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Setting Up a Test for a Population Mean

When we want to make a claim or evaluate a claim about an unknown population mean ( $ \mu $ ), we use a significance test (also known as a hypothesis test). This process allows us to determine if observed sample data provides strong enough evidence to reject a null hypothesis in favor of an alternative hypothesis. The setup phase is crucial for ensuring the validity and interpretability of our results.

The Four-Step Process: A Framework

Setting up a hypothesis test generally follows a structured four-step process, often remembered by acronyms like PANIC or PHANTOMS. This note focuses on the initial steps: Parameters & Hypotheses and Assumptions & Conditions.

Step 1: State the Hypotheses

The first step is to clearly define the null and alternative hypotheses, which are competing claims about the population mean ( $ \mu $ ).

Null Hypothesis ( $ H_0 $ )

The null hypothesis is a statement of “no effect,” “no difference,” or “no change.” It represents the status quo or a previously accepted value. For a population mean, it always takes the form of an equality:

$$ H_0: \mu = \mu_0 $$
Where $ \mu $ is the true population mean we are interested in, and $ \mu_0 $ is the hypothesized value of the population mean based on the existing belief or the specific value we are testing against.

Alternative Hypothesis ( $ H_a $ )

The alternative hypothesis is the claim that we are trying to find evidence for. It challenges the null hypothesis and will be one of three forms, depending on the research question:

1. One-sided (left-tailed): We suspect the true mean is less than the hypothesized value. $$ H_a: \mu < \mu_0 $$ 2. One-sided (right-tailed): We suspect the true mean is greater than the hypothesized value. $$ H_a: \mu > \mu_0 $$ 3. Two-sided: We suspect the true mean is different from (either less than or greater than) the hypothesized value. $$ H_a: \mu \neq \mu_0 $$
   It’s crucial to define $ \mu $ in the context of the problem (e.g., " $ \mu $ is the true mean height of all adult males").

Example

Suppose a cereal company claims their boxes contain, on average, 368 grams of cereal. A consumer group suspects the true mean is less than 368 grams.

• $ H_0: \mu = 368 $ grams (The true mean weight of cereal per box is 368 grams)
• $ H_a: \mu < 368 $ grams (The true mean weight of cereal per box is less than 368 grams)

Step 2: Identify the Significance Level ( $ \alpha $ )

Before collecting data, we must choose a significance level, denoted by $ \alpha $ (alpha). This value represents the probability of making a Potential Errors When Performing Tests#Type I Error|Type I error (rejecting a true null hypothesis). Common values for $ \alpha $ are 0.01, 0.05, or 0.10. If not specified, 0.05 is often used by default.

Step 3: Check Conditions

Before performing any calculations, we must verify that the conditions for inference about a population mean are met. These conditions ensure that the chosen statistical test (typically a t-test) is appropriate.

| Condition | Explanation
The test for a population mean requires specific conditions to be met for the results to be valid:

1. Random Condition: The data must come from a Random Sampling and a Collection|random sample or a Introduction to Experimental Design|randomized experiment. This ensures the sample is representative of the population.
2. Independent Condition (10% Condition): When sampling without replacement, the sample size ( $ n $ ) must be no more than 10% of the population size ( $ N $ ). That is, $ n \le 0.10N $ . This ensures that the probability of success remains approximately constant for each trial and that the standard deviation of the sample mean can be accurately calculated.
3. Normal/Large Sample Condition: The sampling distribution of the sample mean ( $ \bar{x} $ ) must be approximately normal. This can be satisfied in one of three ways:
   • The population distribution is explicitly stated to be normal.
   • The sample size is large enough ( $ n \ge 30 $ ), relying on the The Central Limit Theorem|Central Limit Theorem.
   • If $ n < 30 $ and the population is not normal, a plot of the sample data (e.g., histogram, box plot, normal probability plot) shows no strong skewness or outliers.

Summary Table of Conditions

Next Steps

Once these setup steps are complete, the next phase involves Carrying Out a Test for a Population Mean, where you’ll calculate the test statistic and p-value, and then Concluding a Test for a Population Proportion to make a decision and state your conclusion in context.