Hypothesis Tests for a Population Mean, Known Standard Deviation

Hypothesis testing is a fundamental procedure in statistics that allows researchers to make inferences about population parameters based on sample data. One of the most common applications is testing a claim about a population mean. Specifically, when the population standard deviation is known, the Z-test for a population mean is the appropriate statistical method to use.

The Logic of Hypothesis Testing

At its core, hypothesis testing is an evidence-based decision-making process. We start with a default assumption called the null hypothesis ($H_0$), which represents a statement of no effect or no difference. Alongside it, we propose an alternative hypothesis ($H_1$ or $H_a$), which represents the claim we are trying to find evidence for.

For a population mean $\mu$, the null hypothesis is typically a statement of equality, such as $\mu = \mu_0$, where $\mu_0$ is a specific hypothesized value. The alternative hypothesis can take one of three forms depending on the research question:

• Right-tailed test: We suspect the mean is greater than the hypothesized value ($H_1: \mu > \mu_0$).
• Left-tailed test: We suspect the mean is less than the hypothesized value ($H_1: \mu < \mu_0$).
• Two-tailed test: We suspect the mean is different from the hypothesized value, without specifying a direction ($H_1: \mu \neq \mu_0$).

Conditions for the Z-Test

To validly perform a hypothesis test for a population mean with a known standard deviation, certain conditions must be met. Adhering to these conditions ensures that the sampling distribution of the sample mean is approximately normal, allowing the use of the standard normal (Z) distribution.

• Random Sample: The data must be collected using a simple random sampling method. This ensures that the sample is representative of the population and minimizes bias.
• Independence: The sampled values must be independent of each other. This is generally satisfied if the sample size is less than 10% of the population size (the 10% rule).
• Normality or Large Sample Size: The sampling distribution of $\bar{x}$ must be approximately normal.
  If the population is normally distributed, any sample size will work.
  If the population distribution is unknown or not normal, the Central Limit Theorem applies. Typically, a sample size of $n \ge 30$ is considered sufficient to guarantee approximate normality of the sampling distribution.
• Known Standard Deviation: The population standard deviation ($\sigma$) must be known. In real-world scenarios, this is rare, as $\sigma$ is usually unknown. However, this test serves as a theoretical foundation and is often used in introductory statistics or when previous extensive research provides a reliable $\sigma$.

The Test Statistic

The test statistic is a value calculated from the sample data that is used to decide whether to reject the null hypothesis. For testing a mean with known $\sigma$, we standardize the sample mean $\bar{x}$ using the following formula:

Where:

• $z$ is the test statistic.
• $\bar{x}$ is the sample mean.
• $\mu_0$ is the hypothesized population mean under the null hypothesis.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

This formula tells us how many standard errors the observed sample mean $\bar{x}$ is away from the hypothesized mean $\mu_0$. A large absolute value of $z$ (either positive or negative) indicates that the observed data is unlikely to have occurred if the null hypothesis were true.

Steps of the Hypothesis Test

Conducting the test involves a structured, step-by-step approach to ensure objectivity.

1. State the Hypotheses

Clearly define the null hypothesis ($H_0: \mu = \mu_0$) and the alternative hypothesis ($H_1$). The form of $H_1$ is determined by the context of the problem (left-tailed, right-tailed, or two-tailed).

2. Set the Level of Significance

The significance level, denoted by $\alpha$ (alpha), represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Common choices for $\alpha$ are 0.05 (5%) and 0.01 (1%). This threshold defines the "rejection region."

3. Compute the Test Statistic

Using the sample data (values for $\bar{x}$ and $n$) and the known parameters ($\mu_0$ and $\sigma$), calculate the $z$-score using the formula provided above.

4. Determine the P-value or Critical Value

There are two equivalent methods to make a decision:

• P-value approach: The P-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming $H_0$ is true. For a right-tailed test, it is the area to the right of $z$; for a left-tailed test, it is the area to the left; for a two-tailed test, it is twice the area in the tail beyond $z$.
• Critical value approach: Based on $\alpha$ and the type of test, find the critical Z-value(s) from the standard normal distribution table. For a two-tailed test with $\alpha = 0.05$, the critical values are approximately $-1.96$ and $+1.96$.

5. Make a Decision

Compare the evidence to the threshold.

• Using the P-value: If the P-value $\le \alpha$, reject $H_0$. If the P-value $> \alpha$, fail to reject $H_0$.
• Using Critical Values: If the test statistic $z$ falls within the rejection region (beyond the critical values), reject $H_0$. Otherwise, fail to reject $H_0$.

6. Interpret the Results

Finally, state the conclusion in the context of the original problem. If you reject $H_0$, you have sufficient evidence to support the alternative hypothesis. If you fail to reject $H_0$, you do not have sufficient evidence to support the alternative hypothesis.

Practical Example

Suppose a manufacturer claims that the average life of a specific type of battery is 50 hours. A quality control manager suspects this is an overestimate and wants to test if the true mean is actually less than 50 hours.

• Knowns: Population standard deviation $\sigma = 5$ hours. Sample size $n = 36$. Sample mean $\bar{x} = 48.5$ hours. Significance level $\alpha = 0.05$.
• Step 1: $H_0: \mu = 50$ vs. $H_1: \mu < 50$ (Left-tailed test).
• Step 2: $\alpha = 0.05$.
• Step 3: Calculate $z$. $$ z = \frac{48.5 - 50}{5 / \sqrt{36}} = \frac{-1.5}{5 / 6} = \frac{-1.5}{0.833} \approx -1.80 $$
• Step 4: Find the P-value for $z = -1.80$. Looking at the Z-table, the area to the left of $-1.80$ is approximately $0.0359$.
• Step 5: Since the P-value ($0.0359$) is less than $\alpha$ ($0.05$), we reject the null hypothesis.
• Step 6: There is sufficient evidence at the 0.05 significance level to conclude that the average battery life is less than 50 hours. The manufacturer's claim is likely incorrect.

This structured approach ensures that statistical conclusions are not based on subjective feelings but on a rigorous mathematical framework. While real-world statistics often involve unknown standard deviations (requiring the t-test), understanding the Z-test provides the necessary foundation for grasping the logic of statistical inference.