Hypothesis test for the mean of a normal distribution

Welcome to Hypothesis Testing for the Mean!

In previous chapters, you learned how to test whether a coin is biased or if a certain percentage of people prefer a specific brand. That was for discrete data (things you count). Now, we are moving into the world of continuous data (things you measure), like the height of students, the weight of cereal boxes, or the lifespan of lightbulbs.

Don't worry if this seems a bit "heavier" than Binomial testing; the logic is exactly the same! We are simply using the Normal Distribution as our ruler to see if a result is surprising or "just luck."

1. The Sample Mean as a Random Variable

Imagine you have a giant jar of jellybeans. If you pick out just one bean, its weight might be very high or very low. But if you grab a handful of 20 beans and find their average weight, that average is much less likely to be an extreme value.

In Statistics, we call the average of our sample \(\bar{x}\). Because \(\bar{x}\) changes every time we take a new sample, we treat it as a random variable, denoted by \(\bar{X}\).

The Rule of the "Skinny" Distribution

The syllabus requires you to know a very specific result. If your original population follows a Normal distribution \(X \sim N(\mu, \sigma^2)\), then the sample mean also follows a Normal distribution, but it is "skinnier" (less spread out):

\(\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\)

Where: - \(\mu\) is the population mean. - \(\sigma^2\) is the population variance. - \(n\) is the sample size.

Quick Review: To get the standard deviation of this new distribution (often called the Standard Error), you take the square root of the variance: \(\frac{\sigma}{\sqrt{n}}\).

Did you know?

The larger your sample size (\(n\)), the smaller the variance (\(\frac{\sigma^2}{n}\)) becomes. This makes sense: a large sample gives you a much more reliable estimate of the true average than a small one!

Key Takeaway: When testing the mean, always use the distribution for \(\bar{X}\), which means you must divide the original variance by the sample size \(n\).

2. Setting up the Test

Just like with Binomial tests, we need a starting assumption and an alternative.

The Null Hypothesis (\(H_0\)): This is the "status quo." We assume the mean is exactly what it's supposed to be.
Example: \(H_0: \mu = 50\)

The Alternative Hypothesis (\(H_1\)): This is what we are investigating. - 1-tailed test: We think the mean has increased (\(\mu > 50\)) or decreased (\(\mu < 50\)). - 2-tailed test: We just think the mean has changed (\(\mu \neq 50\)).

Common Mistake to Avoid

Always write your hypotheses using the population parameter \(\mu\). Never use \(\bar{x}\) in your \(H_0\) or \(H_1\). Hypotheses are about the whole population, not just your little sample!

3. The Step-by-Step Process

If you find a method and stick to it, these questions become much easier. Let’s look at the "Critical Value" approach.

Step 1: Hypotheses
State \(H_0\) and \(H_1\) clearly, and define what \(\mu\) represents (e.g., "where \(\mu\) is the mean weight of the chocolate bars").

Step 2: Distribution
State the distribution of the sample mean under the null hypothesis: \(\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\).

Step 3: Significance Level
Identify the significance level (\(\alpha\)) from the question (usually 5% or 1%). If it is a 2-tailed test, remember to split this in half (e.g., 2.5% at each end).

Step 4: Find the Critical Region
Use your calculator's "Inverse Normal" function to find the boundary value (the Critical Value) that corresponds to the significance level for your \(\bar{X}\) distribution.
Example: If \(\alpha = 5\%\) and we are testing \(\mu > 50\), find the value \(c\) such that \(P(\bar{X} > c) = 0.05\).

Step 5: Compare and Conclude
- If your observed sample mean \(\bar{x}\) falls inside the critical region (the "tail"), then it is very unlikely to have happened by chance. Reject \(H_0\). - If it falls outside, then the result isn't surprising enough. Do not reject \(H_0\).

Step 6: Contextual Conclusion
Always write a final sentence in plain English.
Example: "There is sufficient evidence at the 5% level to suggest that the mean weight of chocolate bars has decreased."

4. Using the Z-Transformation

Sometimes, instead of using your calculator on \(\bar{X}\) directly, you might want to use the Standard Normal Distribution \(Z \sim N(0, 1)\). This is especially helpful if you are using the statistical tables provided in your exam formula booklet.

The formula to turn your sample mean into a test statistic (\(z\)) is:

\(z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\)

Think of the \(z\)-score as "How many standard deviations away from the mean is my sample?" If the \(z\)-score is very large (e.g., greater than 1.645 for a 5% 1-tail test), the result is significant!

5. Important Reminders for Success

Memory Aid: The "Tail" Tells the Tale

If the question says "has the mean changed," use a 2-tailed test and split the significance level. If it says "increased" or "decreased," use a 1-tailed test.

Quick Review Box

- Population: \(X \sim N(\mu, \sigma^2)\)
- Sample Mean: \(\bar{X} \sim N(\mu, \frac{\sigma^2}{n})\)
- Hypotheses: Always use \(\mu\)
- Conclusion: Always relate back to the real-world story (e.g., "the lightbulbs," "the students").

Common Pitfalls

1. Forgetting to divide by \(n\): This is the most common error. The sample mean is always more consistent (skinnier distribution) than the population.
2. Using Variance instead of Standard Deviation: In your calculator, "sigma" (\(\sigma\)) is the standard deviation. If the question gives you the variance (\(\sigma^2\)), you must square root it first, then divide by \(\sqrt{n}\).
3. Over-asserting the conclusion: Never say "\(H_0\) is true" or "I have proven the mean has changed." Always use cautious language like "There is evidence to suggest..." or "There is insufficient evidence to suggest..."

Key Takeaway: Hypothesis testing is just a way of asking: "Is my sample average so far away from the expected average that the original average must be wrong?" If the answer is yes, you reject the null!