Copyright 2009, The Johns Hopkins University and John McGready. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site.John McGready Johns Hopkins University Sampling Variability and Confidence IntervalsLecture Topics  Sampling distribution of a sample mean  Variability in the sampling distribution  Standard error of the mean  Standard error vs. standard deviation  Confidence intervals for the population mean µ  Sampling distribution of a sample proportion  Standard error for a proportion  Confidence intervals for a proportion 3The Random Sampling Behavior of a Sample Mean Across Multiple Random Samples Section ARandom Sample  When a sample is randomly selected from a population, it is called a random sample - Technically speaking values in a random sample are representative of the distribution of the values in the population sample, regardless of size  In a simple random sample, each individual in the population has an equal chance of being chosen for the sample  Random sampling helps control systematic bias  But even with random sampling, there is still sampling variability or error 5Sampling Variability of a Sample Statistic  If we repeatedly choose samples from the same population, a statistic will take different values in different samples  If the statistic does not change much if you repeated the study (you get similar answers each time), then it is fairly reliable (not a lot of variability) 6Example: Blood Pressure of Males  Recall, we had worked with data on blood pressures using a random sample of 113 men taken from the population of all men  Assume the population distribution is given by the following: 7 µBP = 125 mmHg σBP = 14 mmHgExample: Blood Pressure of Males  Suppose we had all the time in the world  We decide to do an experiment  We are going to take 500 separate random samples from this population of men, each with 20 subjects  For each of the 500 samples, we will plot a histogram of the sample BP values, and record the sample mean and sample standard deviation  Ready, set, go . . . 8Random Samples  Sample 1: n = 20  Sample 2: n = 20 9 = 125.7 mmHg = 10.9 mmHg = 122.6 mmHg = 12.7 mmHgExample: Blood Pressure of Males  So we did this 500 times: now let’s look at a histogram of the 500 sample means 10 = 125 mmHg = 3.3 mmHgExample: Blood Pressure of Males  We decide to do another experiment  We are going to take 500 separate random samples from this population of me, each with 50 subjects  For each of the 500 samples, we will plot a histogram of the sample BP values, and record the sample mean and sample standard deviation  Ready, set, go . . . 11Random Samples  Sample 1: n = 50  Sample 2: n = 50 12 = 126.7 mmHg = 11.5 mmHg = 125.5 mmHg = 14.0 mmHgExample: Blood Pressure of Males  So we did this 500 times: now let’s look at a histogram of the 500 sample means 13 = 125 mmHg = 1.9 mmHgExample: Blood Pressure of Males  We decide to do one more experiment  We are going to take 500 separate random samples from this population of men, each with 100 subjects  For each of the 500 samples, we will plot a histogram of the sample BP values, and record the sample mean, and sample standard deviation  Ready, set, go . . . 14Random Samples  Sample 1: n = 100  Sample 2: n = 100 15 = 123.3 mmHg = 15.2 mmHg = 125.7 mmHg = 13.2 mmHgExample: Blood Pressure of Males  So we did this 500 times: now let’s look at a histogram of the 500 sample means 16 = 125 mmHg = 1.4 mmHgExample: Blood Pressure of Males  Let’s review the results - Population distribution of individual BP measurements for males: normal - True mean µ = 125 mmHg: σ = 14 mmHg - Results from 500 random samples: 17 Sample Sizes Means of 500 Sample Means SD of 500 Sample Means Shape of Distribution of 500 sample means n = 20 125 mmHg 3.3 mm Hg Approx normal n = 50 125 mmHg 1.9 mm Hg Approx normal n = 100 125 mmHg 1.4 mm Hg Approx normalExample: Blood Pressure of Males  Let’s review the results 18Example 2: Hospital Length of Stay  Recall, we had worked with data on length of stay (LOS) using a random sample of 500 patients taken from sample of all patients discharged in year 2005  Assume the population distribution is given by the following: 19 µLOS = 5.0 days σLOS = 6.9 daysExample 2: Hospital Length of Stay  Boxplot presentation 20 25th percentile: 1.0 days 50th percentile: 3.0 days 75th percentile: 6.0 daysExample 2: Hospital Length of Stay  Suppose we had all the time in the world again  We decide to do another set of experiments  We are going to take 500 separate random samples from this population of patients, each with 20 subjects  For each of the 500 samples, we will plot a histogram of the sample LOS values, and record the sample mean and sample standard deviation  Ready, set, go . . . 21Random Samples  Sample 1: n = 20  Sample 2: n = 20 22 = 6.6 days = 9.5 days = 4.8 days = 4.2 daysExample 2: Hospital Length of Stay  So we did this 500 times: now let’s look at a histogram of the 500 sample means 23 = 5.05 days = 1.49 daysExample 2: Hospital Length of Stay  Suppose we had all the time in the world again  We decide to do one more experiment  We are going to take 500 separate random samples from this population of me, each with 50 subjects  For each of the 500 samples, we will plot a histogram of the sample LOS values, and record the sample mean and sample standard deviation  Ready, set, go . . . 24Random Samples  Sample 1: n = 50  Sample 2: n = 50 25 = 3.3 days = 3.1 days = 4.7 days = 5.1