Introduction to Hypothesis Testing Section 7.1 Introduction to Hypothesis Testing

A statistical hypothesis is a claim about a population. Null hypothesis H0 contains a statement of equality such as ³ , = or £. Alternative hypothesis Ha contains a statement of inequality such as < , ¹ or > Complementary Statements The null hypothesis says things are fine. No problem. In a hypothesis test, you will always assume this is the case (null hypothesis is true). If the evidence shows the null hypothesis is improbable, then you will reject it in support of the alternative hypothesis. You can never actually prove the null hypothesis is true. If I am false, you are true If I am false, you are true

Writing Hypotheses Write the claim about the population. Then, write its complement. Either hypothesis, the null or the alternative, can represent the claim. A hospital claims its ambulance response time is less than 10 minutes. claim A consumer magazine claims the proportion of cell phone calls made during evenings and weekends is at most 60%. Always write the claim first. Then write the complement. Identify the null hypothesis as the statement that contains the = condition. The claim can be represented in either hypothesis. claim

Hypothesis Test Strategy Begin by assuming the equality condition in the null hypothesis is true. This is regardless of whether the claim is represented by the null hypothesis or by the alternative hypothesis. Collect data from a random sample taken from the population and calculate the necessary sample statistics. If the sample statistic has a low probability of being drawn from a population in which the null hypothesis is true, you will reject H0. (As a consequence, you will support the alternative hypothesis.) If the probability is not low enough, fail to reject H0.

Errors and Level of Significance Actual Truth of H0 H0 True H0 False Decision Do not reject H0 Correct Decision Type II Error Type I Error Reject H0 A type I error: Null hypothesis is actually true but the decision is to reject it. The level of significance should be a low probability value. It is usually 0.01, 0.05 or 0.10. A type II error is made when the null hypothesis is false but you fail to reject it. The probability of a type II error is β. 1- β is called the power of the test. Discuss the analogy to the American justice system. Level of significance, Maximum probability of committing a type I error.

Types of Hypothesis Tests Right-tail test Ha is more probable Left-tail test Ha is more probable In the examples, the parameter is mu and the sampling distribution is normal. When testing other parameters, the same 3 types of tests are used. The category for the test is the region in which the null hypothesis is likely to be rejected and the alternative hypothesis is more probable. Ha is more probable Two-tail test

P-values The P-value is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined by the sample data. P-value = indicated area Area in left tail Area in right tail z z For a right tail test For a left tail test Students will encounter results expressed in P-values in professional journals. TI-83 and Minitab results for hypothesis tests are given in P-value form. If z is negative, twice the area in the left tail If z is positive, twice the area in the right tail z z For a two-tail test

Finding P-values: 1-tail Test The test statistic for a right-tail test is z = 1.56. Find the P-value. Area in right tail z = 1.56 The example here is for a right tail test. For a left tail test, calculate the area in the left tail from the cumulative area in the table. The area to the right of z = 1.56 is 1 – .9406 = 0.0594. The P-value is 0.0594.

Finding P-values: 2-tail Test The test statistic for a two-tail test is z = –2.63. Find the corresponding P-value. z = –2.63 For a two tail test, find the standardized test statistic. Then find the area in the tail of the statistic and then double the area. The area to the left of z = –2.63 is 0.0043. The P-value is 2(0.0043) = 0.0086.

Test Decisions with P-values The decision about whether there is enough evidence to reject the null hypothesis can be made by comparing the P-value to the value of , the level of significance of the test. If reject the null hypothesis. In a hypothesis test, the equality condition (contained in the null hypothesis) is assumed to be true. If the P value is less than the level of significance then the probability of obtaining a statistic that is equal to the one in the test or even more extreme is lower than the level of significance. If fail to reject the null hypothesis.

Using P-values The P-value of a hypothesis test is 0.0749. Make your decision at the 0.05 level of significance. Compare the P-value to . Since 0.0749 > 0.05, fail to reject H0. If P = 0.0246, what is your decision if The second example shows why many statisticians use a P-value to report the results of their hypothesis test. 1) Since , reject H0. 2) Since 0.0246 > 0.01, fail to reject H0.

Interpreting the Decision Claim Claim is H0 Claim is Ha There is enough evidence to support the claim. There is enough evidence to reject the claim. Reject H0 Decision You can never prove a claim that is represented by the null hypothesis. You can only say there is not enough evidence to reject it. Use the analogy of a court case. The null hypothesis says the defendant is innocent. An attorney doesn’t need to prove a client’s innocence. It is sufficient to say there is not enough evidence to find the defendant guilty. When you wish to prove a claim, set it up as the alternative hypothesis. If an attorney wants to claim the defendant is guilty, the attorney must show that assuming the defendant innocent, the resulting conditions would be extremely improbable. There is not enough evidence to support the claim. There is not enough evidence to reject the claim. Fail to reject H0

Steps in a Hypothesis Test 1. Write the null and alternative hypothesis. Write H0 and Ha as mathematical statements. Remember H0 always contains the = symbol. 2. State the level of significance. This is the maximum probability of rejecting the null hypothesis when it is actually true. (Making a type I error.) These steps remain the same for all hypothesis tests. Discuss the normal sampling distribution that is the one for the sample mean with a large sample. 3. Identify the sampling distribution. The sampling distribution is the distribution for the test statistic assuming that the equality condition in H0 is true and that the experiment is repeated an infinite number of times.

Perform the calculations to standardize your sample statistic. 4. Find the test statistic and standardize it. Perform the calculations to standardize your sample statistic. 5. Calculate the P-value for the test statistic. The test statistic for the population mean is the sample mean. You will standardize each test statistic to determine probabilities. For a normal distribution, use z-scores. There will be other standardized scores. This is the probability of obtaining your test statistic or one that is more extreme from the sampling distribution.

If the P-value is less than (the level of significance) reject H0. 6. Make your decision. If the P-value is less than (the level of significance) reject H0. If the P value is greater , fail to reject H0. 7. Interpret your decision. If the claim is the null hypothesis, you will either reject the claim or determine there is not enough evidence to reject the claim. If the claim is the alternative hypothesis, you will either support the claim or determine there is not enough evidence to support the claim.