Hypothesis Testing: Definitions and Steps 9/22

Parameter: Any number or symbol that summarizes an entire population. It represents a true, but often unknown, characteristic of the population.
- Example: P represents the population proportion.
Statistic: Any number or symbol that summarizes a sample. It is a calculated value from observed data.
- Example: \hat{p} (p-hat) represents the sample proportion.
Mnemonic: The letter 's' in 'statistic' helps remember that it refers to a 'sample'.
Symbol Correspondence: The parameter symbol used in the alternative hypothesis ( Ha ) will always flow directly into the null hypothesis ( H0 ) and be identical.

Null Hypothesis ( H_0 ):
- Always contains an equal sign ( = ) or a variant of it (e.g., \ge or \le ).
- Two acceptable approaches for the directional symbol in H_0 :
  1. Always use the equal sign ( = ) regardless of the alternative hypothesis's symbol.
  2. Use the opposite direction with an equal sign (e.g., if Ha is < , then H0 would be \ge ).
- Purpose: Hypotheses together map out all possible values for the parameter on a number line (e.g., for proportions, from 0 to 1).
Alternative Hypothesis ( H_a ):
- Must always match the research question.
- Never contains an equal sign (e.g., uses < , > , or \ne ).

Importance: Before proceeding with a hypothesis test, it is crucial to verify that the underlying conditions or assumptions for the chosen test are met.
Problem Identification: Determine if the problem involves proportions or means.
Sample Count: Determine if one sample or two samples were taken.
Conditions for One-Sample Proportion Test:
- The sample is a simple random sample (SRS).
- The population is at least 10 times the sample size (for independence).
- Number of successes and failures: Both must be at least 10:
  - nP \ge 10
  - n(1 - P) \ge 10
- Variables:
  - n : Sample size.
  - P : Population proportion, obtained from the null hypothesis (i.e., the value H_0 assumes P is equal to).
Example: Given n = 60 and P = 0.50 (from H_0: P = 0.50 ):
- 60 \times 0.50 = 30 \ge 10 (condition met)
- 60 \times (1 - 0.50) = 60 \times 0.50 = 30 \ge 10 (condition met)
Theoretical Implication (if conditions pass): If the experiment were repeated many times (e.g., 10,000 times) by taking samples of size n from the population, the distribution of the resulting sample proportions ( \hat{p} values) would be approximately normally distributed.

Formula for the Z-statistic (for a one-sample proportion test):
Z = \frac{\hat{p} - P}{\sqrt{\frac{P(1 - P)}{n}}}
Components Explained:
- \hat{p} (p-hat): The sample proportion calculated from the observed sample data. It must represent the same concept as the population proportion P . For example, if P is the proportion of students who have a credit card, then \hat{p} must also be the proportion of students who have a credit card in the sample.
- P : The population proportion assumed under the null hypothesis ( H_0 ). This is the value that P is hypothesized to be equal to.
- n : The sample size.
Defining P and \hat{p} Carefully:
- P (Population Proportion): Always refers to all of something (e.g.,