Hypothesis Tests for Population Means

Introduction

Key Concepts

Understanding Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. Specifically, hypothesis tests for population means allow researchers to determine whether there is enough evidence to support a particular claim about the mean of a population. This method involves formulating competing hypotheses, selecting an appropriate test, and making a decision based on the evidence provided by the data.

Null and Alternative Hypotheses

In hypothesis testing, two opposing statements are formulated: the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$).

• Null Hypothesis ($H_0$): This hypothesis represents a statement of no effect or no difference and serves as the default or status quo. It implies that any observed difference is due to sampling variability.
• Alternative Hypothesis ($H_a$): This hypothesis represents a statement of an effect, difference, or relationship. It is what the researcher aims to support.

For example, if testing whether a new teaching method affects student performance, the null hypothesis might state that there is no difference in mean scores between the traditional and new methods, while the alternative hypothesis would state that there is a difference.

Types of Errors in Hypothesis Testing

When conducting hypothesis tests, two types of errors can occur:

• Type I Error: Occurs when the null hypothesis is true, but we incorrectly reject it. The probability of making a Type I error is denoted by the significance level ($\alpha$), commonly set at 0.05.
• Type II Error: Occurs when the null hypothesis is false, but we fail to reject it. The probability of making a Type II error is denoted by $\beta$.

Understanding these errors is crucial for interpreting the results of hypothesis tests and for making informed decisions based on statistical evidence.

Significance Level and P-Values

The significance level ($\alpha$) is a predetermined threshold used to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error. Commonly used significance levels are 0.05, 0.01, and 0.10.

The p-value is the probability of obtaining sample results at least as extreme as the observed results, assuming that the null hypothesis is true. A small p-value (typically ≤ $\alpha$) indicates strong evidence against the null hypothesis, leading to its rejection. Conversely, a large p-value suggests weak evidence against the null hypothesis, and it is not rejected.

Mathematically, if $p \leq \alpha$, reject $H_0$; otherwise, do not reject $H_0$.

Test Statistics and the Sampling Distribution

In hypothesis testing, a test statistic measures how far the sample statistic is from the null hypothesis value in units of standard error. The choice of test statistic depends on the sample size and whether the population standard deviation ($\sigma$) is known.

For testing a population mean when $\sigma$ is known and the sample size is large, the z-score is used:

Where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean under $H_0$
• $\sigma$ = population standard deviation
• $n$ = sample size

When $\sigma$ is unknown and the sample size is small, the t-score is used:

Where:

• $s$ = sample standard deviation

The test statistic is then compared to a critical value from the sampling distribution to determine whether to reject $H_0$.

Performing a Hypothesis Test for a Population Mean

The process of performing a hypothesis test for a population mean involves several steps:

1. State the hypotheses: Define $H_0$ and $H_a$ based on the research question.
2. Choose the significance level ($\alpha$): Common choices are 0.05 or 0.01.
3. Select the appropriate test statistic: Use z-test or t-test based on sample size and known parameters.
4. Compute the test statistic: Calculate the z-score or t-score using the sample data.
5. Determine the critical value or p-value: Use statistical tables or software.
6. Make a decision: Compare the test statistic to the critical value or compare p-value to $\alpha$ to decide whether to reject $H_0$.
7. Interpret the results: Relate the statistical decision to the context of the problem.

Confidence Intervals and Hypothesis Testing

Confidence intervals provide a range of plausible values for the population mean and are related to hypothesis testing. If a hypothesized mean value falls outside the confidence interval, it is typically rejected by the corresponding hypothesis test. The 95% confidence interval, for example, is associated with a significance level of 0.05.

Using confidence intervals alongside hypothesis tests can provide a more comprehensive understanding of the data and the precision of estimates.

Assumptions for Hypothesis Testing of Population Means

Several key assumptions must be met to validly perform hypothesis tests for population means:

• Random Sampling: The sample should be randomly selected from the population to ensure representativeness.
• Independence: Observations in the sample should be independent of each other.
• Normality: The sampling distribution of the mean should be approximately normal. This is typically satisfied if the sample size is large (Central Limit Theorem). For small samples, the population distribution should be approximately normal.

Violations of these assumptions can affect the validity of the test results and may require alternative statistical methods or data transformations.

Example Problems

Example 1: A manufacturer claims that the average lifetime of their light bulbs is 800 hours. A sample of 50 bulbs has an average lifetime of 780 hours with a standard deviation of 60 hours. Test the manufacturer's claim at the 0.05 significance level.

Solution:

• Step 1: $H_0: \mu = 800$ hours
  $H_a: \mu \neq 800$ hours
• Step 2: $\alpha = 0.05$
• Step 3: Since $\sigma$ is unknown and $n = 50$ is large, we use the z-test.
• Step 4: Compute the test statistic:
  $$ z = \frac{780 - 800}{60/\sqrt{50}} = \frac{-20}{8.49} \approx -2.358 $$
• Step 5: Determine the p-value: For $z \approx -2.358$, the p-value ≈ 0.018
• Step 6: Since $p < \alpha$, we reject $H_0$.
• Step 7: There is sufficient evidence to conclude that the average lifetime is different from 800 hours.

Example 2: A school administrator claims that the mean test score of students is at least 75. A random sample of 36 students has an average score of 73 with a sample standard deviation of 8. Test the claim at the 0.05 significance level.

Solution:

• Step 1: $H_0: \mu \geq 75$
  $H_a: \mu < 75$
• Step 2: $\alpha = 0.05$
• Step 3: Since $\sigma$ is unknown and $n = 36$ is large, we use the z-test.
• Step 4: Compute the test statistic:
  $$ z = \frac{73 - 75}{8/\sqrt{36}} = \frac{-2}{1.333} \approx -1.5 $$
• Step 5: Determine the critical value: For a one-tailed test at $\alpha = 0.05$, the critical z-value is -1.645.
  Since $z \approx -1.5 > -1.645$, we do not reject $H_0$.
• Step 6: There is insufficient evidence to reject the administrator's claim that the mean test score is at least 75.

Comparison Table

Summary and Key Takeaways