Confidence Intervals Lecture 3 Confidence Intervals for the Population Mean (or percentage) For studies with large samples, “approximately 95% of the time, the population mean will be in the interval given by the sample mean plus or minus two standard errors.” A confidence interval is a range of values WARNING! This DOES NOT imply that for a given experiment the population parameter has a 95% chance of being in the confidence interval. Pregnancy Analogy CORRECT Interpretation: “I am 95% confident that Learning to work with the Gaussian/Normal Distribution What if I want a 90% confidence interval? We need to learn how to calculate probabilities based on the Normal Distribution! For confidence intervals, we are working with the sample mean, a random variable which approximately follows a normal distribution. We will talk now in general about a random variable, Y, which follows a normal distribution with mean μ and standard deviation σ. Framework for the Calculation Define z as the value such that (1-α)100% of the population would fall within z standard deviations of the population mean (as in the Empirical Rule). This is equivalent to saying that a single random variable has a (1-α)100% chance of falling within z standard deviations of the population mean. In the context of confidence intervals (and hypothesis tests), z is called a critical value. Then (1-α)100% of the time, the following statements are true: The Mathematics FACT: Y follows a “standard normal” distribution. The Z-score All normal calculations are based on a z-score: where Y is a normal random variable (which may be a single value, an average, etc.) The z-score measures how many standard deviations (or standard errors) an observed value of a normal random variable is away from its mean. Z-scores follow a standard normal distribution, a normal distribution with Probabilities from the Standard Normal Chapter 4 in textbook & Table A5.2 on p.366 Areas under the Normal curve represent probabilities or proportions of the population. The total area under the curve equals one. Column #2 gives the probability of falling within z standard deviations of the mean, where z is given in column #1. May be used to refer to population distributions or sampling distributions. Application of a Normal Distribution to a Population Distribution of IQ’s Suppose that I am going to measure IQ scores of UMDNJ –School of Public Health students. The national average of IQ scores 100 points and the standard deviation is 16 points. For now assume that the national average equals the average for public health students. According to the Empirical Rule, approximately 68% of scores are between 84 and 116, and 95% of scores are between 68 and 132. IQ Example continued Let’s calculate these percentages exactly. Plus or minus 1 st. dev.: 68.27% = proportion of .6827 Plus or minus 2 st. dev.: 95.45% = proportion of .9545 The probability of a single observed IQ falling within one standard deviation of the mean is equivalent to the proportion of all IQ’s that fall within one standard deviation of the mean. Probabilities for Single Observations Calculate the probability that the IQ score of one public health student will be within 10 points of the national average. Distance between mean and observation=10 Z-score = 10/16 = 0.625 Std. Dev.’s Probability = 0.4679 Probabilities for Sample Means (Sampling Distributions) Suppose I do repeated experiments (studies) in which I draw 10 students for each study. Standard error for a mean of 10 measurements SEM = 16/(square root of 10) = 5.06 Therefore, in approximately 68% of these experiments, the average score will be between (100-5.06=) 94.94 and 105.06, and 95% of these experiments, the average score will be between (100-2*5.06=) 89.88 and 110.12. Equivalently, For any one experiment, we have a 95% chance that the sample mean will fall between 89.88 and 110.12. Sampling Average IQ’s, cont. Calculate the probability that the average IQ score of 10 public health students will be within 10 points of the national average given that the public health distribution of IQ’s is the same as the national distribution. Standard error for a mean of 10 measurements • • SEM = 16/(square root of 10) = 5.060 Z-score = 10/5.060 = 1.976 Probability within ± 1.976 st.dev.s = about 95% Much larger probability than when looking at at single measurement! 95% Large Sample Confidence Interval Suppose that I sample 50 public health students and find that Sample Mean = 110.7 points And the Standard Error Estimated from Sample = SEM = 2.5 points What’s the value of z such that 95% of sample means would fall within z standard errors of the population mean? z = 1.96 Therefore a 95% confidence interval is 110.7 ± 1.96(2.5) = (105.800, 115.600) Interpretation: We are 95% confident that the mean IQ for all public health students falls between 105.8 and 115.6 points. How likely is it that the mean for the public health students is the same as that the national mean (100)? 90% Large Sample Confidence Interval Sample Mean = 110.7 points Standard Error of the Mean = SEM = 2.5 points z = 1.645 Therefore a 90% confidence interval is 110.7 ± 1.645(2.5) = (106.588, 114.813) Interpretation: What Confidence Level should I use? The more confidence we would like to claim in our interval estimate The standard confidence people use is 95%. Smaller confidence levels (such as 90%) may be appropriate, especially for pilot studies or other studies with small sample sizes. Larger confidence levels may also be appropriate when costly reforms rest on the conclusions from those confidence intervals. Large Sample Confidence Interval for a Population Mean Sample mean ± z SEM Where SEM = square root of s2/n Typically, 95% confidence intervals are used, with z =1.96 Here, 1.96 is the 97.5th percentile of the standard normal distribution To be modified for smaller sample sizes shortly Overweight Status and Eating Patterns… (AJPH, 2002) 95% CI for mean BMI of girls is 23.3 +/- 1.96*4.9/sqrt(2000) = (23.09, 23.51) 95% CI for mean BMI of boys is 23.0 +/- 1.96*4.8/sqrt(2000) = (22.79, 23.21) Unfortunately, we have target percentiles for BMI, not target means. Large Sample Confidence Intervals for Population Proportions p ± z (SE) Where the estimated variance of the sample proportion is SE2 = p(1-p)/n Typically, 95% confidence intervals are used, with z =1.96 Actual Percent in the Top Targeted 5th Percentile – 95% Confidence Interval For girls, p = .125, n=2099 SE = sqrt((.125*.875)/2099) = .007219 p ± z (SE) (0.111, .139) (11.1%, 13.9%) For boys, p = .166, n=2141 SE = sqrt((.166*.834)/2141) = .008041 p ± z (SE) (0.150, .182) (15.0%, 18.2%) I am 95% confident that the percent of boys that is in the targeted upper 5th percentile is actually between 15.0% and 18.2% of boys. “Men Carrying Pollutant Have More Boys” For description, see news article. n =101 p = .57 p(1 p) .57(.43) SE = .049 n 101 Thus a 95% confidence interval for the proportion is given by .57 ± 1.96 (.049) Or equivalently (0.473, 0.667) Interpretation for Pollutant – Boys E.g. I am 95 % confident that the true proportion of boy babies born from parents who both have detectable PCB levels in their blood is between 0.473 and 0.667. I.e., if this is one of the 95% of the times that the true parameter falls in the interval, then the mean is between 0.473 and 0.667. If the proportion of of boys in the general population is 0.51, is there a difference in the proportion of boys from parents with detectable PCB levels? Assumptions for “Large Sample CI’s” Random sample • Every individual in the population has an equal chance of being selected. Independent observations • Selecting one subject does not alter the chances of selection of another subject Large Sample Size – so that the Central Limit Theorem “kicks in” and the standard error is accurately estimated • How large is “large”? • Depends! • If population distribution is approximately normal, then “large” is approximately 40 subjects. • The farther the population distribution away from normal, the larger the needed size of the sample. … E.g., binary variables- at least 30 subjects – 5 in each category. Other Assumptions Other Assumptions • • • Study Population = Target Population Variable accurately measures characteristic of interest Values are correctly measured and recorded Quick Re-evaluation of Large Sample CI’s for Population Means Why do we use a z-critical value in the CI? Due to the CLT, we can say that if the sample size is large, the sample mean will fall within the specified number (z) of standard errors of the mean. More specifically, the 95% CI is derived from noting z Y / n falls between –1.96 and +1.96. • The standard deviation is assumed to be known. CI’s for Population Means when the Sample Sizes are Smaller ISSUE: The standard deviation is unknown and, hence, it is estimated. This estimate is generally OK for large sized samples, but not for smaller sized samples. PROBLEM: There is more variation in the z-score with an estimated standard deviation than is allowed for by the Normal distribution. Thanks to a smart person! SOLUTION: • • In 1908, William Sealy Gosset working at the Guinness Brewery in Dublin, Ireland showed, mathematically, that the z-score for the sample average in which the population standard deviation has been replaced by the sample standard deviation follows a “t-distribution” with a specified degrees of freedom (d.f.). I.e., y t s/ n T-Distribution What does it look like? • • Like the Normal distribution With larger variance, depending on the d.f. What are “degrees of freedom.” • • • It is difficult to define “Amount of information about the variance.” Practically, for the sample mean, d.f. = n-1 Using the T-distribution for CI’s Assumptions: • • Approximately normal, Random and independent sample d.f. = n-1 CI: Y t * s n See Table A5.3 on page 368 for a table of critical values for 90%, 95% and 99% CI’s. Cadmium levels (ng/gram) in mothers The sample sizes are 14 (for smoking) and 18 (for non-smoking mothers). Distributions are bimodal. CI using the t critical value Means and standard errors are: • • Smoking: 20.414 & 6.814/sqrt(14) ng/gram Non-smoking: 14.722 & 6.199/sqrt(18) ng/gram t-critical values for 95% confidence intervals are 2.160 and 2.110 respectively for 13 and 17 degrees of freedom Cadmium Confidence Intervals • • • The confidence intervals are: Smoking: 20.414 ± 2.160 (1.821) (16.481, 24.347) Non-smoking: 14.722 ± 2.110 (1.461) (11.639, 17.805) These intervals overlap, suggesting that the mean cadmium levels for the populations of smoking and non-smoking mothers are not different. After the midterm, we will talk about more specific ways to test this hypothesis. (Comparison of two population means) HELP: When do I use which CI? In general for means, use the t-interval. • • • For means (not proportions), the t-interval is robust to violations of the Normality assumption. If the sample distribution is far from normal (as determined by, e.g., histograms), non-parametric or other more “exact” methods may be needed. These are easier to discuss in the context of hypothesis testing. When do I use which CI?, cont… For proportions with large sample sizes, use the z-interval. • “Large” = 30 with between 5 and 25 subjects in the group of interest • Is the pollutant-boys sample size OK? • 101 subjects with at least 43 of each gender. For proportions with small sample sizes, there exist “exact” confidence intervals. See Table A5.1 & pp.16-18 Assumptions for all CI’s presented so far Observations within the sample are independent of one another. Sample consists of randomly selected subjects that are representative of the target population. • In particular, each subject in the study population has an equal chance of being selected.