STAT 3360 Notes

• \(X\) is a discrete random variable;
• \(X\) has \(K\) different possible values;
• \(x_1, x_2, \dots, x_K\) are the \(K\) possible values of \(X\);
• A sample about \(X\) and of size \(n\) is obtained;

• The Goodness-of-Fit test determines whether an assumed (hypothesized) distribution matches the reality. Before applying the test, we should first clarify the hypothesized distribution.

The hypothesized distribution of the discrete random variable \(X\) can be represented by a table, where \(p_1, p_2, \dots, p_K\) are the hypothesized proportions for values \(x_1, x_2, \dots, x_K\), respectively.

• by \(O_i\) the observed count of \(x_i\);
• by \(E_i\) the expected count of \(x_i\).

The above information can be summarized in a table

where \(\boxed{E_i = n \cdot p_i}\) for \(i=1,2,\dots,K\).

• Test Statistic: \(\boxed{T = \sum\limits_{i=1}^{K} \frac{(E_i - O_i)^2}{E_i}}\)
• Degrees of Freedom for the \(\chi^2\) statistic: \(K-1\)
• Critical Value: \(\boxed{\chi^2_{\alpha, K-1}}\)
• Rejection Rule: Reject \(H_0\) if \(\boxed{T > \chi^2_{\alpha, K-1}}\)

• We want to test
  • \(H_0:\) the real distribution of grade is the same as the teacher's assumption
  • \(H_A:\) the reel distribution of grade is NOT hte same as the teacher's assumption

• Let's summarize the information by a table using \(E_i = n \cdot p_i\), where \(n=400\) is the sample size

  Grade	A	B	C	D	F	W	Total
  Hypothesized Proportion	\(p_1=15\%\)	\(p_2=20\%\)	\(p_3=40\%\)	\(p_4=10\%\)	\(p_5=10\%\)	\(p_6=5\%\)	\(100\%\)
  Observed Count	\(O_1=66\)	\(O_2=84\)	\(O_3=148\)	\(O_4=48\)	\(O_5=46\)	\(O_6=8\)	\(n=400\)
  Expected Count	\(E_1=400 \cdot 15\% = 60\)	\(E_2=400 \cdot 20\% = 80\)	\(E_3=400 \cdot 40\% = 160\)	\(E_4=400 \cdot 10\% = 40\)	\(E_5=400 \cdot 10\% = 40\)	\(E_6=400 \cdot 5\% = 20\)	\(n=400\)

• Significance Level: \(\alpha = 5\% = 0.05\)
• Degrees of Freedom for the \(\chi^2\) statistic
  • There are \(K = 6\) different values (categories) of the Grade
  • Thus the degrees of freedom for the \(\chi^2\) statistic is \(K-1=6-1=5\)
• Critical Value: \(\chi^2_{\alpha, K-1} = \chi^2_{0.05, 5} = 11.1\)
• Test Statistic: \(T = \sum\limits_{i=1}^{6} \frac{(E_i - O_i)^2}{E_i} = \frac{(60-66)^2}{60} + \frac{(80-84)^2}{80} + \frac{(160-148)^2}{160} + \frac{(40-48)^2}{40} + \frac{(40-46)^2}{40} + \frac{(20-8)^2}{20} = 11.4\)
• Rejection Rule: Reject \(H_0\) if \(T > \chi^2_{\alpha, K-1}\)
• Decision: Since \(T = 11.4 > 11.1 = \chi^2_{\alpha, K-1}\), we reject \(H_0\). That is, there is sufficient evidence that the teacher's assumption is not true.

• We want to test
  • \(H_0:\) the real distribution of grade is the same as the teacher's assumption
  • \(H_A:\) the reel distribution of grade is NOT hte same as the teacher's assumption
• Significance Level: \(\alpha = 1\% = 0.01\)
• Degrees of Freedom for the \(\chi^2\) statistic
  • There are \(K = 6\) different values (categories) of Grade, the same as in Q1
  • Thus the degrees of freedom for the \(\chi^2\) statistic is \(K-1=6-1=5\), the same as in Q1
• Critical Value: \(\chi^2_{\alpha, K-1} = \chi^2_{0.01, 5} = 15.1\)
• Test Statistic: \(T = 11.4\), the same as in Q1
• Rejection Rule: Rejection \(H_0\) if \(T > \chi^2_{\alpha, K-1}\)
• Decision: Since \(T = 11.4 < 15.1 = \chi^2_{\alpha, K-1}\), we don't reject \(H_0\). That is, there is not sufficient evidence that the teacher's assumption is not true.

• \(X\) and \(Y\) are two discrete random variables;
• \(X\) has \(K\) different possible values;
• \(Y\) has \(L\) different possible values;
• \(x_1, x_2, \dots, x_K\) represents the \(K\) different possible values of \(X\);
• \(y_1, y_2, \dots, y_L\) represents the \(L\) different possible values of \(Y\);
• A sample of size \(n\) (grand total) is obtained, which is simultaneously about the two random variables \(X\) and \(Y\).

We may then summarize the information in the sample by a table as follows, where

• \(O_{ij}\) represents the Observed Counts corresponding to the event [\(X = x_i\) AND \(Y = y_j\)];
• \(R_i\) represents the \(i\)-th row total, \(i=1,2,\dots,K\);
• \(C_j\) represents the \(j\)-th row total, \(j=1,2,\dots,L\).

When \(X\) and \(Y\) are independent, by defintion of independence, we have

\(\boxed{E_{ij}} = \text{Expected Count for } [X = x_i \text{ AND } Y = y_j]\)

\(= \text{Grand Total} \times \text{Theorectial Proportion of the Event } [X = x_i \text{ AND } Y = y_j]\)

\(= \text{Grand Total} \times P(X = x_i \text{ AND } Y = y_j)\)

\(= \text{Grand Total} \times \underbrace{P(X = x_i)}_{\text{population proportion, estimated by sample proportion}} \times \underbrace{P(Y = y_j)}_{\text{population proportion estimated by sample proportion}}\)

\(\approx \text{Grand Total} \times \underbrace{\frac{\text{Total of }x_i}{\text{Grand Total}}}_{\text{sample proportion of } x_i} \times \underbrace{\frac{\text{Total of }y_j}{\text{Grand Total}}}_{\text{sample proportion of }y_j}\)

\(= n \times \frac{\text{Total of }x_i}{n} \times \frac{\text{Total of }y_j}{n}\)

\(= \boxed{ \frac{(\text{Total of }x_i) \times (\text{Total of }y_j)}{n} }\)

If the information are shown by a Table as follows, then we have

\(\boxed{ E_{ij} } = \frac{(i\text{-th row total}) \times (j\text{-th column total})}{n} = \boxed{ \frac{R_i C_j}{n} }\)

• Similar to the Goodness-of-Fit test, the independence test also measures the deviation between the assumption and the reality via the deviation between the expected count and the observed count. The test statistic also follows a \(\chi^2\) distribution.

• Since the test use both expected and observed counts, we may put them together into a single table as follows.

  Expected \(\vert\) Observed	\(y_1\)	\(y_2\)	…	\(y_L\)	Total
  \(x_1\)	\(E_{11} \vert O_{11}\)	\(E_{12} \vert O_{12}\)	…	\(E_{1L} \vert O_{1L}\)	\(R_1\)
  \(x_2\)	\(E_{21} \vert O_{21}\)	\(E_{22} \vert O_{22}\)	…	\(E_{2L} \vert O_{2L}\)	\(R_2\)
  …	…	…	…	…	…
  \(x_K\)	\(E_{K1} \vert O_{K1}\)	\(E_{K2} \vert O_{K2}\)	…	\(E_{KL} \vert O_{KL}\)	\(R_K\)
  Total	\(C_1\)	\(C_2\)	…	\(C_L\)	Grand Total \(=n\)

• \(\boxed{H_0: \text{Random variables } X \text{ and } Y \text{ are INDEPENDENT}}\) (or the distribution of \(X\) is homogeneous across all values of \(Y\)) vs
• \(\boxed{H_A: \text{the two random variables are NOT INDEPENDENT}}\) (or the distribution of \(X\) is NOT homogeneous across all values of \(Y\)),
• Test Statistic: \(\boxed{T = \sum\limits_{j=1}^{L} \sum\limits_{i=1}^{K} \frac{(E_{ij} - O_{ij})^2}{E_{ij}}}\)
• Degrees of Freedom for the \(\chi^2\) statistic: \((K-1)(L-1)\)
• Critical Value: \(\boxed{\chi^2_{\alpha, (K-1)(L-1)}}\)
• Rejection Rule: Reject \(H_0\) if \(\boxed{T > \chi^2_{\alpha, (K-1)(L-1)}}\)