HU Statistics Ch.8 (MATH009)

Hypothesis testing situations have 4 possibilities:

- Ho may or may not be true - a decision is made to reject or not to reject it is based on the data from the sample. -Type I error occurs if you reject the null hypothesis when it is true. -Type II error occurs if you do not reject the null hypothesis when it is false.

The hypothesis testing situations may be likened to a jury trial: 4 possibilities:

- Ho: The defendant is innocent - H1: the defendant is not innocent (guilty) - The defendant may be convicted (acquitted).

Statisticians use 3 arbitrary significance levels:

0.10, 0.05 and 0.01 If the null hypothesis is rejected P(type I error) =10%, 5% or 1% depending in which significance level is used. Alpha = 0.10 implies a 10% chance of rejecting a true null hypothesis.

How do we summarize the results: Outcomes of the hypothesis testing situation

1. Claim is Ho a. Reject Ho- There is enough evidence to reject the claim *Decision when Ho is the claim and Ho is rejected* b. Do not reject Ho- There is not enough evidence to reject the claim 2. Claim is H1 a. Reject Ho- There is enough evidence to support the claim b. Do not reject Ho- There is not enough evidence to support the claim *Decision when H1 is the claim and Ho is not rejected)*

Two types of statistical hypothesis are

1. Null hypothesis, Ho, - there is no difference between a parameter and a specific value. 2. Alternative hypothesis - H1, the existence of a difference between a parameter and a specific value.

Assumptions about Z-tests for a mean when sigma is known:

1. Random sample 2. Either n is greater than or equal to 30 or when the population Is normally distributed if n<30

Assumptions about test for a mean when sigma is unknown:

1. Random sample 2. Either n is greater than or equal to 30 or when the population Is normally distributed if n<30

Solving the hypothesis testing Problems (Traditional Method) steps:

1. State the hypothesis and identify the claim 2. Find the critical value(s) from the appropriate table in appendix C 3. Compute the test value. 4. Make a decision to reject or not to reject the null hypothesis 5. Summarize your results.

Solving hypothesis -testing problems (p-value method) steps:

1. State the hypothesis and identify the claim 2. Compute the test value 3. Find the p-value 4. Make a decision 5. Summarize the results

Steps for solving the hypothesis testing problem:

1. State the hypothesis and identify the claim 2. Find the critical value 3. Compare the test value (z-value) 4. Make a decision to reject or not to reject the null hypothesis 5. Summarize the results

Three methods for testing hypothesis are:

1. The traditional method 2. The P value method 3. The CI method

Statistical Hypothesis may be stated in 3 different ways:

1. Two tailed test: Ho: μ=K and H1: μ≠K 2. Right tailed test: Ho: μ=K and H1: μ>K 3. Left tailed test H0: Ho: μ=K and H1: μ<K

Two tests used for hypothesis concerning means are:

1. Z-test 2. T-test

The traditional method for testing hypothesis

A statistical hypothesis is a conjecture about a population parameter. It may or may not be true.

The mean is computed for sample data and compared to the population mean for what?

For the statistical test

What is hypothesis testing?

It is a decision-making process for evaluating claims about the population

For Z-tests for a mean when sigma is known,

Many hypotheses are tested using a statistical test based on test value where test value=(observed value -expected)/standard error

P-value methods for hypothesis testing:

The P-value (probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. It is the actual area under the standard normal distribution curve

What does it mean when there is a large difference between the sample mean and the hypothesized mean?

The null hypothesis is probably not true

Test value

The numerical value obtained from the statistical test

Non-critical or non-rejection region

The range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should be rejected

Test for a mean when sigma is unknown,

The t-test is a statistical test for the mean of the population and is used when the population is normally distributed or approximately normal and sigma is unknown

How large does the difference need to be to reject the null hypothesis?

We use the level of significance. This is the maximum probability of computing a type I error. The probability is symbolized by alpha. The probability of a type II error is beta. Alpha and beta are related in that they are inversely proportional.

no significant difference

do not reject null hypothesis.

One tailed test indicates what?

indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. It is either (right tailed or left tailed) depending on the direction of the inequality of the alternative hypothesis.

Significant difference

reject null hypothesis

Critical Value

separates critical region and noncritical region

Formula for test for a mean when sigma is unknown

t= (x̄-μ)/(s/√n) d.f=n-1

In a two-tailed test,

the null hypothesis should be rejected when the test is in either of the two critical regions

Critical or rejection region

the range of values of the test value that indicate there is a significant difference and the null hypothesis should be rejected

The decision is made to reject or not to reject the null hypothesis based on what?

the value obtained from the statistical test

After a significant level is chosen a critical value is selected from a table for the appropriate test. If the z-test is used, then what?

the z-table is used for the critical value. The critical value determines the critical and non-critical region.

A statistical test

uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.

Formula for Z-tests for a mean when sigma is known

z= (x̄-μ)/(σ/√n) where x̄=sample mean μ=hypothesized mean σ=population standard deviation n=sample size

P(type I error) =

α

P(type II error)=

β