AP Stats 9.3

Chapter 9: Testing a Claim

Section 9.3: Tests About a Population Mean

Key Concepts

• Random Sampling Conditions: Must check that data comes from a random sample, satisfy the 10% condition (n < 0.10N), and ensure the population is normally distributed or the sample size is large (n ≥ 30).

• Standardized Test Statistic:

  • Formula: [ z = \frac{\bar{x} - \mu_0}{\sigma} ]

  • When population standard deviation ( \sigma ) is unknown, use sample standard deviation ( s_x ) and denote the statistic as t.

Performing a Significance Test

• Four-Step Process:

  1. State: Define hypotheses (Null ( H_0 ) and Alternative ( H_a )) and significance level.

  2. Plan: Identify inference method and check conditions.

  3. Do: If conditions are met, perform calculations, including sample statistic and P-value.

  4. Conclude: Make a conclusion about hypotheses in the context of the problem.

Degrees of Freedom

• For inference about a population mean using a t distribution, degrees of freedom is calculated as:

  • ( df = n - 1 )

• The t distribution is symmetric, bell-shaped, and has heavier tails than the normal distribution.

Power of a Test

• Definition: Probability that a test will correctly reject a false null hypothesis (avoiding Type II error).

• Factors to increase power:

  • Larger sample size (n).

  • Higher significance level (α).

  • Greater distance between null and alternative parameters.

Confidence Intervals and Two-Sided Tests

• A 95% confidence interval that does not capture the null value allows rejection of ( H_0 ) in a two-sided test at α = 0.05.

• Confidence intervals should be used along with significance tests as they provide context for practical importance.

Conclusion Points

• Statistical significance does not imply practical importance; confidence intervals should be used to evaluate practical significance.

• Avoid multiple analyses to prevent misleading results, known as data dredging or P-hacking.

KID EXPLANATIONS

1. Performing a Significance Test

Imagine you're trying to figure out if a new game is better than an old one. To do this, we ask some friends to play each game and tell us how much they enjoy it. We check to see if the new game scores better than the old game based on their answers.

We have a way to do this:

• State: We say what we believe (our guesses) about the games—this is like saying the new game is better (that’s our hope) or that it’s not better (that’s the opposite—the Null Hypothesis).

• Plan: We decide how we will check our guesses and make sure everything is fair.

• Do: We do the testing: We count the scores!

• Conclude: We look at the scores and decide what our guesses were right or wrong.

2. Degrees of Freedom

Degrees of freedom is like how many choices we have when we’re using some numbers to make predictions. If we ask 10 friends to play the game, we have 9 degrees of freedom. (This means if we know 9 friends’ scores, the last score is a bit limited—it can’t be just anything!).

3. Power of a Test

Power of a test is like saying how strong our results are. If our test has a lot of power, it means we’re really good at finding out if the new game is better when it really is! To make our tests stronger, we could invite more friends to play or make our guesses more meaningful.

4. Confidence Intervals & Two-Sided Tests

A confidence interval is like saying, "I am pretty sure the new game’s score is between this number and that number." If the number we think (the Null value) doesn’t fall within that range, it means we can be pretty sure that the new game is better!

In a two-sided test, we check both sides: we want to see if the new game is better or if it’s worse than the old game.

5. Conclusion Points

Just because we found something that looks big and important, it doesn’t mean it’s the coolest thing ever! We also have to check if it really matters in real life. And when we guess a lot of times, we can get confusing answers, so it’s better to stick with our first idea and not change it too much to find what we want.