AP Statistics: Significance Testing - Making Conclusions

Hey there, future AP Stats superstar! ✨ You've crunched the numbers, and now it's time to make sense of it all. Let's break down how to draw conclusions from your significance tests like a pro. Remember, it's all about whether you can reject the null hypothesis or not. Let's get to it!

Significance Test Outcomes: Reject or Fail to Reject

Your journey through hypothesis testing boils down to one of two decisions:

• Reject the null hypothesis (H₀): You have enough evidence to say that the null hypothesis is likely not true.
• Fail to reject the null hypothesis (H₀): You don't have enough evidence to reject the null hypothesis. This does NOT mean you're accepting the null as true. 🙅

The Role of the P-Value

The p-value is your key to making these decisions. It tells you the probability of seeing your results (or more extreme results) if the null hypothesis were true. Think of it as the "surprise factor" of your data.

• Small p-value (typically ≤ α): Your data is surprising if H₀ is true, so you reject H₀. 😠
• Large p-value (typically > α): Your data isn't surprising if H₀ is true, so you fail to reject H₀. 😵‍💫

P-Value in Detail

• A small p-value means your observed results are unlikely to have occurred by chance if the null hypothesis is true. The cutoff is usually 0.05 (or 5%) but it can be different based on the problem.
• If your p-value is below your alpha (α) level, you have statistically significant evidence to reject the null hypothesis.
• If your p-value is above your alpha (α) level, you don't have enough evidence to reject the null hypothesis. This doesn't mean the null is true, just that you don't have enough evidence to say it's false.

The Role of the Z-Score

Another way to make conclusions is by using the z-score. Remember, a z-score shows how many standard deviations your sample statistic is from the mean (assuming the null hypothesis is true).

• Large Z-score (typically > 2 or < -2): This means your sample is far from what's expected under H₀, so you reject H₀. 💯
• Z-score between -2 and 2: This is within the typical range if H₀ is true, so you fail to reject H₀.

Conclusion Template

Here's a template to help you write your conclusions for a one-proportion z-test:

"Since the p-value of [p-value] is </> the alpha level of [α], we reject/fail to reject the null hypothesis. We have/do not have sufficient evidence to conclude that [alternative hypothesis in context]." 📝

The "Big Three" for Conclusions

To get full credit on your AP exam, make sure your conclusion includes these three things: ❗

1. Comparison: Compare the p-value to the significance level (alpha).
2. Decision: State whether you reject or fail to reject the null hypothesis.
3. Context: Explain what your decision means in the context of the problem, referring to the true population parameter.

Practice Problem

Let's put it all together with a practice problem:

A survey is conducted to determine if a new advertising campaign increases brand awareness. The null hypothesis is that the campaign has no effect, and the alternative is that it does increase awareness. A sample of 500 people is split into two groups: one group of 250 sees the ad campaign, and the other 250 does not. The proportion of people in the campaign group aware of the brand is 0.7, and the proportion in the non-campaign group is 0.5. The hypothesis test is conducted at a significance level of α = 0.05. The test statistic is calculated to be z = 2.8. What is the p-value for this hypothesis test?

What is your conclusion about the null hypothesis?

Solution

The p-value for this hypothesis test is 0.0026. This means there is a 0.26% chance of obtaining a test statistic as extreme as 2.8 if the null hypothesis is true. Since the p-value (0.0026) is less than the significance level (α = 0.05), we reject the null hypothesis. This suggests that the advertising campaign is effective at increasing brand awareness! ⭐

Final Exam Focus

Alright, you're in the home stretch! Here's what to focus on for the exam:

• Hypothesis Testing Logic: Make sure you understand the process of setting up hypotheses, checking conditions, calculating test statistics, and making conclusions.
• P-Values: Know how to interpret p-values and use them to make decisions about your null hypothesis.
• Z-Scores: Understand how z-scores relate to p-values and how to use them to draw conclusions.
• Context: Always, always, always include context in your conclusions. This is a huge point-getter!
• Common Mistakes: Be careful not to "accept" the null hypothesis. Remember, we either reject or fail to reject.

Significance tests are a major focus on the AP exam, so make sure you're comfortable with all aspects of the process.

Last-Minute Tips

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back to it later.
• Show Your Work: Even if you don't get the right answer, you can still get partial credit for showing your work.
• Read Carefully: Make sure you understand what the question is asking before you start working on it.
• Stay Calm: You've got this! Take deep breaths and trust your preparation.

Practice Questions

Practice Question

Multiple Choice Questions

1. A researcher is testing the hypothesis $H_0: \mu = 10$ against $H_a: \mu \neq 10$. A sample of size 25 produces a test statistic of $t = 2.3$. The p-value is: (A) 0.01 (B) 0.02 (C) 0.03 (D) 0.04 (E) 0.05

2. A significance test is performed to test the null hypothesis $H_0: p = 0.5$ versus the alternative hypothesis $H_a: p > 0.5$. A sample of size 100 yields a test statistic of $z = 1.8$. The p-value is: (A) 0.036 (B) 0.072 (C) 0.05 (D) 0.01 (E) 0.02

3. A researcher is testing the hypothesis $H_0: \mu_1 = \mu_2$ against $H_a: \mu_1 < \mu_2$. A sample of size 30 from each population produces a test statistic of $t = -2.1$. The p-value is: (A) 0.01 (B) 0.02 (C) 0.03 (D) 0.04 (E) 0.05

Free Response Question

A researcher wants to investigate if the mean height of adult women is different from 64 inches. They collect a random sample of 40 adult women and find a sample mean of 64.8 inches with a sample standard deviation of 2.5 inches. Conduct a complete hypothesis test at a significance level of 0.05. #### Scoring Breakdown:

(a) Hypotheses (1 point)

• Correctly stating both null and alternative hypotheses in symbols and words.

  $H_0: \mu = 64$ (The mean height of adult women is 64 inches.) $H_a: \mu \neq 64$ (The mean height of adult women is different from 64 inches.)

(b) Conditions (1 point)

• Checking the conditions for a one-sample t-test.

  • Random: The sample was randomly collected.
  • Normal: The sample size is 40 which is greater than 30, so the sample distribution is approximately normal.
  • Independent: The sample size is less than 10% of the population of adult women.

(c) Calculations (2 points)

• Correctly calculating the test statistic and p-value.

  Test Statistic: $t = \frac{64.8 - 64}{2.5/\sqrt{40}} = 2.02$ Degrees of freedom = 40 - 1 = 39 P-value = 2 * P(t > 2.02) = 0.05

(d) Conclusion (1 point)

• Correctly interpreting the p-value and making a conclusion in context.

  Since the p-value (0.05) is equal to the significance level (0.05), we reject the null hypothesis. We have sufficient evidence to conclude that the mean height of adult women is different from 64 inches.

You've got this! Go ace that exam! 💪