Confidence Intervals 1 Estimates 2 2 CIs for Population Proportions 3 3 CIs for Population Means 8 4 Minimum Required Sample Sizes www apsu edu jonesmatt 1530 htm 16 1 1 Estimates A point estimate is a one number estimate of a parameter Example 1 The sample mean x is a point estimate for the population mean The sample standard deviation s is a point estimate for the population standard deviation The sample proportion p is a point estimate for the population proportion p A confidence interval estimate CI is an interval paired with a confidence level indicating how sure we are that the parameter is actually within the interval Example 2 We can be 95 confident the average height of all adult men is within the interval 54 90 We can be 99 confident that the proportion of women who smoke is between 1 and 97 www apsu edu jonesmatt 1530 htm 2 2 CIs for Population Proportions Recall that 1 The mean of p is the population proportion p just as the mean of X is the population mean 2 The standard deviation of p is r p p 1 p n p just as n is the standard deviation of X Note that np 1 p is the standard deviation of a Binomial n p random variable 3 By the CLT when the sample size n is large p is distributed p approximately normal p p 1 p n just as X is approximately normal n www apsu edu jonesmatt 1530 htm 3 This means if we take a large simple random sample there is about a 90 probability the actual proportion is within the random interval r p z0 05 p 1 p n r p z0 05 p 1 p n and about a 95 probability the actual proportion is within the random interval r p z0 025 p 1 p n r p z0 025 p 1 p n and about a 1 100 probability the actual proportion is within the random interval r p z 2 p 1 p n www apsu edu jonesmatt 1530 htm r p z 2 p 1 p n 4 CIs for Population Proportions Necessary Assumptions a Simple random sampling b np 15 and n 1 p 15 meaning the number of observed successes is at least 15 and the number of observed failures is at least 15 This is a rule of thumb We can be 1 100 confident the actual population proportion p is between the numbers standard error point estimate z p z r p 1 p z 2 z n margin of error Note p above is no longer random it is the sample proportion computed from collected data Thus the interval above isn t random so the probability this interval contains p is 0 or 1 Thus we use the word confident instead of probability Also p isn t known so we use p in its place www apsu edu jonesmatt 1530 htm 5 Example 3 Of 1010 U S employees 202 play hooky at least once per year Find a 95 CI for the population proportion p of all U S employees who play hooky What is the margin of error Example 4 In a random sample of 345 cans of Vietti Chili 2 were dented during canning Can you find a 97 CI for the actual proportion of dented cans of chili What should you do What is the margin of error www apsu edu jonesmatt 1530 htm 6 What is the Meaning of a CI 95 of all 95 CIs for a population proportion actually contain the true population proportion 71 of all 71 CIs for a population proportion actually contain the true population proportion 64 of all 64 CIs for a population proportion actually contain the true population proportion Example 5 Simulate 100 students guessing on multiple choice exams with 100 questions where each question has 5 possible answers Make one 95 CI per student What is the theoretical proportion of correct answers Approximately what percentage of the CIs should contain the actual proportion of correct answers www apsu edu jonesmatt 1530 htm 7 3 CIs for Population Means If a large sample of size n is taken from a distribution with mean and standard deviation X z 2 1 P z 2 Z z 2 P z 2 n P z 2 X z 2 n n P X z 2 X z 2 n n P X z 2 X z 2 n n This means the random interval X z 2 X z 2 n n will contain with probability approximately 1 www apsu edu jonesmatt 1530 htm 8 t Distribution is rarely known so we fudge a little using s in its place But X S n does not have a standard normal distribution it has a t distribution with n 1 degrees of freedom provided the sample size is large or the population distribution is normal So if a random sample of size n is taken from a distribution with mean and either the population is known to have a normal distribution or n is large S S 1 P X t 2 X t 2 n n This means the random interval S S X t 2 X t 2 n n will contain with probability approximately 1 www apsu edu jonesmatt 1530 htm 9 The t distribution looks similar to the normal but has more spread The total area under the curve is 1 The curve never touches the x axis The curve is symmetric about 0 As the degrees of freedom df increases the density function looks more like that of a standard normal See Excel Demos www apsu edu jonesmatt 1530 htm 10 Recall that is the probability that a standard normal random variable is larger than z Similarly is the probability that a random variable with the t distribution is larger than t www apsu edu jonesmatt 1530 htm 11 Obtain t values with a t table or your TI 84 by doing t invT 1 df Example 6 Calculate the following t0 05 for df 5 t0 05 for df 50 t0 05 for df 100 z0 05 t0 025 for df 5 t0 025 for df 50 t0 025 for df 100 z0 025 www apsu edu jonesmatt 1530 htm 12 CI for a Population Mean First you must satisfy these assumptions 1 Simple random sampling 2 Population has a normal distribution or the sample size is large We can be 1 100 confident that the population mean is between the numbers standard error point estimate z z s x t 2 n z margin of error Remark 1 100 of all 1 CIs will contain the population mean www apsu edu jonesmatt 1530 htm 13 Example 7 The prices in thousands of dollars of thirty six new mobile homes are 53 8 54 4 45 2 42 9 49 9 48 2 41 6 58 9 48 6 53 1 59 4 49 7 43 7 52 7 47 7 41 5 35 3 58 9 35 9 42 5 57 2 45 1 50 3 …