Egwald Statistics: Probability and Stochastic Processes
by
Elmer G. Wiens
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Distributions  Sampling Distributions  Sample Mean  Hypotheses Testing  Random Walk
 References
Hypotheses Testing
Testing the mean  Testing the variance
A. Testing the mean
1. Small sample from a normal population with known variance ø^^{2}
Suppose you know that a random variable X has a normal distribution, but that you only know its variance ø^^{2}. The sample mean of n observations, {X_{1}, ...X_{n}} is
Y = (X_{1} + ... + X_{n})/n
To test that the r.v. X has mean µ, use the fact that
Z = (Y  µ)* sqrt(n) / ø
has the standard normal distribution. If the absolute value of z is greater than 1.96, we can reject the hypotheses, at a 95% level of confidence, that the mean of X, E(X), is equal to µ.
2. Large sample, n >= 30, with unknown mean and (finite) variance
The Central Limit Theorem permits us to approximate ø^^{2} with S^^{2} and assume that the r.v.
Z = (Y  µ) * sqrt(n) / S
has a standard normal distribution.
3. Small sample from a normal population whose mean and (finite) variance are unknown.
If the sample mean = Y, and the sample variance = S^^{2} then the r.v.
Z = (Y  µ) * sqrt(n) / S
has a tdistribution with n1 degrees of freedom.
For example, suppose a sample of 10 observations yields a sample mean of y = 22 and a sample variance of s^^{2} = 7. We want to test the hypotheses that the population mean µ = 20.
z = (22  20)* sqrt(10) / sqrt(7) = 2.39
Since 2.39 > 2.262 we can reject the hypotheses at the 95% confidence level.
B. Testing the difference between means
1. Two independent random samples from two normal populations with known variances
Let {X_{1}, ... , X_{n}} be a sample of size n from the r.v. X with variance ø^^{2}.
Let {W_{1}, ... , W_{m}} be a sample of size m from the r.v. W with variance ö^^{2}. Compute their means as:
Y = [X_{1}+ ... + X_{n}] / n
Z = [W_{1}+ ... + W_{m}] / m
Suppose we want to test the hypotheses:
H_{0}: y  z = þ
against
H_{1}: y  z != þ
The r.v. V defined by:
V = [Y  Z  þ] / sqrt[(ø^^{2}/n) + (ö^^{2}/m)]
has a standard normal distribution. We can reject the hypotheses if, for two samples, the value of V is larger than 1.96, or less than 1.96.
2. Two large, independent random samples (n,m >= 30) from two populations (not necessarily normal) with unknown variances
The Central Limit Theorem permits us to approximate ø^^{2} with the sample variance S^^{2}, and ö^^{2} with the sample variance R^^{2}. Then
V = [Y  Z  þ] / sqrt[(S^^{2}/n) + (R^^{2}/m)]
has a standard normal distribution.
For example: if for sample 1, y = 195, n = 35, s^^{2} = 45, and for sample 2, z = 200, m = 40, r^^{2} = 55. Suppose we want to test that the two populations have the same mean. Then
v = [195  200  0] / sqrt[(45/35) + (55/40)] = 5 / 1.63 = 3.06
Using the standard normal distribution, we can reject the hypotheses, H_{0}: y  z = 0, at the 95% confidence level.
3. Two small, independent random samples from two normal populations with the same variance ø^^{2}
Suppose for sample 1, the sample mean = Y, the sample variance = S^^{2}, and the sample size = n; for sample 2, the sample mean = Z, the sample variance = R^^{2}, and the sample size = m. Then, the distribution of the r.v.
T = [Y  Z  þ] / {sqrt[(n  1)*S^^{2} + (m  1)*R^^{2}] * sqrt[1 / (n + m  2)] * sqrt[(1/n) + 1/m)]}
is the Studentt distribution with (n + m  2) degrees of freedom.
