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Chapter 12 More About Confidence Intervals Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Recall: • A parameter is a population characteristic – value is usually unknown. We estimate the parameter using sample information. • A statistic, or estimate, is a characteristic of a sample. A statistic estimates a parameter. • A confidence interval is an interval of values computed from sample data that is likely to include the true population value. • The confidence level for an interval describes our confidence in the procedure we used. We are confident that most of the confidence intervals we compute using a procedure will contain the true population value. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 2 12.1 Examples of Different Estimation Situations Situation 1. Estimating the proportion falling into a category of a categorical variable. Example research questions: What proportion of American adults believe there is extraterrestrial life? In what proportion of British marriages is the wife taller than her husband? Population parameter: p = proportion in the population falling into that category. Sample estimate: p̂ = proportion in the sample falling into that category. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 3 More Estimation Situations Situation 2. Estimating the mean of a quantitative variable. Example research questions: What is the mean time that college students watch TV per day? What is the mean pulse rate of women? Population parameter: m (spelled “mu” and pronounced “mew”) = population mean for the variable Sample estimate: x = the sample mean for the variable Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 More Estimation Situations Situation 3. Estimating the difference between two populations with regard to the proportion falling into a category of a qualitative variable. Example research questions: How much difference is there between the proportions that would quit smoking if taking the antidepressant buproprion (Zyban) versus if wearing a nicotine patch? How much difference is there between men who snore and men who don’t snore with regard to the proportion who have heart disease? Population parameter: p1 – p2 = difference between the two population proportions. Sample estimate: pˆ1 pˆ 2 = difference between the two sample proportions. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 More Estimation Situations Situation 4. Estimating the difference between two populations with regard to the mean of a quantitative variable. Example research questions: How much difference is there in average weight loss for those who diet compared to those who exercise to lose weight? How much difference is there between the mean foot lengths of men and women? Population parameter: m1 – m2 = difference between the two population means. Sample estimate: x1 x2 = difference between the two sample means. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 6 Independent Samples Two samples are called independent samples when the measurements in one sample are not related to the measurements in the other sample. • Random samples taken separately from two populations and same response variable is recorded. • One random sample taken and a variable recorded, but units are categorized to form two populations. • Participants randomly assigned to one of two treatment conditions, and same response variable is recorded. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 7 Paired Data: A Special Case of One Mean Paired data (or paired samples): when pairs of variables are collected. Only interested in population (and sample) of differences, and not in the original data. • Each person measured twice. Two measurements of same characteristic or trait are made under different conditions. • Similar individuals are paired prior to an experiment. Each member of a pair receives a different treatment. Same response variable is measured for all individuals. • Two different variables are measured for each individual. Interested in amount of difference between two variables. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 8 12.2 Standard Errors Rough Definition: The standard error of a sample statistic measures, roughly, the average difference between the statistic and the population parameter. This “average difference” is over all possible random samples of a given size that can be taken from the population. Technical Definition: The standard error of a sample statistic is the estimated standard deviation of the sampling distribution for the statistic. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 9 Standard Error of a Sample Proportion s.e.( pˆ ) pˆ 1 pˆ , n pˆ sample proportion Example 12.1 Intelligent Life on Other Planets Poll: Random sample of 935 Americans Do you think there is intelligent life on other planets? Results: 60% of the sample said “yes”, p̂ = .60 .61 .6 s.e. pˆ .016 935 The standard error of .016 is roughly the average difference between the statistic, p̂ , and the population parameter, p, for all possible random samples of n = 935 from this population. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 10 Standard Error of a Sample Mean s s.e.( x ) , s sample standard deviation n Example 12.2 Mean Hours Watching TV Poll: Class of 175 students. In a typical day, about how much time to you spend watching television? Variable N Mean Median TrMean StDev TV 175 2.09 2.000 1.950 1.644 SE Mean 0.124 s 1.644 s.e.x .124 n 175 Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 11 Standard Error of the Difference Between Two Sample Proportions pˆ1 1 pˆ1 pˆ 2 1 pˆ 2 s.e.( pˆ pˆ ) 1 2 n1 n2 Example 12.3 Patches vs Antidepressant (Zyban)? Study: n1 = n2 = 244 randomly assigned to each treatment Zyban: 85 of the 244 Zyban users quit smoking p̂1 = .348 Patch: 52 of the 244 patch users quit smoking p̂2= .213 So, pˆ1 pˆ 2 .348 .213 .135 .3481 .348 .2131 .213 and s.e.( pˆ1 pˆ 2 ) .040 244 244 Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 12 Standard Error of the Difference Between Two Sample Means s.e.( x1 x2 ) s12 s22 n1 n2 Example 12.4 Lose More Weight by Diet or Exercise? Study: n1 = 42 men on diet, n2 = 47 men on exercise routine Diet: Lost an average of 7.2 kg with std dev of 3.7 kg Exercise: Lost an average of 4.0 kg with std dev of 3.9 kg So, x1 x2 7.2 4.0 3.2 kg and s.e.( x1 x2 ) 3.7 2 3.92 42 Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 47 0.81 13 12.3 Approximate 95% CI For sufficiently large samples, the interval Sample estimate 2 Standard error is an approximate 95% confidence interval for a population parameter. Note: The 95% confidence level describes how often the procedure provides an interval that includes the population value. For about 95% of all random samples of a specific size from a population, the confidence interval captures the population parameter. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 14 Necessary Conditions Sample Size Requirements: • For one proportion: Both npˆ and n1 pˆ are at least 5, preferably at least 10. • For one mean: n is greater than 30. • For two proportions: npˆ and n1 pˆ are at least 5 (preferably 10) for each sample. • For two means: n1 and n2 are each greater than 30. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 15 Necessary Conditions Other Requirements: • The samples are randomly selected. In practice, it is sufficient to assume that samples are representative of the population for the question of interest. • For the confidence intervals for the difference between two proportions or two means, the two samples must be independent of each other. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 16 Example 12.1 Intelligent Life? (cont) Poll: Random sample of 935 Americans Do you think there is intelligent life on other planets? Results: 60% of the sample said “yes”, p̂ = .60 .61 .6 s.e. pˆ .016 935 Approximate 95% Confidence Interval: .60 2(.016) => .60 .032 => .568 to .632 Note: For about 95% of all random samples from the population, the corresponding confidence interval captures the population parameter. We don’t know if particular interval does or does not capture the population value. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 17 Example 12.2 Watching TV (cont) Poll: Class of 175 students. In a typical day, about how much time do you spend watching television? The sample mean was 2.09 hours and the sample standard deviation was 1.644 hours. s 1.644 s.e.x .124 n 175 Approximate 95% Confidence Interval: 2.09 2(.124) => 2.09 .248 => 1.842 to 2.338 hours Note: We are 95% confident that the mean time that Penn State students spend watching television per day is somewhere between 1.842 and 2.338 hours. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 18 Example 12.3 Patch vs Antidepressant (cont) Study: n1 = n2 = 244 randomly assigned to each group Zyban: 85 of the 244 Zyban users quit smoking p̂1 = .348 Patch: 52 of the 244 patch users quit smoking p̂2= .213 So, pˆ1 pˆ 2 .348 .213 .135 and s.e.( pˆ1 pˆ 2 ) .040 Approximate 95% Confidence Interval: .135 2(.040) => .135 .080 => .055 to .215 Note: Zyban had a higher success rate and the interval does not include the value 0, so it supports a difference between the success rates of the two methods. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 19 Example 12.4 Diet vs Exercise (cont) Study: n1 = 42 men on diet, n2 = 47 men exercise Diet: Lost an average of 7.2 kg with std dev of 3.7 kg Exercise: Lost an average of 4.0 kg with std dev of 3.9 kg So, x1 x2 7.2 4.0 3.2 kg and s.e.( x1 x2 ) 0.81 kg Approximate 95% Confidence Interval: 3.2 2(.81) => 3.2 1.62 => 1.58 to 4.82 kg Note: We are 95% confident the interval 1.58 to 4.82 kg covers the increased mean population weight loss for dieters compared to those who exercise. The interval does not cover 0, so a real difference is likely to hold for the population. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 20 12.4 General CI for One Mean or Paired Data A Confidence Interval for a Population Mean s x t s.e.x x t n * * where the multiplier t* is the value in a t-distribution with degrees of freedom = df = n - 1 such that the area between -t* and t* equals the desired confidence level. (Found from Table A.2.) Conditions: • Population of measurements is bell-shaped and a random sample of any size is measured; OR • Population of measurements is not bell-shaped, but a large random sample is measured, n 30. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 21 Example 12.5 Mean Forearm Length Data: Forearm lengths (cm) for a random sample of n = 9 men 25.5, 24.0, 26.5, 25.5, 28.0, 27.0, 23.0, 25.0, 25.0 Note: Dotplot shows no obvious skewness and no outliers. s 1.52 x 25.5, s 1.52, and s.e.x .507 n 9 Multiplier t* from Table A.2 with df = 8 is t* = 2.31 95% Confidence Interval: 25.5 2.31(.507) => 25.5 1.17 => 24.33 to 26.67 cm Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 22 Example 12.6 What Students Sleep More? Q: How many hours of sleep did you get last night, to the nearest half hour? Class N Stat 10 (stat literacy) 25 Stat 13 (stat methods) 148 Mean StDev SE Mean 7.66 1.34 0.27 6.81 1.73 0.14 Note: Bell-shape was reasonable for Stat 10 (with smaller n). Notes: Interval for Stat 10 is wider (smaller sample size) Two intervals do not overlap => Stat 10 average significantly higher than Stat 13 average. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 23 Paired Data Confidence Interval Data: two variables for n individuals or pairs; use the difference d = x1 – x2. Population parameter: md = mean of differences for the population = m1 – m2. Sample estimate: d = sample mean of the differences Standard deviation and standard error: sd = standard deviation of the sample of differences; sd s.e.d n Confidence interval for md: d t * s.e. d , where df = n – 1 for the multiplier t*. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 24 Example 12.7 Screen Time: Computer vs TV Data: Hours spent watching TV and hours spent on computer per week for n = 25 students. Task: Make a 90% CI for the mean difference in hours spent using computer versus watching TV. Note: Boxplot shows no obvious skewness and no outliers. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 25 Example 12.7 Screen Time: Computer vs TV Results: sd 15.24 d 5.36, sd 15.24, and s.e.d 3.05 n 25 Multiplier t* from Table A.2 with df = 24 is t* = 1.71 90% Confidence Interval: 5.36 1.71(3.05) => 5.36 5.22 => 0.14 to 10.58 hours Interpretation: We are 90% confident that the average difference between computer usage and television viewing for students represented by this sample is covered by the interval from 0.14 to 10.58 hours per week, with more hours spent on computer usage than on television viewing. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 26 12.5 General CI for Difference Between Two Means (Indep) A CI for the Difference Between Two Means (Independent Samples): x1 x2 t * s12 s22 n1 n2 where t* is the value in a t-distribution with area between -t* and t* equal to the desired confidence level. The df used depends on if equal population variances are assumed. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 27 Necessary Conditions • Two samples must be independent. Either … • Populations of measurements both bell-shaped, and random samples of any size are measured. or … • Large (n 30) random samples are measured. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 28 Degrees of Freedom The t-distribution is only approximately correct and df formula is complicated (Welch’s approx): Statistical software can use the above approximation, but if done by-hand then use a conservative df = smaller of n1 – 1 and n2 – 1. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 29 Example 12.8 Effect of a Stare on Driving Randomized experiment: Researchers either stared or did not stare at drivers stopped at a campus stop sign; Timed how long (sec) it took driver to proceed from sign to a mark on other side of the intersection. No Stare Group (n = 14): 8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8, 7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7 Stare Group (n = 13): 5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5, 6.1, 4.8, 4.9, 4.5, 7.2, 5.8 Task: Make a 95% CI for the difference between the mean crossing times for the two populations represented by these two independent samples. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 30 Example 12.8 Effect of a Stare on Driving Checking Conditions: Boxplots show … • No outliers and no strong skewness. • Crossing times in stare group generally faster and less variable. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 31 Example 12.8 Effect of a Stare on Driving Note: The df = 21 was reported by the computer package based on the Welch’s approximation formula. The 95% confidence interval for the difference between the population means is 0.14 seconds to 1.93 seconds . Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 32 Equal Variance Assumption Often reasonable to assume the two populations have equal population standard deviations, or equivalently, equal population variances: 12 22 2 Estimate of this variance based on the combined or “pooled” data is called the pooled variance. The square root of the pooled variance is called the pooled standard deviation: Pooled standard deviation s p Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. n1 1s12 n2 1s22 n1 n2 2 33 Pooled Standard Error Pooled s.e.( x1 x2 ) s 2p n1 s 2p n2 1 1 s n1 n2 2 p sp 1 1 n1 n2 Note: Pooled df = (n1 – 1) + (n2 – 1) = (n1 + n2 – 2). Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 34 Pooled Confidence Interval Pooled CI for the Difference Between Two Means (Independent Samples): 1 1 x1 x2 t s p n1 n2 * where t* is found using a t-distribution with df = (n1 + n2 – 2) and sp is the pooled standard deviation. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 35 Example 12.9 Male and Female Sleep Times Q: How much difference is there between how long female and male students slept the previous night? Data: The 83 female and 65 male responses from students in an intro stat class. Task: Make a 95% CI for the difference between the two population means sleep hours for females versus males. Note: We will assume equal population variances. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 36 Example 12.9 Male and Female Sleep Times Two-sample T for sleep [with “Assume Equal Variance” option] Sex Female Male N 83 65 Mean 7.02 6.55 StDev 1.75 1.68 SE Mean 0.19 0.21 Difference = mu (Female) – mu (Male) Estimate for difference: 0.461 95% CI for difference: (-0.103, 1.025) T-Test of difference = 0 (vs not =): T-Value = 1.62 P = 0.108 DF = 146 Both use Pooled StDev = 1.72 Notes: • Two sample standard deviations are very similar. • Sample mean for females higher than for males. • 95% confidence interval contains 0 so cannot rule out that the population means may be equal. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 37 Example 12.9 Male and Female Sleep Times Pooled standard deviation and pooled standard error “by-hand”: Pooled std dev s p n1 1s12 n2 1s22 n1 n2 2 83 11.752 65 11.682 83 65 2 2.957 1.72 Pooled s.e.( x1 x2 ) s p 1.72 1 1 n1 n2 1 1 0.285 83 65 Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 38 Pooled or Unpooled? • If sample sizes are equal, the pooled and unpooled standard errors are equal. If sample standard deviations similar, assumption of equal population variance is reasonable and pooled procedure can be used. • If sample sizes are very different, pooled test can be quite misleading unless sample standard deviations are similar. If the smaller standard deviation accompanies the larger sample size, we do not recommend using the pooled procedure. • If sample sizes are very different, the standard deviations are similar, and the larger sample size produced the larger standard deviation, the pooled procedure is acceptable because it will be conservative. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 39 12.5 The Difference Between Two Proportions (Indep) A CI for the Difference Between Two Proportions (Independent Samples): p1 p2 z * p1 1 p1 p2 1 p2 n1 n2 where z* is the value of the standard normal variable with area between -z* and z* equal to the desired confidence level. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 40 Necessary Conditions • Condition 1: Sample proportions are available based on independent, randomly selected samples from the two populations. • Condition 2: All of the quantities – n1 pˆ1 , n1 1 pˆ1 , n2 pˆ 2 , and n2 1 pˆ 2 – are at least 5 and preferably at least 10. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 41 Example 12.10 Snoring and Heart Attacks Q: Is there a relationship between snoring and risk of heart disease? Data: Of 1105 snorers, 86 had heart disease. Of 1379 nonsnorers, 24 had heart disease. 86 24 pˆ 1 .0778, pˆ 2 .0174, and pˆ 1 pˆ 2 .0604 1105 1379 .07781 .0778 .01741 .0174 and s.e.( pˆ1 pˆ 2 ) .0088 1105 1379 Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 42 Example 12.10 Snoring and Heart Attacks Note: the higher the level of confidence, the wider the interval. It appears that the proportion of snorers with heart disease in the population is about 4% to 8% higher than the proportion of nonsnorers with heart disease. pˆ1 .0778 4.5 pˆ 2 .0174 Risk of heart disease for snorers is about 4.5 times what the risk is for nonsnorers. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 43