Section 7 1 Introduction to Hypothesis Testing LarsonFarber

Section 7. 1 Introduction to Hypothesis Testing Larson/Farber 4 th ed. 1

Section 7. 1 Objectives • State a null hypothesis and an alternative hypothesis • Determine whether to use a one-tailed or two-tailed statistical test and find a p-value • Make and interpret a decision based on the results of a statistical test • Write a claim for a hypothesis test Larson/Farber 4 th ed. 2

Hypothesis Tests Hypothesis test • A process that uses sample statistics to test a claim about the value of a population parameter. • For example: An automobile manufacturer advertises that its new hybrid car has a mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the sample mean differs enough from the advertised mean, you can decide the advertisement is wrong. Larson/Farber 4 th ed. 3

Hypothesis Tests Statistical hypothesis • A statement, or claim, about a population parameter. • Need a pair of hypotheses • one that represents the claim • the other, its complement • When one of these hypotheses is false, the other must be true. Larson/Farber 4 th ed. 4

Stating a Hypothesis Null hypothesis • A statistical hypothesis that contains a statement of equality such as , =, or . • Denoted H 0 read “H subzero” or “H naught. ” Alternative hypothesis • A statement of inequality such as >, , or <. • Must be true if H 0 is false. • Denoted Ha read “H sub -a. ” complementary statements Larson/Farber 4 th ed. 5

Stating a Hypothesis • To write the null and alternative hypotheses, translate the claim made about the population parameter from a verbal statement to a mathematical statement. • Then write its complement. H 0: μ ≤ k Ha : μ > k H 0: μ ≥ k Ha : μ < k H 0: μ = k Ha : μ ≠ k • Regardless of which pair of hypotheses you use, you always assume μ = k and examine the sampling distribution on the basis of this assumption. Larson/Farber 4 th ed. 6

Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. 1. A university publicizes that the proportion of its students who graduate in 4 years is 82%. Solution: H 0: p = 0. 82 Equality condition (Claim) Ha: p ≠ 0. 82 Complement of H 0 Larson/Farber 4 th ed. 7

Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. 2. A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2. 5 gallons per minute. Solution: H 0: μ ≥ 2. 5 gallons per minute Ha: μ < 2. 5 gallons per minute Larson/Farber 4 th ed. Complement of Ha Inequality (Claim) condition 8

Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. 3. A cereal company advertises that the mean weight of the contents of its 20 -ounce size cereal boxes is more than 20 ounces. Solution: H 0: μ ≤ 20 ounces Ha: μ > 20 ounces Larson/Farber 4 th ed. Complement of Ha Inequality (Claim) condition 9

• No matter which hypothesis represents the claim, always begin the hypothesis test assuming that the equality condition in the null hypothesis is true. • At the end of the test, one of two decisions will be made: § reject the null hypothesis § fail to reject the null hypothesis Larson/Farber 4 th ed. 10

Level of Significance Level of significance § Denoted by , the lowercase Greek letter alpha. • By setting the level of significance at a small value, you are saying that you want the probability of rejecting a true null hypothesis to be small. • Commonly used levels of significance: § = 0. 10 = 0. 05 = 0. 01 Larson/Farber 4 th ed. 11

Statistical Tests • After stating the null and alternative hypotheses and specifying the level of significance, a random sample is taken from the population and sample statistics are calculated. • The statistic that is compared with the population parameter in the null hypothesis is called the test statistic. Population parameter Test statistic μ p σ2 Larson/Farber 4 th ed. s 2 Standardized test statistic z (Section 7. 2 n 30) t (Section 7. 3 n < 30) z (Section 7. 4) χ2 (Section 7. 5) 12

P-values P-value (or probability value) • The probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. • Depends on the nature of the test. You will understand this better on examples for 7. 2 Larson/Farber 4 th ed. 13

Nature of the Test • Three types of hypothesis tests § left-tailed test § right-tailed test § two-tailed test • The type of test depends on the region of the sampling distribution that favors a rejection of H 0. • This region is indicated by the alternative hypothesis. Larson/Farber 4 th ed. 14

Left-tailed Test • The alternative hypothesis Ha contains the less-than inequality symbol (<). H 0: μ k Ha : μ < k P is the area to the left of the test statistic. z -3 -2 -1 0 1 2 3 Test statistic Larson/Farber 4 th ed. 15

Right-tailed Test • The alternative hypothesis Ha contains the greaterthan inequality symbol (>). H 0: μ ≤ k P is the area Ha : μ > k to the right of the test statistic. z -3 -2 -1 0 1 2 3 Test statistic Larson/Farber 4 th ed. 16

Two-tailed Test • The alternative hypothesis Ha contains the not equal inequality symbol (≠). Each tail has an area of ½P. H 0: μ = k Ha : μ k P is twice the area to the right of the positive test statistic. P is twice the area to the left of the negative test statistic. z -3 Larson/Farber 4 th ed. -2 -1 Test statistic 0 1 2 Test statistic 3 17

Example: Identifying The Nature of a Test For each claim, state H 0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. 1. A university publicizes that the proportion of its students who graduate in 4 years is 82%. Solution: H 0: p = 0. 82 ½ P-value Ha: p ≠ 0. 82 area Two-tailed test Larson/Farber 4 th ed. -z 0 z z 18

Example: Identifying The Nature of a Test For each claim, state H 0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. 2. A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2. 5 gallons per minute. Solution: P-value H 0: μ ≥ 2. 5 gpm area μ < 2. 5 gpm Ha : Left-tailed test Larson/Farber 4 th ed. -z 0 z 19

Example: Identifying The Nature of a Test For each claim, state H 0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. 3. A cereal company advertises that the mean weight of the contents of its 20 -ounce size cereal boxes is more than 20 ounces. Solution: P-value H 0: μ ≤ 20 oz area Ha: μ > 20 oz Right-tailed test Larson/Farber 4 th ed. 0 z z 20

Making a Decision Rule Based on P-value • Compare the P-value with . § If P , then reject H 0. § If P > , then fail to reject H 0. Important for conclusions!! Decision Claim is H 0 Claim is Ha Reject H 0 There is enough evidence to reject the claim There is enough evidence to support the claim Fail to reject H 0 There is not enough evidence to reject the claim There is not enough evidence to support the claim Larson/Farber 4 th ed. 21

Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H 0? If you fail to reject H 0? 1. H 0 (Claim): A university publicizes that the proportion of its students who graduate in 4 years is 82%. Please, refer to the table in the previous slide to answer these examples (print it if possible) Larson/Farber 4 th ed. 22

Solution: Interpreting a Decision • The claim is represented by H 0. • If you reject H 0 you should conclude “there is sufficient evidence to indicate that the university’s claim is false. ” • If you fail to reject H 0, you should conclude “there is insufficient evidence to indicate that the university’s claim (of a four-year graduation rate of 82%) is false. ” Larson/Farber 4 th ed. 23

Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H 0? If you fail to reject H 0? 2. Ha (Claim): Consumer Reports states that the mean stopping distance (on a dry surface) for a Honda Civic is less than 136 feet. Solution: • The claim is represented by Ha. • H 0 is “the mean stopping distance…is greater than or equal to 136 feet. ” Larson/Farber 4 th ed. 24

Solution: Interpreting a Decision • If you reject H 0 you should conclude “there is enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet. ” • If you fail to reject H 0, you should conclude “there is not enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet. ” Larson/Farber 4 th ed. 25

Steps for Hypothesis Testing 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. H 0: ? Ha : ? 2. Specify the level of significance. This sampling distribution is based on the assumption α= ? that H 0 is true. 3. Determine the standardized sampling distribution and draw its graph. z 0 4. Calculate the test statistic and its standardized value. Add it to your sketch. z 0 Test statistic Larson/Farber 4 th ed. 26

Steps for Hypothesis Testing 5. Find the P-value. 6. Use the following decision rule. Is the P-value less than or equal to the level of significance? No Fail to reject H 0. Yes Reject H 0. 7. Write a statement to interpret the decision in the context of the original claim. Larson/Farber 4 th ed. 27