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Setting Up a Test for a Population Proportion
When we want to make a claim or decision about the true proportion of a certain characteristic within a population, we often use a Hypothesis Test. This process allows us to assess the evidence against a null hypothesis in favor of an alternative hypothesis. This note page focuses on the crucial initial steps of setting up such a test for a population proportion.
The Four-Step Process (Introduction)
A hypothesis test generally follows a four-step process, often remembered by mnemonics like “PHANTOMS” or “PANDA” (though specific steps vary slightly). The setup involves the first three parts:
- Parameters: Define the population parameter of interest.
- Hypotheses: State the null and alternative hypotheses.
- Assumptions/Conditions: Check the conditions required for the test.
- Name the test and Test statistic: Identify the appropriate test and calculate its statistic (covered in Carrying Out a Test for a Population Proportion).
- Obtain p-value: Determine the probability of observing our data (or more extreme) if the null hypothesis were true (covered in Interpreting p-Values).
- Make a decision and State a conclusion: Compare the p-value to the significance level and draw a conclusion in context (covered in Concluding a Test for a Population Proportion).
1. Defining the Parameter (P)
First, clearly define the population parameter you are interested in. For a test about a population proportion, this is usually denoted by $ p $ .
- Let $ p $ = the true proportion of [contextual event/group] in the [defined population].
Example:
- Let $ p $ = the true proportion of U.S. adults who support a new healthcare policy.
2. Stating Hypotheses (H)
This is a critical step where you articulate the claim you are testing. Every hypothesis test involves two competing statements: the null hypothesis ( $ H_0 $ ) and the alternative hypothesis ( $ H_a $ ).
Null Hypothesis ( $ H_0 $ )
The null hypothesis represents the status quo, a statement of no effect, no difference, or no change. It always includes an equality sign.
- $ H_0: p = p_0 $ * where $ p_0 $ is a specific hypothesized value for the population proportion.
Alternative Hypothesis ( $ H_a $ )
The alternative hypothesis is the statement we are trying to find evidence for. It reflects the claim or suspicion that something has changed or is different. It can be one-sided (greater than or less than) or two-sided (not equal to).
Types of Alternative Hypotheses:
| Type | Notation | Description |
|---|---|---|
| Two-Sided | $ H_a: p \ne p_0 $ | Used when we are interested in whether the true proportion is simply different from $ p_0 $ . |
| One-Sided | $ H_a: p > p_0 $ | Used when we are interested in whether the true proportion is greater than $ p_0 $ . |
| One-Sided | $ H_a: p < p_0 $ | Used when we are interested in whether the true proportion is less than $ p_0 $ . |
Choosing the Alternative: The choice of $ H_a $ depends on the specific question being asked or the claim being investigated.
Example:
- A company claims that 80% of its customers are satisfied. A consumer group suspects this percentage is lower.
- $ H_0: p = 0.80 $
- $ H_a: p < 0.80 $
3. Checking Conditions (A)
Before performing a hypothesis test for a population proportion, three key conditions must be met to ensure the sampling distribution of the sample proportion ( $ \hat{p} $ ) is approximately normal. These are often referred to as the “Random, Normal, Independent” conditions. This builds upon concepts from Sampling Distributions for Sample Proportions.
a. Random Condition
The data must come from a Random Sampling and a Collection or a Randomized Experiment. This ensures the sample is representative of the population.
- Check: Was the sample obtained using a simple random sample (SRS) or a well-designed random process? If it’s an experiment, were subjects randomly assigned?
b. Normal Condition (Large Counts Condition)
The sampling distribution of $ \hat{p} $ is approximately normal if the expected number of successes and failures are both at least 10. Crucially, we use the hypothesized population proportion ( $ p_0 $ ) from $ H_0 $ for this check, not the sample proportion ( $ \hat{p} $ ).
- $ np_0 \ge 10 $
- $ n(1 - p_0) \ge 10 $
Where:
- $ n $ = sample size
- $ p_0 $ = hypothesized population proportion from $ H_0 $
c. Independent Condition (10% Condition)
When sampling without replacement, the observations must be independent. This is ensured if the sample size is no more than 10% of the population size.
- $ n \le 0.10N $ * Where $ N $ = population size.
- Check: Is the sample size ( $ n $ ) less than 10% of the total population size? If we assume the population is very large relative to the sample (e.g., millions of people), this condition is usually met.
Once these three steps are completed, you have successfully set up your hypothesis test for a population proportion and are ready to proceed with calculating the test statistic and p-value.