20. The term degree of freedom is used with reference to: A) Test of goodness of fit only B) Z-test and test of goodness of fit C) T-test and test of goodness of fit D) T-test and Z-test

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The term "degrees of freedom" (df) is a fundamental concept in statistics that refers to the number of independent values or quantities which can be assigned to a statistical distribution. It plays a crucial role in various statistical tests, primarily those that involve estimating parameters from sample data.

**Here's how degrees of freedom apply to the options provided:**

1. **T-test:**
   - **One-sample t-test:** Typically has $$ n-1 $$ degrees of freedom, where $$ n $$ is the sample size.
   - **Two-sample t-test:** Depends on the sample sizes of both groups, often calculated as $$ n_1 + n_2 - 2 $$.

2. **Test of Goodness of Fit:**
   - This is usually a chi-square test, which calculates degrees of freedom based on the number of categories minus the number of parameters estimated from the data minus one.

3. **Z-test:**
   - The Z-test is based on the standard normal distribution, which technically has infinite degrees of freedom because the population variance is known and not estimated from the sample.

**Analyzing the Options:**

- **Option A:** "Test of goodness of fit only" — This is incorrect because degrees of freedom are also used in other tests like the t-test.
  
- **Option B:** "Z-test and test of goodness of fit" — Incorrect because the Z-test does not typically involve finite degrees of freedom.
  
- **Option C:** "T-test and test of goodness of fit" — Correct. Both the t-test and the test of goodness of fit utilize degrees of freedom in their calculations.
  
- **Option D:** "T-test and Z-test" — Incorrect because, as mentioned, the Z-test does not involve finite degrees of freedom.

**Answer:**  
C) T-test and test of goodness of fit
Degrees of freedom are used in both T-tests and tests of goodness of fit.

The term degree of freedom is used with reference to:
A) Test of goodness of fit only
B) Z-test and test of goodness of fit
C) T-test and test of goodness of fit
D) T-test and Z-test

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