Chapter 1

Hypothesis testing

1.1. Introduction

Inferential statistics consists of methods that use sample results to help make decisions or predictions about a population. The point and interval estimation procedures are forms of statistical inference. Another type of statistical inference is hypothesis testing. In hypothesis testing we begin by stating a hypothesis about a population characteristic. This hypothesis, called the null hypothesis, is assumed to be true unless sufficient evidence can be found in a sample to reject it. The situation is quite similar to that in a criminal trial. The defendant is assumed to be innocent; if sufficient evidence to the contrary is presented, however, the jury will reject this hypothesis and conclude that the defendant is guilty.

In statistical hypothesis testing, often the null hypothesis is an assumption about the value of a population parameter. A sample is selected from the population, and a point estimate is computed. By comparing the value of the point estimate to the hypothesized value of the parameter we draw a conclusion with respect to whether or not there is a sufficient evidence to reject the null hypothesis. A decision is made and often a specific action is taken depending upon whether or not the null hypothesis about the population parameter is accepted or rejected.

1.1.2. Concepts of hypothesis testing

Let us consider example about coffee cans. A company may claim that, on average, its cans contain 100 grams of coffee. A government agency may want to test whether or not such cans contain, on average, 100 grams of coffee.

Suppose we take a sample of 50 cans of the coffee under investigation. We then find out that the mean amount of coffee in these 50 cans is 97 grams. Based on these results, can we state that on average, all such cans contain less than 100 grams of coffee and that the company is lying to the public?

Not until we perform a test of hypothesis. The reason is that the mean grams is obtained from the sample. The difference between 100 grams (the required amount for the population) and 97 grams (the observed average amount for the sample) may have occurred only because of the sampling error. Another sample of 100 cans may give us a mean of 105 grams. Therefore, we make a test of hypothesis to find out how large the difference between 100 grams and 97 grams is and to investigate whether or not this difference has occurred as a result of chance alone. If 97 grams is the mean of all cans and not for only 100 cans, then we do not need to make a test of hypothesis. Instead, we can immediately state that the mean amount of coffee in all such cans is less than 100 grams. We perform a test of hypothesis only when we are making a decision about a population parameter based on the value of a sample statistic.

1.1.3. The null and alternative hypothesis

We will begin our general discussion by using to denote a population probability distribution parameter of interest, such as the mean, variance, or proportion. Our discussion begins with a hypothesis about the parameter that will be maintained unless there is strong contrary evidence. In statistical language it is called the null hypothesis.

For example, we might initially accept company’s claim that on average, the contests of the cans weight at least 100 grams. Then after

collecting sample data this hypothesis can be tested. If the null hypothesis is not true, then some alternative must be true. In carrying out a hypothesis test the investigator defines an alternative hypothesis against which the null hypothesis is tested.

For this coffee cans example a likely alternative is that on average can’s weights are less than 100 grams. These hypotheses are chosen such that one or the other must be true. The null hypothesis will be denoted as and the alternative hypothesis as .

Definition: A null hypothesis is a claim (or statement) about a population parameter that is assumed to be true until it is declared false.

Definition: An alternative hypothesis is a claim about population parameter that will be true if the null hypothesis is false.

Our analysis will be designed with the objective of seeking strong evidence to reject the null hypothesis and accept the alternative hypothesis. We will only reject the null hypothesis when there is a small probability that the null hypothesis is true. Thus rejection will provide strong evidence against and in favor of the alternative hypothesis, . If we fail to reject then either is true or our evidence is not sufficient to reject and hence accept . Thus we will be more comfortable with our decision if we reject and accept .

A hypothesis, whether null or alternative, might specify a single value, say , for the population parameter . In that case, the hypothesis is said to be a simple hypothesis designated as

That is read as, “The null hypothesis is that the population parameter is equal to the specific value ”.

Alternatively, a range of values might be specified for unknown parameter. We define such hypothesis as a composite hypothesis, and it will hold true for more than one value of the population parameter. In many applications, a simple null hypothesis, say

is tested against a composite alternative. One possibility would be to test the null hypothesis against the general two-sided hypothesis

In other cases, only alternatives on one side of the null hypothesis are of interest. For example, a government agency would be perfectly happy if the mean weight of coffee cans greater than 100 grams. Then we could write the null hypothesis as

and the alternative hypothesis of interest might be

We call these hypothesis one- sided composite alternatives.

Example:

A company intends to accept the product unless it has evidence to suspect that more than 10% of products are defective. Let denote the population proportion of defectives. The null hypothesis is that the proportion is less than 0.1, that is

and the alternative hypothesis is

The null hypothesis is that the product is of adequate quality overall, while the alternative is that the product is not adequate quality. In this case the product would only be rejected if there is strong evidence that there are more than 10% defectives.

Once we have specified a null hypothesis and alternative hypothesis and collected sample data, a decision concerning the null hypothesis must be made. We can either accept the null hypothesis or reject it in favor of the alternative. For good reasons many statisticians prefer not to use the term “accept the null hypothesis” and instead say “fail to reject”. When we accept or fail to reject the null hypothesis, then either the hypothesis is true or our test procedure was not strong enough to reject and we have committed an error. When we use the term accept a null hypothesis that statement can be considered shorthand for failure to reject.

From our discussion of sampling distributions, we know that the sample mean is different from the population mean. With only a sample mean we can not be certain of the value of the population mean. Thus the decision rule we adopt will have some chance of reaching an erroneous conclusion. One error we call Type I error. Type I error is defined as the rejection of the null hypothesis when the null hypothesis is true. We will see that our decision rules will be defined so that the probability of rejecting a true null hypothesis, denoted as , is “small”. The probability, , is defined as the significance level of the test. Since the null hypothesis is either accepted or rejected, it follows that the probability of accepting the null hypothesis when it is true is . The other possible error, called Type II error, arises when false null hypothesis is accepted. We say that for a particular decision rule, the probability of making such an error when the null hypothesis is false is denoted . Then, the probability of rejecting a false null hypothesis is which is called the power of test.

Type I error

A type I error occurs when a true null hypothesis is rejected. The value represents the probability of committing this type of error, that is

The value represents the significance level of the test.

Type II error

A Type II error occurs when a false null hypothesis is not rejected. The value represents the probability of committing a Type II error,

that is

The value is called the power of the test. It represents the probability of not making a Type II error.