Hypothesis Testing: Definitions and Steps 9/22

Hypothesis Testing: Key Definitions and Steps

1. Parameter Versus Statistic

• 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 ($H<em>a$) will always flow directly into the null hypothesis ($H</em>0$) and be identical.

2. Formulating Hypotheses ($H<em>0$ and $H</em>a$)

• 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 $H<em>a$ is < , then $H</em>0$ 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 $\ne$).

3. Checking Conditions and Assumptions

• 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.

4. Calculating the Test Statistic

• 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.,