Industrial Data Analytics for Diagnosis and Prognosis. Yong Chen

is the sample variance . The sample mean X̄ follows N(μ, σ²/n) and (n − 1)s²/σ² follows a χ² distribution with n − ¹ degrees of freedom. Consequently, under H₀ the t statistic in (3.18) follows a Student’s t-distribution with n − ¹ degrees of freedom. We reject H₀ at significance level α and conclude that μ is not equal to μ₀ if |t|>t_α/2,n−1, where t_α/2,n−1 denotes the upper 100(α/2)th percentile of the t-distribution with n − 1 degrees of freedom. Intuitively, |t|>t_α/2,n−1 indicates that we only have a small probability to observe |t| if we sample from the Student’s t-distribution with n − 1 degrees of freedom. Thus, it is very likely the null hypothesis H₀ is not correct and we should reject H₀.

      The test based on a fixed significance level α, say α = 0.05, has the disadvantage that it gives the decision maker no idea about whether the observed value of the test statistic is just barely in the rejection region or if it is far into the region. Instead, the p-value can be used to indicate how strong the evidence is in rejecting the null hypothesis H₀. The p-value is the probability that the test statistic will take on a value that is at least as extreme as the observed value when the null hypothesis is true. The smaller the p-value, the stronger the evidence we have in rejecting H₀. If the p-value is smaller than α, H₀ will be rejected at the significance level of α. The p-value based on the t statistic in (3.18) can be found as

      where T(n − 1) denotes a random variable following a t distribution with n − 1 degrees of freedom.

      We can define the 100(1 − α)% confidence interval for μ as

      It is easy to see that the null hypothesis H₀ is not rejected at level α if and only if μ₀ is in the 100(1 − α)% confidence interval for μ. So the confidence interval consists of all those “plausible” values of μ₀ that would not be rejected by the test of H₀ at level α.

      To see the link to the test statistic used for a multivariate normal distribution, we consider an equivalent rule to reject H₀, which is based on the square of the t statistic:

(3.19)

We reject H₀ at significance level α if t²>(t_α/2,n−1)².

      For a multivariate distribution with unknown mean μ and known Σ, we consider testing the following hypotheses:

(3.20)

      Let X₁, X₂,…, X_n denote a random sample from a multivariate normal population. The test statistic in (3.19) can be naturally generalized to the multivariate distribution as

(3.21)

      where X̄ and S are the sample mean vector and the sample covariance matrix of X₁, X₂,…, X_n. The T² statistic in (3.19) is called Hotelling’s T² in honor of Harold Hotelling who first obtained its distribution. Assuming H₀ is true, we have the following result about the distribution of the T²-statistic:

      where Fp,n−p denotes the F-distribution with p and n − p degrees of freedom. Based on the results on the distribution of T², we reject H₀ at the significance level of α if

(3.22)

      where Fp,n−p denotes the upper (100α)th percentile of the F-distribution with p and n − p degrees of freedom. The p-value of the test based on the T²-statistic is

      where F(p,n − p) denotes a random variable distributed as F_p,n−p.

      The T² statistic can also be written as