Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 5, Section 2 (pp. 397-408)
The standard deviation found in 4.71 is s, the population's standard deviation. The s.d. in 4.72(b) should be the same as that in 4.71 divided by the square root of 2 (since n = 2). You could decrease variability even more by including more observations in the calculation of the mean, illustrating the first sentence of Example 5.15.
Actually, no matter what shape the underlying distribution has, the sampling distribution for the sample mean will become more and more normal for larger and larger values of n, with distribution N( m, s/n1/2). This is what the central limit theorem says. In fact, the authors go on to say that any random variable that, like a mean, is the sum of many small influences will have approximately a normal distribution. (This would explain why so many biological traits height, weight, etc. are normally-distributed.)