Hypothesis testingChapter 9

Introduction to Statistical Tests

Stating Hypotheses

• Null hypothesis H0: This is the statement that is under investigation or being tested. Usually the null hypothesis represents a statement of “no effect”, “no difference”, or, put another way, “things haven’t changed”.• Alternate hypothesis H1: This is the statement you will adopt in the

situation in which the evidence (data) is so strong tat you reject H0. A statistical test is designed to assess the strength of the evidence (data) against the null hypothesis.• Null hypotheses are always of the form H0: some parameter = some

value.

Types of Tests

A statistical test is:

left-tailed if H1 states hat the parameter is less than the cited value claimed in H0.

right-tailed if H1 states hat the parameter is greater than the value claimed in H0.

two-tailed if H1 states that the parameter is different from the value claimed in H0.

Hypothesis Tests of µ, Given that x is Normal and σ is Known• Requirements• The x distribution is normal with known standard deviation σ. Then

has a normal distribution. The standard test statistic is where = mean of a random sample, σ = value stated in H0, and n = sample size.

The P-value of a Statistical Test

• P-value• Assuming H0 is true, the probability that the test statistic will take on

values as extreme as or more extreme than the observed test statistic is called the P-value of the test. The smaller the P-value computed from sample data, the stronger the evidence against H0.• See pages 393, 394

Types of Errors

• A type I error occurs if H0 is true but we reject H0

• A type II error occurs if H0 is false but we do not reject H0

• The level of significance α is the probability of rejecting H0 when it is true. This is the probability of a type I error.• The probability of making a type II error is denoted by the Greek letter

β• The quantity 1-β is called the power of a test and represents the

probability of rejecting H0 when it is false• See table 9-3

Concluding a Statistical Test

• If P-value ≤ α, we reject the null hypothesis and say the data are statistically significant at the level α• If P-value > α, we do not reject the null hypothesis

Testing the Mean µ

Testing µ when σ is Known

• Requirements• Let x be a random variable appropriate to your application. Obtain a

simple random sample (of size n) of x values from which you compute the sample mean . The value of σ is already known. If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size

Testing µ when σ is Known

• Procedure1. In the context of the application, state the null and alternate hypotheses

and set the level of significance α2. Use the known σ, sample size n, the value of from the sample, and from

the null hypothesis to compute the standardized sample test statistic 3. Use the standard normal distribution and the type of test, one-tailed or

two-tailed, to find the P-value corresponding to the test statistic

4. Conclude the test. If , then reject H0. If P-value > α, then do not reject H0

5. Interpret your conclusion in the context of the application

Testing µ when σ is Unknown

• Requirements• Let x be a random variable appropriate to your application. Obtain a

simple random sample (of size n) of x values from which you compute the sample mean and the sample standard deviation s. If you can assume that x has a normal distribution or simply a mound-shaped and symmetrical distribution, then any sample size n will work. If you cannot assume this, then use a sample size

Testing µ when σ is Unknown

• Procedure1. In the context of the application, state the null and alternate

hypotheses and set the level of significance α

2. Use , s, and n from the sample, with from the H0 to compute the sample test statistic with degrees of freedom d.f. = n - 1

3. Use Student’s t distribution and the type of test, one-tailed or two-tailed, to find the P-value corresponding to the test statistic

4. Conclude the test. If , then reject H0. If P-value > α, then do not reject H0

5. Interpret your conclusion in the context of the application

Testing µ Using Critical Regions(Traditional Method)• Requirements• Let x be a random variable appropriate to your application. Obtain a

simple random sample (of size n) of x values from which you compute the sample mean . The value of σ is already known. If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size . Then follows a distribution that is normal or approximately normal

Testing µ Using Critical Regions(Traditional Method)• Procedure1. In the context of the application, state the null and alternate hypotheses and set

the level of significance α. We use the most popular choices, or 2. Use the known σ, the sample size n, the value of from the sample, and from the

null hypothesis to compute the standardized sample test statistic 3. Show the critical region and critical value(s) on a graph of the sampling

distribution. The level of significance α and the alternate hypothesis determine the locations of critical regions and critical values

4. Conclude the test. If the test statistic z computed in step 2 is in the critical region, then reject H0. If the test statistic z is not in the critical region, then do not reject H0

5. Interpret your conclusion in the context of the application

Testing a Proportion p

How to test a Proportion p

• Requirements• Consider a binomial distribution experiment with n trials, where p

represents the population probability of success and q = 1 – p represents the population probability of failure. Let r be a random variable that represents the number of successes out of the n binomial trials. The number of trials n should be sufficiently large so that both np > 5 and nq > 5. In this case, can be approximated b the normal distribution.

How to test a Proportion p

• Procedure1. In the context of the application, state the null and alternate

hypotheses and set the level of significance α.2. Compute the standardized sample test statistic where p is the value

specified in H0 and q = 1 - p3. Use the standard normal distribution and the type of test, one-tailed or

two-tailed, to find the P-value corresponding to the test statistic

4. Conclude the test. If , then reject H0. If P-value > α, then do not reject H0

5. Interpret your conclusion in the context of the application