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The standard error of the sample mean and confidence intervals. How far is the average sample mean from the population mean? In what interval around mu can we expect to find 95% or 99% or sample means. An introduction to random samples.
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How far is the average sample mean from the population mean?
In what interval around mu can we expect to find 95% or 99% or sample means
Population is 1320 students taking a test.
is 72.00, = 12
Let’s randomly sample one student at a time and see what happens.We’ll create a random sample with 8 students’ scores in the sample.
sigma2X-bar = sigmaX-bar.
sigmaX-bar is called the standard error of the sample mean or, more briefly, the standard error of the mean. Let’s look at the formulae:
sigma2X-bar = sigma2/n
sigmaX-bar = sigma/
We want to define two intervals around mu:One interval into which 95% of the sample means will fall. Another interval into which 99% of the sample means will fall.
95% of sample means will fall in a symmetrical interval around mu that goes from 1.960 standard errors below mu to 1.960 standard errors above mu
CI.95: mu + 1.960 sigmaX-bar or
CI.95: mu - 1.960 sigmaX-bar < X-bar < mu + 1.960 sigmaX-bar
As I said, 95% of sample means will fall in a symmetrical interval around mu that goes from 1.960 standard errors below mu to 1.960 standard errors above mu
CI.95: mu + 1.960 sigmaX-bar = 500+9.80 or
CI.95: 490.20 < X-bar < 509.20
2.576 (5.00)= 12.88
CI.99: mu + 2.576 sigmaX-bar = 500+12.88 or
CI.99: 487.12 < the sample mean < 512.88
1.960 (3.00)= 5.88 points
CI.95: mu + 1.960 sigmaX-bar = 100+5.88 or
CI.95: 94.12 < X-bar < 105.88
99% of the sample means (n=25) can be expected to fall in the interval 100 + (2.576)(3.00) = 100 + 7.73
CI.99: 100+7.73 or
CI.99: 92.27 < X-bar < 107.73
Here is another example. This time we start with an even smaller population (N=4) and take all possible samples of size 3. There are 64 of them. Let’s see that again the means form a normal curve around mu and the standard error equals sigma divided by the square root of the sample size (3).