Sample MeansThe sample mean from a group of observations is an estimate of the population mean. Given a sample of sizen, consider n independent random variables X1,X2, …, Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be.By the properties of means and variances of random variables, the mean and variance of the sample mean are the following:
Although the mean of the distribution of is identical to the mean of the population distribution, the variance is much smaller for large sample sizes.
For example, suppose the random variable X records a randomly selected student’s score on a national test, where the population distribution for the score is normal with mean 70 and standard deviation 5 (N(70,5)). Given asimple random sample (SRS) of 200 students, the distribution of the sample mean score has mean 70 and standard deviation 5/sqrt(200) = 5/14.14 = 0.35.
Distribution of the Sample MeanWhen the distribution of the population is normal, then the distribution of the sample mean is also normal. For a normal population distribution with mean and standard deviation, the distribution of the sample mean is normal, with mean and standard deviation.
This result follows from the fact that any linear combination of independent normal random variables is also normally distributed. This means that for two independent normal random variablesX and Y and any constants a and b, aX + bY will be normally distributed. In the case of the sample mean, the linear combination is =(1/n)*(X1 + X2 + … Xn).
For example, consider the distributions of yearly average test scores on a national test in two areas of the country. In the first area, the test scoreX is normally distributed with mean 70 and standard deviation 5. In the second area, the yearly average test scoreY is normally distributed with mean 65 and standard deviation 8. The differenceX – Y between the two areas is normally distributed, with mean 70-65 = 5 and variance 5 + 8 = 25 + 64 = 89. The standard deviation is the square root of the variance, 9.43. The probability that areaX will have a higher score than area Y may be calculated as follows:P(X > Y) = P(X – Y > 0) = P(((X – Y) – 5)/9.43 > (0 – 5)/9.43) = P(Z > -0.53) = 1 – P(Z < -0.53) = 1 – 0.2981 = 0.7019. Area X will have a higher average score than area Y about 70% of the time.
The Central Limit TheoremThe most important result about sample means is the Central Limit Theorem. Simply stated, this theorem says that for a large enough sample sizen, the distribution of the sample mean will approach a normal distribution.This is true for a sample of independent random variables from any population distribution, as long as the population has a finite standard deviation.
A formal statement of the Central Limit Theorem is the following:
If is the mean of a random sampleX1, X2, … , Xn of size n from a distribution with a finite mean and a finite positive variance, then the distribution ofW = isN(0,1) in the limit as n approaches infinity.
This means that the variable is distributedN().
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