mean completion time. Let m1 denote the population mean completion time for production method 1 and m2 denote the population mean completion time

Inferences About the Difference Between Two Population Proportions

Letting p1 denote the proportion for population 1 and p2 denote the proportion for population 2, we next consider inferences about the difference between the two population proportions: P1 – P2. To make an inference about this difference, we will select two independent random samples consisting of n1 units from population 1 and n2

Inferences About a Population Variance

The sample variance is the point estimator of the population variance s2. In using the sample variance as a basis for making inferences about a population variance, the sampling distribution of the quan­tity (n – 1)s1/s2 is helpful. This sampling distribution is described as follows. Figure 11.1 shows some possible forms of the sampling

Inferences About Two Population Variances

In some statistical applications we may want to compare the variances in product quality resulting from two different production processes, the variances in assembly times for two assembly methods, or the variances in temperatures for two heating devices. In making comparisons about the two population variances, we will be using data collected from two

Testing the Equality of Population Proportions for Three or More Populations

In this section, we show how the chi-square (^2) test statistic can be used to make statist­ical inferences about the equality of population proportions for three or more populations. Using the notation the hypotheses for the equality of population proportions for k > 3 populations are as follows: If the sample data and the

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Test of Independence

An important application of a chi-square test involves using sample data to test for the independence of two categorical variables. For this test we take one sample from a pop­ulation and record the observations for two categorical variables. We will summarize the data by counting the number of responses for each combination of a

Goodness of Fit Test

In this section we use a chi-square test to determine whether a population being sam­pled has a specific probability distribution. We first consider a population with a histor­ical multinomial probability distribution and use a goodness of fit test to determine if new sample data indicate there has been a change in the population distribution

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An Introduction to Experimental Design and Analysis of Variance

As an example of an experimental statistical study, let us consider the problem facing Chemitech, Inc. Chemitech developed a new filtration system for municipal water supplies. The components for the new filtration system will be purchased from several suppliers, and Chemitech will assemble the components at its plant in Columbia, South Carolina. The industrial

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