Chapter 10, "Hypothesis Tests Regarding a Parameter,"

Statistical Hypothesis

A verbal statement, or claim, about a population parameter.

2 important points to remember about hypotheses:

First, hypotheses must be written prior to collecting any data in the experiment. A researcher should be well-versed in the field of study, and should be knowledgeable enough to make predictions about what will happen in the experiment. The second important point involves interpretation of our hypothesis testing. In science, we can never prove anything, we can only disprove it. Therefore, we can never prove that our research hypothesis is true, we can only disprove the null hypothesis.

Type I Error

Occurs if the null hypothesis is rejected when it is true. - In other words, we make a Type I error if we indicate that a significant difference exists between two population means when such a difference does not actually exist. If we commit a Type I error, we would say that significant differences exist when they really do not. We rejected the null hypothesis when we really should have failed to reject it.

When you fail to reject the null hypothesis, group differences are due to ________.

chance

Hypothesis test

involves gathering data from a sample population to test the statistical hypothesis, the claim being made about a population.

To test the statistical hypothesis, the researcher needs two hypotheses—

one that states the claim is true (the research, or alternate, hypothesis), and another that states the claim is false (the null hypothesis).

The hypothesis, "The consumption of sugar makes children more hyperactive for a short period of time," is ________.

one-tailed

A statement or claim about a population parameter *A statistical hypothesis* A claim that any differences observed between the groups are due to chance, not actual differences *A null hypothesis* A statement of what the researcher wants or expects to see happen in the research study *An alternate hypothesis*

...

A statistical hypothesis makes a claim about a population parameter.

...

Hypotheses must be written prior to collecting any data in the experiment.

...

Reject the Null Hypothesis - Type I Error - t subscript a b t end subscript space greater or equal than space t subscript c r i t end subscript Fail to Reject the Null Hypthesis - t subscript a b t end subscript space less or equal than space t subscript c r i t end subscript - Type II Error

...

The t-table is split into categories for one-tailed and two-tailed tests.

...

A one sample t-test is used to test differences between means.

....

One-Sample t- Test

A one-sample t-test is used to test mean differences between a sample and the population from which it was chosen. The use of this inferential test of the hypothesis requires that we know both the sample mean and the population mean. We also need to have either the sample standard deviation or the population standard deviation. The formula used to calculate a one-sample t is t space equals space fraction numerator X with bar on top space minus space mu over denominator S subscript x with bar on top end subscript end fraction w h e r e S subscript x with bar on top end subscript space equals space fraction numerator S over denominator square root of n end fraction The denominator should look familiar to you. It's the estimated standard error of the mean. The numerator is the difference between the two means.

Hypothesis Test

A process that uses sample statistics to test a claim about the value of a population parameter. -In real-life situations, the values of population parameters are often impossible to obtain. For example, when businesses or politicians make a certain claim or when a pharmaceutical company declares that the new allergy drug can last for 36 hours, their claims need to be verified. To test these claims, we use the statistical technique called hypothesis testing. A hypothesis test is a process that uses sample statistics to test a claim about the value of a population parameter. For instance, an allergy medicine claims that its effect lasts longer than the effects of the other brands available on the market. Conducting a hypothesis test for the medicine will involve testing the claim with a sample of medicine users.

Null Hypothesis and Alternate Hypothesis

A statement or claim about a population parameter is called a statistical hypothesis. For example, the statement "The consumption of alcohol will increase the amount of time it takes laboratory rats to complete a task" is an example of a statistical hypothesis. To test a hypothesis, you need to present a pair of hypotheses - one that supports the claim as true and the other that supports the claim as false. The pair of hypotheses consists of the following: the null hypothesis and the research or alternate hypothesis. Null hypotheses are commonly notated as "begin mathsize 11px style H subscript 0 end style", alternate hypotheses as "begin mathsize 11px style H subscript 1 end style".

Alternate Hypothesis

Complement of the null hypothesis. It is a statement of inequality, such as >, ?, or <, that must be true if Ho is false.

Review

Hypothesis testing is a test of the null hypothesis. If enough evidence exists to reject the null hypothesis, then the conclusion is that any changes are the effect of the hypothesis and are not due to chance. The research, or alternate, hypothesis can be one-tailed or two-tailed. A hypothesis that indicates one mean will be greater than or less than the other mean is a one-tailed hypothesis. It will contain a greater than (>) or a less than (<) symbol. The following is an example of a one-tailed research hypothesis. Dogs treated with the preventative will have fewer ticks than dogs left untreated. H subscript 1 colon space X with bar on top subscript 1 space less than space X with bar on top subscript 2 Specifying just that there will be a change or that the means will be different is a two-tailed research hypothesis. The following is an example of a two-tailed research hypothesis. The flea-and-tick preventive will affect the numbers of ticks on dogs. H subscript 1 colon space X with bar on top subscript 1 space not equal to space X with bar on top subscript 2

Type I and Type II Errors

Hypothesis testing is done on a sample and not on the entire population; therefore, there is always a possibility of making an error in decision-making. These errors are categorized as Type I or Type II errors.

Summary 2

Inferential statistics allows you to make inferences about a population based on sample statistics. You can use hypothesis testing to determine if a claim can be reasonably made about the population. Since you cannot prove a hypothesis is true, testing is done to reject or to fail to reject the null hypothesis. You must write all hypotheses before any data are collected so that you do not influence any of the hypotheses. Hypotheses are predictions about what will happen in an experiment and should be based on prior knowledge about the field of study.

Type II Error

Occurs if the null hypothesis is not rejected when it is false. - Remember that the null hypothesis says that there are no differences between the group means. If we commit a Type II error, we would say that no significant differences exist when they really do. We failed to reject the null hypothesis when we really should have rejected it.

Intro

Once a statistical hypothesis and the corresponding null and alternate hypotheses are stated, the next step is to test the null hypothesis. Since you are making inferences about a population based on data from a sample population, the research hypothesis cannot actually be proved. There would be no way to test every dog to see if dogs treated with the flea-and-tick preventative have fewer ticks than dogs that are left untreated. Therefore, testing focuses on the null hypothesis. Only two outcomes are possible: -Reject the null hypothesis -Fail to reject the null hypothesis Rejecting the null hypothesis is the desired outcome. Rejection indicates that any changes are the effect of the hypothesis. Failure to reject the null hypothesis implies that any differences are due to chance and not to the experiment.

The t-Table

Once we calculate t (often abbreviated as begin mathsize 11px style t subscript a b t end subscript end style), we need to determine our critical value for t (abbreviated tcrit) in order to make a decision about the null hypothesis. We will use the Table of t-distributions that we used when constructing confidence intervals. This time, however, we must use the appropriate t Table: the one-tailed table for one-tailed hypotheses or the two-tailed table for two-tailed hypotheses.

Outcomes of Hypothesis Testing

Remember that inferential statistical tests use sample statistics to make inferences about population parameters. In other words, we are using information gathered from the sample to answer some question about the population, and we are using hypothesis testing to do so. Hypothesis testing is actually a test of the null hypothesis, and only allows for two possible outcomes: -Reject the null hypothesis -Fail to reject the null hypothesis When we reject the null hypothesis, we are indicating that our groups are different for reasons that have to do with our experiment and not just by chance. Failing to reject the null hypothesis indicates that any group differences we see are due to chance and have nothing to do with our experimental conditions.

Inferential Statistical Tests

Remember that inferential statistical tests use sample statistics to make inferences about population parameters. In other words, we are using information gathered from the sample to answer some question about the population, and we do so with hypothesis testing. There are several inferential tests that we can use for hypothesis testing. Which one we use depends on the specific hypotheses we are attempting to test. In other words, what is the research question that we wish to answer? The hypotheses of a research study are derived from the research question, and they give us the information we need to determine which statistical test to use. We will cover several inferential statistical methods used for hypothesis testing throughout this statistics course. We will begin with an inferential method used for hypothesis testing with one sample, a one sample t-test.

Null Hypothesis

Statistical hypothesis that states that there is no difference between the parameters of two populations.

One-Tailed Test

The alternate hypothesis contains a greater-than, >, or less-than, <, symbol. - Research hypotheses can be one-tailed or two-tailed. When we specify the direction of the difference between two groups, the hypothesis is one-tailed. Our laboratory rat example is a one-tailed hypothesis: "The consumption of alcohol will increase the amount of time it takes laboratory rats to complete a task." We made a specific prediction that alcohol would increase the amount of time for the task, so our research hypothesis written in appropriate notation would be H subscript 1 colon space mu subscript 1 space less than space mu subscript 2 This notation indicates that the time collected from the first group, that is, the rats before they consumed alcohol, would be smaller than the time collected from the second group, or the rats after they consumed alcohol. Because we used the symbol, <, we specified the direction that our second group would change. We predicted that the second group would have a greater time. This would be an example of a one-tailed test of the hypothesis.

Two-Tailed Test

The alternative hypothesis contains the not-equal-to symbol, ?. -If we had worded our research question differently, our research hypothesis could be two-tailed. Assume that the research hypothesis is "The consumption of alcohol will affect the amount of time it takes laboratory rats to complete a task." Using notation, our research hypothesis would be H subscript 1 colon space mu subscript 1 space not equal to space mu subscript 2 The symbol "does not equal" in this case means that the time for group one will be different from the time for group two. "Different" in this case means "statistically significantly different," which will be explained very soon in this lesson. The researcher predicts that the two groups will differ in some way, but no prediction is made as to which group will have the greater time. This is an example of a two-tailed test of the hypothesis. Difference is predicted, but the direction of the change is not specified.

The Null Hypothesis, H0

The null hypothesis is a claim that any differences observed between the groups are due to chance, not actual differences. The null hypothesis indicates that one group mean is equal to the other group mean, and is often written using the following notations: H subscript 0 colon space mu subscript 1 space equals space mu subscript 2 Using our laboratory example mentioned earlier, the null hypothesis would be "The consumption of alcohol has no affect on the amount of time it takes laboratory rats to complete a task." What this null hypothesis would mean is that the rats' time to complete a task would not change at all once they had consumed the alcohol.

Review

The pair of hypotheses needed to test a statistical hypothesis is the null hypothesis (H0) and the research, or alternate, hypothesis (H1). The null hypothesis states that any observed differences between the groups are due to chance and that one group mean is equal to the other group mean. H subscript 0 colon space X with bar on top subscript 1 space equals space X with bar on top subscript 2 The research, or alternate, hypothesis states what the researcher expects to happen. This hypothesis has three options: H subscript 1 colon space X with bar on top subscript 1 space less than space X with bar on top subscript 2 The mean of one group is less than the mean of the other group. H subscript 1 colon space X with bar on top subscript 1 space greater than space X with bar on top subscript 2 The mean of one group is greater than the mean of the other group. H subscript 1 colon space X with bar on top subscript 1 space not equal to space X with bar on top subscript 2 The two means are not equal. For the statistical hypothesis that a popular flea-and-tick preventative kills and prevents ticks, the researcher could use the following hypotheses to test the claim. Null hypothesis: Dogs left untreated will have the same number of ticks as dogs that receive the preventative. Research hypothesis: Dogs treated with the preventative will have fewer ticks than dogs left untreated.

The Research or Alternate Hypothesis, H1

The research or alternate hypothesis is a complement of the null hypothesis. It is a statement of what the researcher wants or expects to see happen in the research study. For the laboratory example mentioned earlier, our research hypothesis might be, "The consumption of alcohol will increase the amount of time it takes laboratory rats to complete a task." In other words, the researcher expects the alcohol to slow down the rats' work, thus, increasing the amount of time they need to complete their task. The researcher would base the research hypothesis on other studies of the effects of alcohol. Thus, the research hypothesis is an educated guess about what might happen in our study. There are three options for writing research hypotheses using the appropriate notation: begin mathsize 11px style H subscript 1 colon space mu subscript 1 space less than space mu subscript 2 space space space space space space o r H subscript 1 colon space mu subscript 1 space greater than space mu subscript 2 space space space space space space o r H subscript 1 colon space mu subscript 1 space not equal to space mu subscript 2 end style In other words, the researcher can predict that the mean for group one will be less than the mean for group two, the mean for group one will be greater than the mean for group two, or that the two means will be different from one another.

Critical Value for t

To use the table, we must know our degrees of freedom and our desired level of significance. The degrees of freedom is our sample size minus one (n — 1). The level of significance is set prior to collecting data, and is most often .05. We move down the left column of the appropriate t-table to the value closest to the degrees of freedom without going over, and then move across the columns to the value that is under the .05 level of significance. The value for t that we obtain from the table is our critical value for t (begin mathsize 11px style t subscript c r i t end subscript end style). We use the critical value for t to determine if we can reject our null hypothesis or if we fail to reject our null hypothesis. We use the following rule: If t subscript a b t subscript blank end subscript space greater or equal than space t subscript c r i t end subscript, we reject the null hypothesis; our group means ARE significantly different from one another If we obtain a t that is less than the critical value for t, then we fail to reject the null hypothesis. In that situation, we do not have significant mean differences. Note: When comparing tobt to tam we only consider the absolute value of tobt, that is, the positive value of tobt.

Disproving the Null Hypothesis

Why is this the case? Perhaps we had 1,000 incidences in which we could show that our research hypothesis was true. We had 1,000 different samples that all turned out the same way: the laboratory rats took longer to do their task once they had consumed the alcohol. Our research hypothesis must be true, right? We just proved that alcohol slows down the rats, right? Wrong!! Yes, we found this to be true for 1,000 samples, but it still doesn't mean that it is true all the time. All we need is one sample to prove us wrong - one sample in which the rats perform at the same rate or even slightly faster after consuming alcohol. Therefore, in science and research, we set out to disprove the null hypothesis, because disproving that lends support to our research hypothesis. We still can't say that we proved anything!! Rejecting the null hypothesis does not prove our research hypothesis. And failing to reject our null hypothesis does not prove our null hypothesis to be true. Instead, we are essentially finding the exception to the rule, the time when the null hypothesis does not hold true, which lends empirical support to our research hypothesis.

Review 2

Working with inferential statistics always includes a chance for errors in deciding whether to reject the null hypothesis or fail to reject it. Errors can be classified as Type I or Type II. Type I errors occur when you reject the null hypothesis when you should have failed to reject it, meaning the null hypothesis is actually true. Type II errors occur when you fail to reject the null hypothesis when it really should be rejected, meaning the null hypothesis is actually false. Because you are using statistics gathered from a sample of the population, hypothesis testing is used to answer questions about the population. Several inferential tests are used for hypothesis testing. For hypothesis testing with one sample, you can use the one sample t-test. To use this method, you need to know both sample and population means and either the sample or population standard deviation. t space equals space fraction numerator X with bar on top space minus space mu over denominator S subscript x with bar on top end subscript end fraction space where space S subscript x with bar on top end subscript space equals space fraction numerator S over denominator square root of n end fraction Once you find the value of t (tobt), you need to find the critical value for t (tcrit) using the table of t-distributions. Use the one-tailed table for one-tailed hypotheses. Use the two-tailed table for two-tailed hypotheses. If tobt ≥ tcrit, reject the null hypothesis; the group means are significantly different. If t subscript o b t space end subscript less than space t subscript c r i t times end subscript you fail to reject the null hypothesis; the group means are not significantly different.

When you reject the null hypothesis, group differences are due to ________.

conditions of the experiment

Hypothesis testing

is used to determine if certain claims can be made about the population in general.

Inferential statistics

rely on data collected from a sample of a larger population. -When working with inferential statistics, errors are always possible. Minimizing the chance for error will result in more reliable conclusions about the population. - involves gathering data from a sample group that is part of a larger population. The data are then used to draw conclusions about the population. Answers to specific questions about a population rely on hypothesis testing where an inferential test is conducted to determine if you should reject the null hypothesis or fail to reject the null hypothesis. -Note that anytime sample statistics are used to make inferences about a population, errors are possible.

The hypothesis, "Walking a dog everyday will change its expected lifespan," is ________.

two-tailed