The confidence interval is used to estimate the probability that the true population mean lies within a range around a sample mean. For continuous data with large sample size, we use this formula: Mu equals X bar plus or minus Z sub Alpha divided by 2 times Sigma over the square root of n, where Mu is the true population mean which is unknown, X bar is the sample mean, n is the number of samples, Sigma is the population standard deviation, and Z sub Alpha over 2 is the normal distribution value for the desired confidence interval. The average of 100 samples is 18. With the population standard deviation of 6, calculate the 95 percent confidence interval for the population mean. The Z sub Alpha over 2 value from the Z table is 1.96 plus or minus. Plugging into the formula yields: 18 plus or minus 1.176. The result gives us a confidence interval or a range that the true population mean could exist with 95 percent confidence between 16.82 and 19.18. For confidence intervals for the mean of continuous data small sample, we use this formula: Mu equals X bar plus or minus t sub Alpha over 2 times S over the square root of n, where Mu is the true population mean which is unknown, X bar is the sample mean, n is the number of samples, S is the sample standard deviation, and t sub Alpha over 2 is the t-distribution value for the desired confidence interval with degrees of freedom n minus 1. In this example, the average of 25 samples is 18, with the population standard deviation of 6, calculate the 95 percent confidence interval for the population mean. The t sub Alpha over 2 value from the t-table using df equal 24 and t sub 0.025 is plus or minus 2.064. Plugging into the formula yields: 18 plus or minus 2.48. The result gives us a confidence interval or range that the true population mean could exist with 95 percent certainty between the values of 15.52 and 20.48. The confidence intervals for the mean were symmetrical around the average, as seen in our last two examples. This is not true for variance, since it is based on the chi-square distribution. We use this formula: n minus 1 times S squared over chi-square to the Alpha over 2 and n minus 1 degrees of freedom is less than or equal Sigma squared, which is less than or equal 2, n minus 1 times S squared all over chi-square to the 1 minus Alpha over 2 with n minus 1 degrees of freedom, where n is the number of samples, S squared is the point estimates of the variance, and both chi-square 1 minus Alpha over 2 and Alpha over 2 are the table values for n minus 1 degrees of freedom. Let's take an example. The sample variance for a set of 25 samples was found to be 36. Calculate the 90 percent confidence interval for the population variance. The chi-square Alpha over 2 value for chi-square 0.05 from the chi-square table using df equal 24 yields 36.415. The chi-square sub 1 minus Alpha over 2 value or chi-square 0.95 from the chi-square table using df equal 24 yields 13.848. Plugging into the formula yields: a range between 23.72 and 62.38. This result gives us the confidence interval or a range that the true population variance could exist, with 90 percent certainty. The confidence intervals for proportions is needed when we take a sample, the proportion of the sample into categories. This is typically expressed as a percent, such as "my class has 40 percent females". We would need a large sample size of n times p and n times 1 minus p to be greater than or equal to 4. We can calculate the confidence interval to help estimate the entire population proportion for all classes. Assuming a normal distribution and random sampling, we use this formula: p plus or minus Z sub Alpha over 2 times the square root of p times 1 minus p over n, where p is the population proportion estimate, n is the number of samples, and Z sub Alpha over 2 is the appropriate confidence level from the Z table. For example, if 16 defective parts were found in a sample size of 200 units, calculate the 90 percent confidence interval for the population proportion. So we calculate p equal 16 over 200 equal 0.08 and both n times p and n times 1 minus p are greater than 4, so we can use our formula. At a 90 percent or an Alpha equal 0.10, two-tailed Z sub Alpha over 2 from the Z-table yields 1.645. We plug it into our formula, and we get 0.08 plus or minus point 0.032. So we have a range now of 0.048 is less than p, is less than 0.112. With the confidence interval between these values, with 90 percent certainty, that the true population proportion could exist.