ch20 - STAT 2023 - Holbrook Testing Hypotheses About...

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Unformatted text preview: STAT 2023 - Holbrook Testing Hypotheses About Proportions Chapter 20 20 - 1 STAT 2023 - Holbrook Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Estimation 20 - 2 Hypothesis Testing STAT 2023 - Holbrook Hypothesis Testing Concepts 20 - 3 STAT 2023 - Holbrook What’s a Hypothesis? 1. A Belief about a Belief Population Parameter Population Parameter Is Parameter Population Mean, Population Proportion, Variance Proportion, I believe the population believe proportion of SFCC students who have tattoos is .25! who Must Be Stated Before Analysis © 1984-1994 T/Maker Co. 20 - 4 STAT 2023 - Holbrook Population 20 - 5 Hypothesis Testing STAT 2023 - Holbrook Population 20 - 6 Hypothesis Testing I believe the population proportion is .7 (hypothesis). STAT 2023 - Holbrook Population Hypothesis Testing I believe the population proportion is .7 (hypothesis). Random Random sample sample Prop. ^ = .3 p 20 - 7 STAT 2023 - Holbrook Population Hypothesis Testing I believe the population proportion is .7 (hypothesis). Random Random sample sample Prop. ^ = .3 p 20 - 8 Reject Reject hypothesis! hypothesis! Not close. Not close. STAT 2023 - Holbrook Basic Idea Sampling Distribution µ p = 50 .7 20 - 9 H0 Sample Mean Sample proportion Basic Idea STAT 2023 - Holbrook Sampling Distribution It is unlikely It that we would get a sample proportion of this value ... this .3 20 - 10 10 µ p = 50 .7 H0 Sample Mean Sample proportion Basic Idea STAT 2023 - Holbrook Sampling Distribution It is unlikely It that we would get a sample proportion of this value ... this ... if in fact this were the population prop. .3 20 - 11 11 µ p = 50 .7 H0 Sample Mean Sample proportion Basic Idea STAT 2023 - Holbrook Sampling Distribution It is unlikely It that we would get a sample proportion of this value ... this ... therefore, ... we reject the hypothesis that p = .7 .7 ... if in fact this were the population prop. .3 20 - 12 12 µ p = 50 .7 H0 Sample Mean Sample proportion STAT 2023 - Holbrook Null Hypothesis 1. Has Serious Outcome If Incorrect Has Decision Made Decision 2. Always Has Equality Sign: = , ≤ , or ≥ Always or 3. Designated H0 (Pronounced H-oh) 4. Specified as H0: p = Some Numeric Value 20 - 13 13 Specified with = Sign Even if ≤ , or ≥ Specified or Example, H0: p = .3 STAT 2023 - Holbrook Alternative Hypothesis 1. Opposite of Null Hypothesis 2. Put what we are trying to prove in the Put alternative hypothesis alternative 3. 4. 5. 5. Also called the research hypothesis Always Has Inequality Sign: ≠ , < , or or > Designated Ha Specified Ha: p < Some Value 20 - 14 14 Example, Ha: p < .3 .3 Identifying Hypotheses Steps STAT 2023 - Holbrook 1. Example Problem: Test That the Population Example proportion Is Not .3 proportion 2. Steps State the Question Statistically (p ≠ .3) State State the Opposite Statistically (p = .3) Select the Alternative Hypothesis (p ≠ .3) Select 20 - 15 15 Must Be Mutually Exclusive & Exhaustive Has the ≠ , <, or > Sign Has or Sign State the Null Hypothesis (p = .3) What Are the Hypotheses? STAT 2023 - Holbrook Is the population proportion of SFCC students who ever drink alcohol .8 ? students State the question statistically: p = .8 State .8 State the opposite statistically: p ≠ .8 State Select the alternative hypothesis: Ha: p ≠ .8 State the null hypothesis: H0: p = .8 State 20 - 16 16 What Are the Hypotheses? STAT 2023 - Holbrook Is the population proportion of SFCC students who ever drink alcohol different from .8 ? from State the question statistically: p ≠ .8 State State the opposite statistically: p =.8 State Select the alternative hypothesis: Ha: p ≠ .8 State the null hypothesis: H0: p = .8 State 20 - 17 17 What Are the Hypotheses? STAT 2023 - Holbrook Is the proportion of SFCC students who have ever smoked marijuana less than or equal to .5 ? equal State the question statistically: p ≤ .5 State State the opposite statistically: p > .5 State Select the alternative hypothesis: Ha: p > .5 Select State the null hypothesis: H0: p = .5 State 20 - 18 18 What Are the Hypotheses? STAT 2023 - Holbrook Is the proportion of SFCC students who have pierced their ears greater than .7 ? have State the question statistically: p > .7 State State the opposite statistically: p ≤ .7 State Select the alternative hypothesis: Ha: p > .7 State the null hypothesis: H0: p = .7 State 20 - 19 19 STAT 2023 - Holbrook Hypothesis Testing Steps 20 - 20 20 STAT 2023 - Holbrook Step 1 s State H0 s State Ha s Choose n s Collect data 20 - 21 21 H0 Testing Steps STAT 2023 - Holbrook Step 1 H0 Testing Steps Steps 2 through 5 s (2) Compute the test (2) statistic. statistic. s State H0 s State Ha s (3) Find/interpret the p-value. s Choose n s (4) Make statistical decision. s Collect data • s 20 - 22 22 Reject or Fail to Reject H0 (5) Express conclusion in the (5) context of the problem. context STAT 2023 - Holbrook Z Test for Proportion 20 - 23 23 One­Sample Z Test for Proportion STAT 2023 - Holbrook 1. Assumptions Two Categorical Outcomes Normal Approximation can be used 20 - 24 24 ˆ ˆ np > 10 and n (1 − p) > 10 One­Sample Z Test for Proportion STAT 2023 - Holbrook 1. Assumptions Two Categorical Outcomes Normal Approximation can be used ˆ ˆ np > 10 and n (1 − p) > 10 2. Z-test statistic for proportion 2. Z≅ 20 - 25 25 p − p0 p0 ⋅ (1 − p0 ) n Hypothesized Hypothesized population proportion population STAT 2023 - Holbrook Observed Level of Significance: p­value 20 - 26 26 STAT 2023 - Holbrook p­Value 1. Probability of Obtaining a Test Statistic Probability More Extreme (≤ or ≥ ) than Actual More Sample Value Given H0 Is True Sample 2. Called Observed Level of Significance 2. Smallest Value of α H0 Can Be Rejected Smallest 3. Used to Make Rejection Decision 20 - 27 27 If p-Value < α, Reject H0 If If p-Value ≥ α, Do Not Reject H0 If STAT 2023 - Holbrook Examples 20 - 28 28 STAT 2023 - Holbrook One Proportion Z Test Example The present packaging The system produces 10% 10% defective cereal boxes. Using a new system, a random sample of 200 200 boxes had 11 defects. boxes 11 Does the new system produce fewer defects? fewer 20 - 29 29 Checking the Assumption* STAT 2023 - Holbrook ˆ np > 10 or 200(0.055) = 11 > 10 ˆ and n (1 − p) > 10 or 200(1 − 0.055) = 189 > 10 So the sample size is large enough. Note: Our way of checking the assumptions is slightly different from the book. is 20 - 30 30 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .10 Ha: p < .10 n = 200 Test Statistic: P-value: P-value: Decision: Decision: Conclusion: 20 - 31 31 STAT 2023 - Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20 - 32 32 p − p0 = p0 ⋅ (1 − p0 ) n 11 −.10 200 = −2.12 .10 ⋅ (1−.10) 200 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .10 Ha: p < .10 n = 200 P-value: P-value: Test Statistic: Z ≅ −2.12 Decision: Decision: Conclusion: 20 - 33 33 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .10 Ha: p < .10 n = 200 Test Statistic: Z ≅ −2.12 P-value: = 0.0170 P-value: 0.0170 1.7% of the time we 1.7% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Conclusion: 20 - 34 34 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .10 Ha: p < .10 n = 200 Test Statistic: Z ≅ −2.12 P-value: = 0.0170 P-value: 0.0170 1.7% of the time we 1.7% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 20 - 35 35 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .10 Ha: p < .10 n = 200 Test Statistic: Z ≅ −2.12 P-value: = 0.0170 P-value: 0.0170 1.7% of the time we 1.7% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 20 - 36 36 Enough evidence that the Enough population proportion of defects is < .10 defects STAT 2023 - Holbrook One Proportion Z Test Thinking Challenge You’re an accounting You’re manager. A year-end audit showed 4% of transactions 4% had errors. You implement new procedures. A random sample of 500 transactions 500 had 25 errors. Has the 25 proportion of incorrect proportion transactions changed? changed 20 - 37 37 Checking the Assumption* STAT 2023 - Holbrook ˆ np > 10 or 500(0.05) = 25 > 10 ˆ and n (1 − p) > 10 or 500(1 − 0.05) = 475 > 10 So the sample size is large enough. 20 - 38 38 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .04 Ha: p ≠ .04 n = 500 Test Statistic: P-value: P-value: Decision: Decision: Conclusion: 20 - 39 39 STAT 2023 - Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20 - 40 40 p − p0 = p0 ⋅ (1 − p0 ) n 25 −.04 500 = 1.14 .04 ⋅ (1−.04 ) 500 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .04 Ha: p ≠ .04 n = 500 P-value: P-value: Test Statistic: Z ≅ 1.14 Decision: Decision: Conclusion: 20 - 41 41 STAT 2023 - Holbrook Two­Tailed Z Test p­Value Solution p-value is P(Z ≤ -1.14 or Z ≥ 1.14) p-value 1/2 p-Value 1/2 p-Value -1.14 -1.50 0 1.14 1.50 20 - 42 42 Z Z value of sample value statistic (observed) statistic STAT 2023 - Holbrook Two­Tailed Z Test p­Value Solution p-value is P(Z ≤ -1.14 or Z ≥ 1.14) p-value .1271 1/2 p-Value 1/2 p-Value -1.14 -1.50 0 1.14 1.50 20 - 43 43 Z Z value of sample value statistic (observed) statistic STAT 2023 - Holbrook Two­Tailed Z Test p­Value Solution Due to symmetry .1271 1/2 p-Value .1271 1/2 p-Value -1.14 -1.50 0 1.14 1.50 20 - 44 44 Z Z value of sample value statistic (observed) statistic STAT 2023 - Holbrook Two­Tailed Z Test p­Value Solution So the p-value = .1271 + .1271 = .2542 So .1271 1/2 p-Value .1271 1/2 p-Value -1.14 -1.50 0 1.14 1.50 20 - 45 45 Z Z value of sample statistic (observed) statistic STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .04 Ha: p ≠ .04 n = 500 Test Statistic: Z ≅ 1.14 P-value: = 0.2542 P-value: 0.2542 25.42% of the time we 25.42% would have seen data like this if H0 was true. like (not a rare occurrence) (not Decision: Conclusion: 20 - 46 46 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .04 Ha: p ≠ .04 n = 500 Test Statistic: Z ≅ 1.14 P-value: = 0.2542 P-value: 0.2542 25.42% of the time we 25.42% would have seen data like this if H0 was true. like (not a rare occurrence) (not Decision: Fail to Reject H0 Conclusion: 20 - 47 47 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .04 Ha: p ≠ .04 n = 500 Test Statistic: Z ≅ 1.14 P-value: = 0.2542 P-value: 0.2542 25.42% of the time we 25.42% would have seen data like this if H0 was true. like (not a rare occurrence) (not Decision: Fail to Reject H0 Conclusion: 20 - 48 48 Not enough evidence that the Not population proportion of errors is different from .04 is STAT 2023 - Holbrook One Proportion Z Test Example Is drug use running rampant? Is 200 students at SFCC were 200 asked if they have ever smoked marijuana and 140 140 responded yes. Is the population proportion of SFCC students who have ever smoked marijuana more than 60% ? 60% 20 - 49 49 Checking the Assumption* STAT 2023 - Holbrook ˆ np > 10 or 200(0.7) = 140 > 10 ˆ and n (1 − p) > 10 or 200(1 − 0.7) = 60 > 10 So the sample size is large enough. 20 - 50 50 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .6 Ha: p > .6 n = 200 Test Statistic: P-value: P-value: Decision: Decision: Conclusion: 20 - 51 51 STAT 2023 - Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20 - 52 52 ˆ p − p0 = p 0 ⋅ (1 − p 0 ) n 140 − .60 200 = 2.89 .60 ⋅ (1 − .60) 200 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .6 Ha: p > .6 n = 200 Test Statistic: Z ≅ 2.89 P-value: P-value: Decision: Decision: Conclusion: 20 - 53 53 One­Tailed Z Test p­Value Solution STAT 2023 - Holbrook p-Value is P(Z ≥ 2.89) p-Value Use Use alternative hypothesis to find direction direction p-Value 0 1.50 2.89 20 - 54 54 Z Z value of sample statistic statistic STAT 2023 - Holbrook One­Tailed Z Test p­Value Solution p-Value is P(Z ≥ 2.89) = 1 - .9981 = .0019 p-Value Use Use alternative hypothesis to find direction direction .9981 .0019 p-Value 0 1.50 2.89 20 - 55 55 Z Z value of sample statistic statistic STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .6 Ha: p > .6 n = 200 Test Statistic: Z ≅ 2.89 P-value: = 0.0019 P-value: 0.0019 0.19% of the time we 0.19% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Conclusion: 20 - 56 56 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .6 Ha: p > .6 n = 200 Test Statistic: Z ≅ 2.89 P-value: = 0.0019 P-value: 0.0019 0.19% of the time we 0.19% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 20 - 57 57 STAT 2023 - Holbrook One Proportion Z Test Solution H0: p = .6 Ha: p > .6 n = 200 Test Statistic: Z ≅ 2.89 P-value: = 0.0019 P-value: 0.0019 0.19% of the time we 0.19% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 20 - 58 58 Enough evidence that the population Enough proportion of students who have ever smoked marijuana is more than .6 or 60% .6 Hypothesis Test Solution STAT 2023 - Holbrook 1. Using the TI-83 Hit “STAT” scroll to “TESTS” then “5”, hit “ENTER” “1-PropZTest“ Scroll to p0 and enter the hypothesized value For example: p0=0.6 Enter x and n. For example: x=140 and n=200 Scroll and select (by hitting “ENTER”) one of the ”) following: two tail (≠ ), lower tail (<) or upper tail (>) following: ), alternative hypothesis. alternative 20 - 59 59 For example: >p0 Select “Calculate” and hit “ENTER” End of Chapter Any blank slides that follow are blank intentionally. ...
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