Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Symbols & graphs quiz • Look through book and be ready to ask questions. Page 367 1-36 Types of Errors • A type I error reject 𝐻0 , and 𝐻0 is true. • A type II error fail to reject 𝐻0 , and 𝐻0 is false. Larson/Farber 4th ed. 3 EXAMPLE • Claim : It will snow 3 inches tonight. • 𝐻0 = 3 inches • 𝐻𝑎 ≠ 3inches • Type I error: we reject 𝐻0 and then it snows 3in • Type II error: we fail to reject 𝐻0 and it does not snow 3in Example • Claim: a company claims their parachutes failure rate is not more than 1%. • 𝐻0 : p ≤ .01 • 𝐻𝑎 : p > .01 • Type I error: reject p ≤ .01 and it is true • Type II error: fail to reject p ≤ .01 and it is false • Unusual events: p ≤ .05 or 5% • Reject null hypothesis when sample statistic is unusual. • Because samples vary , there is always a chance you reject 𝐻0 when it is true. • Can decrease the probability of this happening by lowering the “level of significance” • Level of significance: maximum probability that we make a type I error (reject 𝐻0 ) • Symbol : α ( alpha) 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 • Smaller the value of α says you want a small probability of rejecting a true 𝐻0 . 7 • P(type II error) = β (beta) • Power of test: 1 – β ( probability of rejecting 𝐻0 when it is false) Statistical Tests • 1.After stating the 𝐻0 and 𝐻𝑎 and specifying the level of significance, • 2. then a random sample is taken from the population and find sample statistics ( x-bar, 𝑠 2 , s, p, etc). • Then pick a statistic that is compared with the parameter in the null hypothesis is called the test statistic. Population parameter Test statistic μ x p σ2 p̂ s2 9 Standardized test statistic z (Section 7.2 n 30) t (Section 7.3 n < 30) z (Section 7.4) χ2 (Section 7.5) P-values P-value (or probability value) • The probability, if the null hypothesis is true, of getting a sample statistic with a value as or more extreme than the one determined from the sample data. • So if you did another sample : P-value is probability the sample statistic would be the same or farther out than the 1st statistic you got. Larson/Farber 4th ed. 10 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 H0. • This region is indicated by the Ha . Larson/Farber 4th ed. 11 Left-tailed Test • The alternative hypothesis Ha contains the less-than inequality symbol (<). H0: μ k Ha: μ < k P is the area to the left of the test statistic. z -3 -2 -1 0 Test statistic Larson/Farber 4th ed. 12 1 2 3 Right-tailed Test • The alternative hypothesis Ha contains the greater-than inequality symbol (>). H0: μ ≤ k Ha: μ > k P is the area to the right of the test statistic. z -3 -2 -1 0 1 2 Test statistic Larson/Farber 4th ed. 13 3 Two-tailed Test • The alternative hypothesis Ha contains the not equal inequality symbol (≠). Each tail has an area of ½P. H0: 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 -2 -1 Test 0 14 1 Test 2 3 • The smaller the p-value, then the more evidence to reject H0 • A small P-value shows an unusual event. • Still can’t say 𝐻0 is false for sure. Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, righttailed, 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: H0: p = 0.82 Ha: p ≠ 0.82 ½ P-value area Two-tailed test -z 16 ½ P-value area 0 z z Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, righttailed, 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: H0: μ ≥ 2.5 gpm Ha: μ < 2.5 gpm P-value area -z Left-tailed test Larson/Farber 4th ed. 17 0 z Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, righttailed, 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: H0: μ ≤ 20 oz Ha: μ > 20 oz P-value area 0 Right-tailed test Larson/Farber 4th ed. 18 z z Making a Decision Decision Rule Based on P-value • Compare the P-value with . If P , then reject H0. (probability less than or = to level of significance) If P > , then fail to reject H0. (probability greater than level of significance) *failing to reject 𝐻0 does not mean it is true, just tells you there is not enough evidence to say it is false. 19 Making a Decision Decision Rule Based on P-value Claim Decision Claim is H0 Claim is Ha Reject H0 There is enough evidence to reject the claim There is enough evidence to support the claim Fail to Reject H0 There is not enough evidence There is not enough evidence to reject the claim to support the claim Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0? 1. H0 (Claim): A university publicizes that the proportion of its students who graduate in 4 years is 82%. Larson/Farber 4th ed. 21 Solution: Interpreting a Decision • The claim is represented by H0. • If you reject H0 you should conclude “there is sufficient evidence to indicate that the university’s claim is false.” • If you fail to reject H0, 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 4th ed. 22 Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0? 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. • H0 is “the mean stopping distance…is greater than or equal to 136 feet.” Larson/Farber 4th ed. 23 Solution: Interpreting a Decision • If you reject H0 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 H0, 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 4th ed. 24 Steps for Hypothesis Testing using P-value 1. State the H0: ? Ha: ? 2. Specify the level of significance. α= ? 3. Determine the standardized sampling distribution and draw its graph. 4. Calculate the test statistic and its standardized value. Add it to your sketch. Larson/Farber 4th ed. This sampling distribution is based on the assumption that H0 is true. z 0 0 z Test statistic 25 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? Fail to reject H0. No Yes Reject H0. 7. Write a statement to interpret the decision in the context of the original claim. Larson/Farber 4th ed. 26 ASSIGNMENT • Page 367 37-51, 57,58.