ch23 - STAT 2023 - Holbrook Inferences About Means Chapter...

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Unformatted text preview: STAT 2023 - Holbrook Inferences About Means Chapter 23 23 - 1 STAT 2023 - Holbrook Confidence Interval Estimate for the Mean (σ Known) 23 - 2 STAT 2023 - Holbrook Recall: Confidence Limits for Population Mean Parameter = Statistic ± Error Error = Y − µ or Y + µ (3) Y − µ Error Z= = σy σy (4) 23 - 3 µ = Y ± Error (2) © 1984-1994 T/Maker Co. (1) Error = Zσy (5) µ = Y ± Zσy STAT 2023 - Holbrook Confidence Interval Mean (σ Known) 1. Assumptions 23 - 4 Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by If Normal Distribution (n ≥ 30) Normal Confidence Interval Mean (σ Known) STAT 2023 - Holbrook 1. Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by If Normal Distribution (n ≥ 30) Normal 2. Confidence Interval Estimate σ σ Y − Z⋅ ≤µ ≤ Y + Z⋅ n n 23 - 5 Important! STAT 2023 - Holbrook Rarely do we ever know what σ (the Rarely population standard deviation) is. population 23 - 6 STAT 2023 - Holbrook Example: Confidence Interval Estimate for the Mean (σ Known) 23 - 7 Thinking Challenge STAT 2023 - Holbrook You’re a Q/C inspector for You’re Gallo. The σ for 2-liter bottles is .05 liters. A random sample .05 of 100 bottles showedY = 100 1.99 liters. What is the 90% 90% confidence interval estimate of the true mean amount in 2-liter mean bottles? 2 liter 2 liter © 1984-1994 T/Maker Co. 23 - 8 STAT 2023 - Holbrook Confidence Interval Solution* σ σ Y −Z⋅ ≤µ≤ Y + Z⋅ n n .05 .05 1.99 −1.645 ⋅ ≤ µ ≤1.99 +1.645 ⋅ 100 100 1.982 ≤ µ ≤ 1.998 23 - 9 Interpretation STAT 2023 - Holbrook I’m 90% confident that the population I’m mean (µ) amount of Gallo wine found in 2mean liter bottles is between 1.982 and 1.998. 23 - 10 10 STAT 2023 - Holbrook Confidence Interval Estimate for the Mean (σ Unknown) 23 - 11 11 STAT 2023 - Holbrook Confidence Interval Mean (σ Unknown) 1. Assumptions Population Standard Deviation Is Unknown Population is Normally Distributed Population Normally If Not Normal, Can Be Approximated by Normal If Distribution (n ≥ 30) Distribution 2. Use t Distribution 23 - 12 12 Confidence Interval Mean (σ Unknown) STAT 2023 - Holbrook 1. Assumptions Population Standard Deviation Is Unknown Population is Normally Distributed Population Normally If Not Normal, Can Be Approximated by Normal If Distribution (n ≥ 30) Distribution 2. Use t Distribution 3. Confidence Interval Estimate S S Y − t n −1 ⋅ ≤ µ ≤ Y + t n −1 ⋅ n n 23 - 13 13 Checking Assumptions STAT 2023 - Holbrook 1. When the book asks you to check the assumptions this When is what you need to do: is Check and see if the population of whatever you are Check measuring is Normally Distributed. measuring If you don’t know or can’t determine if the population is If normally distributed, check the sample size. normally 23 - 14 14 If it is normal (or if the histogram of your sample data is “mound If shaped”) then you can conduct the test. can If n ≥ 3 0 then central limit theorem applies and we can conduct If can the test. the If n < 30 and the population is not normal, then we can not conduct If can the test. the STAT 2023 - Holbrook An Aside: Practice Using the T Table (Appendix D) 23 - 15 15 Find the T Value STAT 2023 - Holbrook For a 90% C.I. and a sample of size n=3, For find the t value. find 23 - 16 16 STAT 2023 - Holbrook Degrees of Freedom (df) df = n­1 1. Number of Observations that Are Free to Number Vary After Sample Statistic Has Been Calculated Calculated 2. Example 23 - 17 17 Sum of 3 Numbers Is 6 X1 = 1 (or Any Number) (or X2 = 2 (or Any Number) (or X3 = 3 (Cannot Vary) (Cannot Sum = 6 degrees of freedom degrees = n -1 = 3 -1 -1 =2 T Table STAT 2023 - Holbrook 90% C.I. 95% C.I. df Two tail probability (what we use for C.I.’s) .20 .10 .05 .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 t values 23 - 18 18 One tail probability One Assume: n=3 df = n - 1 = 2 90% C.I. So, α = .10 So, T Table STAT 2023 - Holbrook 90% C.I. 95% C.I. 90% df Two tail probability (what we use for C.I.’s) .20 .10 .05 .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 23 - 19 19 Assume: n=3 df = n - 1 = 2 90% C.I. So, α = .10 So, T Table STAT 2023 - Holbrook 90% C.I. 95% C.I. 90% df Two tail probability (what we use for C.I.’s) .20 .10 .05 .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 23 - 20 20 Assume: n=3 df = n - 1 = 2 90% C.I. So, α = .10 So, .10 T Table STAT 2023 - Holbrook 90% C.I. 95% C.I. 90% df Two tail probability (what we use for C.I.’s) .20 .10 .05 .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 23 - 21 21 Assume: n=3 df = n - 1 = 2 90% C.I. So, α = .10 So, Find the T Value Solution STAT 2023 - Holbrook So the t value for a 90% C.I. and a sample So of size n=3 is t = 2.92. 2.92 23 - 22 22 Note about the T Table STAT 2023 - Holbrook Note that the infinity line of the T Table Note represents the Z-values we used for confidence intervals in Chapter 19 and Chapter 22. Chapter 23 - 23 23 STAT 2023 - Holbrook Example: Confidence Interval Estimate for the Mean (σ unknown) 23 - 24 24 C.I. For Mean (σ unknown) STAT 2023 - Holbrook How well do students perform How academically? 101 students at 101 SFCC were asked what their GPA was. The sample mean was 3.13 and the sample 3.13 standard deviation was 0.4767. Calculate and interpret a 95% confidence 95% interval estimate for population mean. mean. 23 - 25 25 Assumptions STAT 2023 - Holbrook Note: We don’t know if the population is normal, but n ≥ 30 so the central limit theorem applies and we can conduct the test. test. 23 - 26 26 Find the T Value STAT 2023 - Holbrook For a 95% C.I. and a sample of size For n=101, find the t value. n=101, 23 - 27 27 Find the T Value Solution STAT 2023 - Holbrook The t value for a 95% C.I. and a sample of The size n=101, or df=100 is t = 1.984. 1.984 23 - 28 28 STAT 2023 - Holbrook Confidence Interval Solution* s s Y−t⋅ ≤µ≤Y+t⋅ n n 0.4767 0.4767 3.13 − 1.984 ⋅ ≤ µ ≤ 3.13 + 1.984 ⋅ 101 101 3.036 ≤ µ ≤ 3.224 23 - 29 29 Interpretation STAT 2023 - Holbrook I’m 95% confident that the population I’m mean (µ) GPA of all SFCC students is mean GPA between 3.036 and 3.224. between 23 - 30 30 Confidence Interval Solution STAT 2023 - Holbrook 1. Using the TI-83 Hit “STAT” scroll to “TESTS” then “8”, hit ”, “ENTER” “TInterval“ Select “Stats” hit “ENTER” then enter the then appropriate statistics appropriate For example: y = 3.13, S=0.4767, n=101 23 - 31 31 Choose a “.95” “C-Level” Select “Calculate” and hit “ENTER” STAT 2023 - Holbrook Hypothesis Test for the Mean (σ Known) 23 - 32 32 Important! STAT 2023 - Holbrook Rarely do we ever know what σ (the Rarely population standard deviation) is. population 23 - 33 33 STAT 2023 - Holbrook Hypothesis Test for Mean (σ Known) 1. Assumptions 23 - 34 34 Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by If Normal Distribution (n ≥ 30) Normal STAT 2023 - Holbrook Hypothesis Test for Mean (σ Known) 1. Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by If Normal Distribution (n ≥ 30) Normal 2. Z-Test Statistic Y − µy Y −µ Z= = σ σy n 23 - 35 35 Note: STAT 2023 - Holbrook We will not work examples for this type of We not hypothesis test because it is so rare that we know what σ is (and that none of your homework problems give you !!) !!) 23 - 36 36 STAT 2023 - Holbrook Hypothesis Test of the Mean (σ Unknown) 23 - 37 37 STAT 2023 - Holbrook Hypothesis Test for Mean (σ Unknown) 1. Assumptions 23 - 38 38 Population Standard Deviation Is Unknown Population is Normally Distributed Population Normally If Not Normal, Can Be Approximated by Normal If Distribution (n ≥ 30) Distribution STAT 2023 - Holbrook Hypothesis Test for Mean (σ Unknown) 1. Assumptions Population Standard Deviation Is Unknown Population is Normally Distributed Population Normally If Not Normal, Can Be Approximated by Normal If Distribution (n ≥ 30) Distribution 2. Use t Distribution 23 - 39 39 STAT 2023 - Holbrook Hypothesis Test for Mean (σ Unknown) 1. Assumptions Population Standard Deviation Is Unknown Population is Normally Distributed Population Normally If Not Normal, Can Be Approximated by Normal If Distribution (n ≥ 30) Distribution 2. Use t Distribution 3. t Test statistic 23 - 40 40 Y −µ t= s n Checking Assumptions (Same steps as for Confidence Intervals) STAT 2023 - Holbrook 1. When the book asks you to check the assumptions this When is what you need to do: is Check and see if the population of whatever you are Check measuring is Normally Distributed. measuring If you don’t know or can’t determine if the population is If normally distributed, check the sample size. normally 23 - 41 41 If it is normal (or if the histogram of your sample data is “mound If shaped”) then you can conduct the test. can If n ≥ 3 0 then central limit theorem applies and we can conduct If can the test. the If n < 30 and the population is not normal, then we can not conduct If can the test. the STAT 2023 - Holbrook Note: Step 3 – finding the p­value 1. We must use our TI-83 to find p-values (the Z We table will not work as it did in Chapter 20). table 23 - 42 42 STAT 2023 - Holbrook Example: Hypothesis Test for the Mean (σ Unknown) 23 - 43 43 STAT 2023 - Holbrook Hypothesis Test for the Mean Example Is the average capacity of Is batteries at least 140 ampereamperehours? A random sample of 20 hours? 20 batteries had a mean of 138.47 138.47 & a standard deviation of 2.66. 2.66 Assume that the population of battery capacities are normally distributed. distributed. 23 - 44 44 STAT 2023 - Holbrook H0: µ = 140 Ha: µ < 140 n = 20 df = 20 – 1 = 19 Test Statistic: t Test Solution P-value: P-value: Decision: Decision: Conclusion: 23 - 45 45 STAT 2023 - Holbrook t Test Solution Step 2: Test Statistic Y − µ 138.47 − 140 t= = = −2.57 S 2.66 n 20 23 - 46 46 STAT 2023 - Holbrook t Test Solution H0: µ = 140 Ha: µ < 140 n = 20 df = 20 – 1 = 19 Test Statistic: t = −2.57 P-value: Decision: Conclusion: 23 - 47 47 t Test Solution : Finding the p­value STAT 2023 - Holbrook 1. Using the TI-83 Hit “2nd” and then “VARS” (Bringing up “DISTR”) Scroll down to 5, “tcdf(“ and hit “ENTER” Scroll “tcdf(“ should appear Hit “-” then “1” then “2nd” then “,” then “99” 23 - 48 48 The default is tcdf(lower t value, upper t value, df) This is the Lower bound = -1 x 1099 (negative infinity) Separate lower bound by a “,” Enter “-2.57” separated by a “,” Enter “19” for df and then “)” Screen should read tcdf(-1E99, -2.57, 19) Screen tcdf(-1 STAT 2023 - Holbrook t Test Solution H0: µ = 140 Ha: µ < 140 n = 20 df = 20 – 1 = 19 Test Statistic: t = −2.57 P-value: =0.0094 P-value: =0.0094 0.94% of the time we 0.94% would have seen data like this if H0 data was true. (rarely) was Decision: Conclusion: 23 - 49 49 STAT 2023 - Holbrook t Test Solution H0: µ = 140 Ha: µ < 140 n = 20 df = 20 – 1 = 19 Test Statistic: t = −2.57 P-value: =0.0094 P-value: =0.0094 0.94% of the time we 0.94% would have seen data like this if H0 data was true. (rarely) was Decision: Reject H0 Conclusion: 23 - 50 50 STAT 2023 - Holbrook t Test Solution H0: µ = 140 Ha: µ < 140 n = 20 df = 20 – 1 = 19 Test Statistic: t = −2.57 P-value: =0.0094 P-value: =0.0094 0.94% of the time we 0.94% would have seen data like this if H0 data was true. (rarely) was Decision: Reject H0 Conclusion: 23 - 51 51 Enough evidence that the Enough population mean battery capacity is less than 140 ampere hours. is STAT 2023 - Holbrook Example: Hypothesis Test for the Mean (σ Unknown) 23 - 52 52 STAT 2023 - Holbrook Two­Tailed t Test Example Does an average box of Does cereal contain 368 368 grams of cereal? A random sample of 36 36 boxes had a mean of 372.5 & a standard 372.5 deviation of 12 grams. deviation 12 grams. 368 gm. 23 - 53 53 STAT 2023 - Holbrook H0: µ = 368 Ha: µ ≠ 368 n = 36 df = 36 – 1 = 35 Test Statistic: t Test Solution P-value: P-value: Decision: Decision: Conclusion: 23 - 54 54 STAT 2023 - Holbrook t Test Solution Step 2: Test Statistic Y − µ 372.5 − 368 t= = = 2.25 S 12 n 36 23 - 55 55 STAT 2023 - Holbrook H0: µ = 368 Ha: µ ≠ 368 n = 36 df = 36 – 1 = 35 Test Statistic: t = 2.25 t Test Solution P-value: P-value: Decision: Decision: Conclusion: 23 - 56 56 STAT 2023 - Holbrook Two­Tailed t Test p­Value Solution p-value is P(t ≤ -2.25 or t ≥ 2.25) p-value 1/2 p-Value 1/2 p-Value -2.25 -1.50 0 2.25 1.50 23 - 57 57 Zt t value of sample value statistic (observed) statistic t Test Solution : Finding the p­value STAT 2023 - Holbrook 1. Using the TI-83 (LOWER TAIL ONLY!) Hit “2nd” then “VARS” then “5” “tcdf(“ should appear Hit “-” then “1” then “2nd” then “,” then “99” 23 - 58 58 This is the Lower bound = -1 x 1099 (negative infinity) Separate lower bound by a “,” Enter “-2.25” separated by a “,” Enter “35” for df and then “)” Screen should read tcdf(-1E99, -2.25, 35) Screen tcdf(-1 STAT 2023 - Holbrook Two­Tailed t Test p­Value Solution p-value is P(t ≤ -2.25 or t ≥ 2.25) p-value .0154 1/2 p-Value 1/2 p-Value -2.25 -1.50 0 2.25 1.50 23 - 59 59 Zt t value of sample value statistic (observed) statistic STAT 2023 - Holbrook Two­Tailed t Test p­Value Solution Due to symmetry . 0154 1/2 p-Value 1/2 p-Value . 0154 - 2.25 -1.50 0 2.25 1.50 23 - 60 60 Zt t value of sample value statistic (observed) statistic STAT 2023 - Holbrook Two­Tailed t Test p­Value Solution So the p-value = .0154 + .0154 = .0308 So .0154 1/2 p-Value 1/2 p-Value .0154 - 2.25 -1.50 0 2.25 1.50 23 - 61 61 Zt t value of sample statistic (observed) statistic STAT 2023 - Holbrook H0: µ = 368 Ha: µ ≠ 368 n = 36 df = 36 – 1 = 35 Test Statistic: t = 2.25 t Test Solution P-value: =0.0308 P-value: =0.0308 3.08% of the time we 3.08% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Conclusion: 23 - 62 62 STAT 2023 - Holbrook H0: µ = 368 Ha: µ ≠ 368 n = 36 df = 36 – 1 = 35 Test Statistic: t = 2.25 t Test Solution P-value: =0.0308 P-value: =0.0308 3.08% of the time we 3.08% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 23 - 63 63 STAT 2023 - Holbrook H0: µ = 368 Ha: µ ≠ 368 n = 36 df = 36 – 1 = 35 Test Statistic: t = 2.25 t Test Solution P-value: =0.0308 P-value: =0.0308 3.08% of the time we 3.08% would have seen data like this if H0 was true. like (rarely) (rarely) Decision: Reject H0 Conclusion: 23 - 64 64 Enough evidence that the Enough population mean amount of cereal found in Cereal O’s is different from 368 grams. Hypothesis Test Solution Alternate Method STAT 2023 - Holbrook 1. Using the TI-83 Hit “STAT” scroll to “TESTS” then “2”, hit “ENTER” “T-Test“ Select “Stats” then hit “ENTER“ Scroll to μ0 and enter the hypothesized value Scroll the For example: μ0=368 For Enter the appropriate statistics For example: y = 372.5, S=12, n=36 Scroll and select (by hitting “ENTER”) one of the ”) following: two tail (≠ ), lower tail (<) or upper tail (>) following: ), alternative hypothesis. alternative For example: ≠ μ0 For Select “Calculate” and hit “ENTER” 23 - 65 65 STAT 2023 - Holbrook Finding Sample Sizes (Note: we skipped this topic with respect to proportions.) 23 - 66 66 STAT 2023 - Holbrook (1) (2) (3) Finding Sample Sizes for Estimating µ Y − µ Error Z= = σy σy σ s Error = Zσ y = Z ≅Z n n 22 Zs n= 2 (Error ) Error Is Also Called Bound, B Error 23 - 67 67 I don’t want to sample too much or too little! Note STAT 2023 - Holbrook • Note: This formula and process of finding n (the samples size) is slightly different (and easier) and more conservative than the formula and procedure described in your book on Page 603. (Page 442 – First Edition.) (Page 23 - 68 68 Sample Size Example STAT 2023 - Holbrook What sample size is needed to be 90% What confident of being correct within a bound or error of ± 5? A pilot study suggested that the standard deviation is 45. that 23 - 69 69 Sample Size Example Solution STAT 2023 - Holbrook (1.645) ( 45) = 219.2 ≅ 220 Zs n= = 2 2 Error ( 5) 22 2 2 Always round up! (Yes, I know it goes against everything that you’ve learned!) 23 - 70 70 End of Chapter Any blank slides that follow are blank intentionally. ...
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This note was uploaded on 12/07/2011 for the course STA 2023 taught by Professor Staff during the Fall '11 term at Santa Fe College.

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