In this situation, there are different formula for the mean and standard error, but the same logic and procedure for solving problems remains the same.
When (as is usually the case) the population standard deviations are unknown, the standard error must be estimated and a multiplier from a t-distribution should be used. The textbook presents two ways of doing this, neither of which is currently believed to be the best way to handle the problem. These notes will demonstrate an alternative method.
Examples will demonstrate the methodology.
and SE() =
The shape will be approximately normal for sufficiently large samples. For most practical applications, this will hold if each sample has at least 25 or 30 observations.
When the population standard deviations are known, and the samples are sufficiently large, plug into the general confidence expression
(estimate) +/- (reliability coefficient)(standard error)
The estimate for the difference in population means is the difference in sample means, 19.2 - 15.5 = 3.7. The exact standard error is
SE = sqrt( (8.6^2 / 20) + (7.8^2 / 25) ) = 2.476Since we know the standard error for the population, and do not need to estimate it from sample data, we can use the reliability coefficient 1.96 from the normal table.
3.7 +/- (1.96)(2.476)We are 95% confident that the difference in means is in the interval
3.7 +/- 4.9Example 2 (unknown population SDs)
A method to assess the effectiveness of a drug is to measure its concentration in the urine after a period of time. Twenty people are given the first brand and twenty-five are given the second brand. For the two samples, the mean concentrations are 19.2 and 15.6 (mg%) with standard deviations of 8.6 and 7.8 (mg%) respectively, one hour after ingestion. Give a 95% confidence interval for ().
The estimate for the difference in population means is the difference in sample means, 19.2 - 15.5 = 3.7. The estimated standard error, found by replacing population standard deviations with sample standard deviations is
SE = sqrt( (8.6^2 / 20) + (7.8^2 / 25) ) = 2.476The estimated number of degrees of freedom is
df = 2.476^4 / ( (1/19)(8.6^2 / 20)^2 + (1/24)(7.8^2 / 25)^2 ) = 38.9With a table, we'll round down to 35 degrees of freedom.
The multiplier for 35 degrees of freedom is 2.0301. (With software, we could have done better and found 2.0229.)
3.7 +/- (2.0301)(2.476)We are 95% confident that the difference in means is in the interval
3.7 +/- 5.0
Bret Larget, email@example.com