Confidence Intervals for Population Proportions: Lessons from WMU Statistics - Prof. Jung , Study notes of Business Statistics

Download Confidence Intervals for Population Proportions: Lessons from WMU Statistics - Prof. Jung and more Study notes Business Statistics in PDF only on Docsity! Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Chapter 7. Confidence Intervals J.C. Wang Department of Statistics Western Michigan University Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Goal and Objectives Goal: to learn confidence intervals Objectives: To understand that each interval has two end-points (lower and upper bound) and Interpret the confidence interval To compute the confidence interval: a point estimate ± the margin of error To determine the sample size Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Outline 1 Introduction Applications, Definations and Notation 2 Confidence Intervals Computation z-Confidence Intervals t-Confidence Intervals Sample Size Determination An Example 3 Comparing Two Populations Comparing Means of Two Independent Populations Comparing Means of Two Dependent Populations 4 Confidence Interval for Proportion Confidence Interval for Population Proportion Sample Size Determination Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Outline 1 Introduction Applications, Definations and Notation 2 Confidence Intervals Computation z-Confidence Intervals t-Confidence Intervals Sample Size Determination An Example 3 Comparing Two Populations Comparing Means of Two Independent Populations Comparing Means of Two Dependent Populations 4 Confidence Interval for Proportion Confidence Interval for Population Proportion Sample Size Determination Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Applications of Estimation in Business examples Store inventory value Manufacture process Distribution process Drug delivery Auditor Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Applications of Estimation in Business examples Store inventory value Manufacture process Distribution process Drug delivery Auditor Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Applications of Estimation in Business examples Store inventory value Manufacture process Distribution process Drug delivery Auditor Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Definitions Sample statistic: a value computed from the sample (i.e., from data). Point estimate (pt.est): a single sample statistic that estimates the population parameter, such as, the mean or proportion. Interval estimate of the true population parameter takes into account the sampling distribution of the point estimate where we have an upper bound and a lower bound. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Notations to be discussed and used later CI — confidence interval CVal — critical value ME — margin of error SE — standard error SD — standard deviation pt.est. — point estimate Zα/2 — normal distribution critical value (use invnorm) tn−1 — students t distribution critical value with n-1 degrees of freedom (use math solver or the invT) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computation of confidence intervals pt.est ± ME Where the point estimate estimates population mean µ (by x) or population proportion p (by p̂) marginOfError = criticalValue × standardError In other words, ME = (CVal)(SE). Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Standard Error Most of the time we will not have the SD of population mean, but we can compute sample SE of the mean: SEx = s√ n Also, we will not have the SD of population proportion, but we can compute sample proportion SE: SEp̂ = √ p̂(1− p̂) n Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Standard Error Most of the time we will not have the SD of population mean, but we can compute sample SE of the mean: SEx = s√ n Also, we will not have the SD of population proportion, but we can compute sample proportion SE: SEp̂ = √ p̂(1− p̂) n Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Critical Value z for normal distribution t for students t-distribution The students t-distribution has n − 1 degrees of freedom, df = n − 1 Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion z-Critical Value Notation: zα/2 = upper (100× α/2)th standard normal percentile That is: P(Z > zα/2) = α/2 ≡ P(Z ≤ zα/2) = 1− α/2 So, zα/2 = invNorm(1− α/2) Example 95% confidence interval will give 2.5% in each tail of the bell-shaped curve; therefore, the z-CVal, zcv = z.025 = invNorm(1−.025) = invNorm(.975) = 1.96. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion z-Critical Value Notation: zα/2 = upper (100× α/2)th standard normal percentile That is: P(Z > zα/2) = α/2 ≡ P(Z ≤ zα/2) = 1− α/2 So, zα/2 = invNorm(1− α/2) Example 95% confidence interval will give 2.5% in each tail of the bell-shaped curve; therefore, the z-CVal, zcv = z.025 = invNorm(1−.025) = invNorm(.975) = 1.96. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion t-Critical Value using TI calculators 1 math −→ solver −→ tcdf(L,U,D)− A/T , where L = tcv (to be solved) U = 9999 D = df = n − 1 A = α (error rate) T = number of tails = 2 for c.i. 2 or use invT(1− α/2,df ) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Cereal Box Packaging Example Consider a cereal packaging plant in Battle Creek that is concerned with putting 368 gram of cereal into a box. What are the costs associated with putting too much cereal in a box? What are the costs associated with putting too little cereal in a box? Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Cereal Box Packaging Example Consider a cereal packaging plant in Battle Creek that is concerned with putting 368 gram of cereal into a box. What are the costs associated with putting too much cereal in a box? What are the costs associated with putting too little cereal in a box? Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Cereal Box Packaging Example continued, confidence interval Since the confidence interval is the pt.est ± ME CI = 365 ± 5.88 = (359.12, 370.88). Therefore, we are 95% confident that the population mean is between 359 and 371. Since 368, the value that is printed on the box indicates the manufacturing process is working properly (is within the interval), there is no reason to conclude that anything is wrong with the process. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion z-Confidence Interval Using TI Calculators example Let’s use TI calculator: Do this: STAT→ TESTS→ Zinterval→ STATS ↓ σ:15 ↓ x:365 ↓ n:25 ↓ C-Level:.95 ↓ CALCULATE READOUT: Zinterval (359.12, 370.88) x = 365 n = 25 Since 368, the target of the package, is within the interval; production should continue. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion z-Confidence Interval Using TI Calculators example Let’s use TI calculator: Do this: STAT→ TESTS→ Zinterval→ STATS ↓ σ:15 ↓ x:365 ↓ n:25 ↓ C-Level:.95 ↓ CALCULATE READOUT: Zinterval (359.12, 370.88) x = 365 n = 25 Since 368, the target of the package, is within the interval; production should continue. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Note on z-Confidence Intervals The value of z selected for constructing such a confidence interval is called the critical value for the distribution. There are different critical values for each level of confidence (or confidence level, CL), 1− α, where α = significance level, SL (or error rate). Frequently Used zcv : SL CL 2-tailed CVal 10% 90% 1.645 5% 95% 1.96 1% 99% 2.58 Note: There is a trade off between the width of the confidence interval and the level of confidence. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Problem When SD is Unknown We have been dealing with N(µ, σ) where σ (population or process SD) is known. What happens when standard deviation (σ) is not from a population or process SD? Is this requirement rigid? Can we compute standard deviation from the sample? Let us review some history first. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion History of the Student t Distribution William Gosset, an employee of Guinness Breweries in Ireland, had a preoccupation with making statistical inferences about the mean when SD was unknown. Since the employees of the company were not allowed to publish their scientific work under their own name. He chose the pseudonym “Student.” Therefore, his contribution is still known as Student’s t-Distribution. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion t-Confidence Interval for the Mean using TI calculators Do this: STAT→ TESTS ↓ TInterval→ STATS ↓ x:25 ↓ Sx:10.777 ↓ n:14 ↓ C-Level:.95 ↓ CALCULATE READOUT: Tinterval (18.778, 31.222) x = 25 n = 14 We are 95% confident that the true mean quiz score is between 18.8 and 31.2. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion t-Confidence Interval for the Mean using TI calculators Do this: STAT→ TESTS ↓ TInterval→ STATS ↓ x:25 ↓ Sx:10.777 ↓ n:14 ↓ C-Level:.95 ↓ CALCULATE READOUT: Tinterval (18.778, 31.222) x = 25 n = 14 We are 95% confident that the true mean quiz score is between 18.8 and 31.2. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion t-Confidence Interval for the Mean using TI calculators Do this: STAT→ TESTS ↓ TInterval→ STATS ↓ x:25 ↓ Sx:10.777 ↓ n:14 ↓ C-Level:.95 ↓ CALCULATE READOUT: Tinterval (18.778, 31.222) x = 25 n = 14 We are 95% confident that the true mean quiz score is between 18.8 and 31.2. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example page 100, problem #1 21 20 22 7 9 14 23 9 10 25 15 17 11 (a) x = 15.6154 and s = 6.1310 (b) population average and SD: not possible. (c) the sample average will miss the population average by the SE. (d) SEx = s/ √ n = 6.1310/ √ 13 = 1.7 (e) ME = CVal ×SE = t.975,13−1× 1.7 = 2.1788× 1.7 = 3.704 (f) 95% CI is 15.6154± 3.704 = (11.91,19.32) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example page 100, problem #1 21 20 22 7 9 14 23 9 10 25 15 17 11 (a) x = 15.6154 and s = 6.1310 (b) population average and SD: not possible. (c) the sample average will miss the population average by the SE. (d) SEx = s/ √ n = 6.1310/ √ 13 = 1.7 (e) ME = CVal ×SE = t.975,13−1× 1.7 = 2.1788× 1.7 = 3.704 (f) 95% CI is 15.6154± 3.704 = (11.91,19.32) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example page 100, problem #1 21 20 22 7 9 14 23 9 10 25 15 17 11 (a) x = 15.6154 and s = 6.1310 (b) population average and SD: not possible. (c) the sample average will miss the population average by the SE. (d) SEx = s/ √ n = 6.1310/ √ 13 = 1.7 (e) ME = CVal ×SE = t.975,13−1× 1.7 = 2.1788× 1.7 = 3.704 (f) 95% CI is 15.6154± 3.704 = (11.91,19.32) Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example continued (f) (continued) can also do this (assuming data have been entered into list 1, L1): STAT→ TESTS ↓ tInterval→ DATA ↓ List:L1 ↓ CALCULATE (g) If the confidence level is reduced to 90%, the new interval will be shorter. (h) 90% CI→ (12.585, 18.646) (i) Interpret the 95% CI.: We are 95 percent confident that the true (population average) is between 12 and 19. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example continued (f) (continued) can also do this (assuming data have been entered into list 1, L1): STAT→ TESTS ↓ tInterval→ DATA ↓ List:L1 ↓ CALCULATE (g) If the confidence level is reduced to 90%, the new interval will be shorter. (h) 90% CI→ (12.585, 18.646) (i) Interpret the 95% CI.: We are 95 percent confident that the true (population average) is between 12 and 19. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example continued (f) (continued) can also do this (assuming data have been entered into list 1, L1): STAT→ TESTS ↓ tInterval→ DATA ↓ List:L1 ↓ CALCULATE (g) If the confidence level is reduced to 90%, the new interval will be shorter. (h) 90% CI→ (12.585, 18.646) (i) Interpret the 95% CI.: We are 95 percent confident that the true (population average) is between 12 and 19. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Slow Wave Sleep Example continued (j) Does the 95% CI suggest that elderly men over 60 spend 20% of their sleep in REM? No, since 20 (%) is not in the 95% CI. (k) What sample size should we use if we change the ME to 2.5? n = CVal2 × SD2 ME2 = 1.962 × 6.132 2.52 = 23.10 use= 24 Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Outline 1 Introduction Applications, Definations and Notation 2 Confidence Intervals Computation z-Confidence Intervals t-Confidence Intervals Sample Size Determination An Example 3 Comparing Two Populations Comparing Means of Two Independent Populations Comparing Means of Two Dependent Populations 4 Confidence Interval for Proportion Confidence Interval for Population Proportion Sample Size Determination Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Comparing Means of two independent populations We are not limited to comparing an average to a constant. Suppose we want to compare the means of two independent populations. Parameter of interest: δ = µ1 − µ2 Recall: CI is pt.est ± ME pt .est = d = x1 − x2, ME = CVal × SE where CVal = tn1+n2−2, SE = √ SE21 + SE 2 2 Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Example battery example A statistics student designed an experiment to see if there was any real difference in battery life between brand-name AA batteries and generic AA batteries. He used six pairs of AA alkaline batteries from two major battery manufactures: a well known brand name and a generic brand. He measured the length of battery life while playing a CD player continuously. He recorded the time (in minutes) when the sound stopped. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Battery Example continued Generic Brand Name x 206 187.4 S 10.3 14.6 n 6 6 Want 95% CI (a) What is the standard error? (b) What is the 95% CVal? (c) What is the ME? (d) What is the 95% CI? (e) Does this confidence interval suggest that generic AA batteries will last longer than brand-name AA batteries? (f) Interpret the 95% CI. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Battery Example continued Generic Brand Name x 206 187.4 S 10.3 14.6 n 6 6 Want 95% CI (a) What is the standard error? (b) What is the 95% CVal? (c) What is the ME? (d) What is the 95% CI? (e) Does this confidence interval suggest that generic AA batteries will last longer than brand-name AA batteries? (f) Interpret the 95% CI. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Battery Example continued, answers (e) Does this confidence interval suggest that generic AA batteries will last longer than brand-name AA batteries? Yes, because zero is not within the interval (f) Interpret the 95% CI. We are 95% confident that the true mean difference is between 2 and 35. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Battery Example continued, using TI calculator Do this: STAT→ TESTS ↓ 2-SampTInt→ STATS ↓ x1:206.0 ↓ Sx1:10.3 ↓ n1:6 ↓ x2:187.4 ↓ Sx2:14.6 ↓ n2:6 ↓ C-Level:.95 ↓ Pooled:Yes ↓ CALCULATE READOUT: 2-sampTInt (2.3471, 34.853) df=10 : Sxp: 12.6342788 : Zero is not within this interval, we can conclude that there is a difference between the two means. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Battery Example continued, using TI calculator Do this: STAT→ TESTS ↓ 2-SampTInt→ STATS ↓ x1:206.0 ↓ Sx1:10.3 ↓ n1:6 ↓ x2:187.4 ↓ Sx2:14.6 ↓ n2:6 ↓ C-Level:.95 ↓ Pooled:Yes ↓ CALCULATE READOUT: 2-sampTInt (2.3471, 34.853) df=10 : Sxp: 12.6342788 : Zero is not within this interval, we can conclude that there is a difference between the two means. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Comparing Means of two related groups We are not limited to comparing two averages of independent populations. Suppose we want to compare the means of two related populations. Recall CI is pt .est ±ME pt .est = x1 − x2 ME = CVal × SE where CVal = tα/2,n−1 = invT(1− α 2 ,n − 1) SE = sdiff√ n Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Comparing Means of two related groups We are not limited to comparing two averages of independent populations. Suppose we want to compare the means of two related populations. Recall CI is pt .est ±ME pt .est = x1 − x2 ME = CVal × SE where CVal = tα/2,n−1 = invT(1− α 2 ,n − 1) SE = sdiff√ n Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Example computer stock prices We want to compare January 2002 prices vs. January 2003 prices of computer companies, see page 92. Computer Stock Prices Jan. 02 Jan. 03 Diff. x 25.91 17.96 7.946 s 6.34 5.65 6.1426 size n 5 5 5 Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computer Stock Prices Example continued What is Standard Error? What is 95% Critical Value? What is 95% Margin of Error? What is a 95% Confidence Interval? Does this confidence interval suggest a difference in stock prices between Jan. 2002 and Jan. 2003? Interpret the 95% CI Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computer Stock Prices Example answers SE = sdiff√ n = 6.1426√ 5 = 2.7471 CVal = t.025,n−1 = invT(1− .025,4) = 2.7764 ME = 2.7764× 2.7471 = 7.6271 95%CI −→ (0.3189, 15.573) Does this confidence interval suggest a difference in stock prices between Jan. 2002 and Jan. 2003? Yes, because zero is NOT within CI. Interpret the 95% CI: We are 95% confident that the true difference is between .3 and 15.6. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computer Stock Prices Example answers SE = sdiff√ n = 6.1426√ 5 = 2.7471 CVal = t.025,n−1 = invT(1− .025,4) = 2.7764 ME = 2.7764× 2.7471 = 7.6271 95%CI −→ (0.3189, 15.573) Does this confidence interval suggest a difference in stock prices between Jan. 2002 and Jan. 2003? Yes, because zero is NOT within CI. Interpret the 95% CI: We are 95% confident that the true difference is between .3 and 15.6. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computer Stock Prices Example answers using TI calculators Do this: STAT→ EDIT and Enter data into L1 and L2 then place cursor on L3, do 2nd2 − 2nd1 (i.e.,L2 − L1)→ STAT→ TESTS ↓ tInterval→ DATA ↓ List:L3 ↓ C-Level:.95 ↓ CALCULATE READOUT: TInterval (0.3189, 15.573) x = 7.946 Sx = 6.1426 n = 5 Zero is not within this interval, we can conclude that there is a difference between the two means. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion Computer Stock Prices Example answers using TI calculators Do this: STAT→ EDIT and Enter data into L1 and L2 then place cursor on L3, do 2nd2 − 2nd1 (i.e.,L2 − L1)→ STAT→ TESTS ↓ tInterval→ DATA ↓ List:L3 ↓ C-Level:.95 ↓ CALCULATE READOUT: TInterval (0.3189, 15.573) x = 7.946 Sx = 6.1426 n = 5 Zero is not within this interval, we can conclude that there is a difference between the two means. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion West Michigan Telecom Example problem 13 on page 104 Some stock market analysts have speculated that parts of West Michigan Telecom might be worth more that the whole. For example, the company’s communication systems in Ann Arbor and Detroit can be sold to other communications companies. Suppose that a stock market analyst chose nine (9) acquisition experts and asked each to predict the return (in percent) on investment (ROI) in the company held to the year 2003 if (i) it does business as usual, or (ii) if it breaks up its communication system and sells all its parts. Their predictions follow: Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion West Michigan Telecom Example, continued Expert 1 2 3 4 5 6 7 8 9 Not Break 12 21 8 20 16 5 18 21 10 Break Up 15 25 12 17 17 10 21 28 15 SE = sdiff/ √ n = 2.8626/ √ 9 = 0.9542 CVal = tα/2,n−1 = t.025,8 = invT(1− .025,8) = 2.3060 ME = 2.306× .9542 = 2.2004 95%CI −→ (1.0218, 5.4226) Does this confidence interval suggest a difference between breaking up the company or not? Yes, because zero is NOT within CI. Interpret the 95% CI: We are 95% confident that the true difference among the experts is between 1.0 and 5.4. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion West Michigan Telecom Example continued, using TI calculators STAT→ EDIT and Enter data into L1 and L2 and place cursor on L3, do 2nd2 − 2nd1 (i.e., L2 − L1) then do STAT→ TESTS ↓ tInterval→ Data ↓ List:L3 ↓ C-Level:.95 ↓ CALCULATE READOUT: TInterval (1.02, 5.42) x = 3.22 Sx: 2.8626 n = 9 Zero is not within this interval, we can conclude that there is a difference between the two means. Introduction Confidence Intervals Comparing Two Populations Confidence Interval for Proportion West Michigan Telecom Example continued, using TI calculators STAT→ EDIT and Enter data into L1 and L2 and place cursor on L3, do 2nd2 − 2nd1 (i.e., L2 − L1) then do STAT→ TESTS ↓ tInterval→ Data ↓ List:L3 ↓ C-Level:.95 ↓ CALCULATE READOUT: TInterval (1.02, 5.42) x = 3.22 Sx: 2.8626 n = 9 Zero is not within this interval, we can conclude that there is a difference between the two means.