7.1 Introducing Statistics: Should I Worry About Error?
In Unit 6, we used the Normal distribution because the standard error of a proportion is completely determined by p and n. But what happens when we look at quantitative data (means)?
The Standard Error Problem: The true standard deviation of the sample mean is σ / √n. However, in reality, we almost never know the population standard deviation (σ). Instead, we have to estimate it using the sample standard deviation (s).
Because s varies from sample to sample, using it introduces extra variability (extra error!). To account for this extra "fuzziness," we can't use the standard Normal (Z) distribution anymore. We must use the Student's t-distribution.
Degrees of Freedom (df)
The shape of the t-distribution changes based on sample size. We define this using degrees of freedom:
df = n − 1.
As df increases, the t-curve gets closer and closer to the Standard Normal (Z) curve.
7.2 Constructing a Confidence Interval for a Population Mean
To estimate a population mean (μ) when σ is unknown, we use a 1-Sample t-Interval.
Formula: 1-Sample t-Interval for μ
Where t* is the critical value for n − 1 degrees of freedom.
The Four-Step Process (PANIC)
Remember PANIC for Free Response questions.
| Step | What to do |
|---|---|
| P | Parameter: Define μ in context (e.g., "μ = the true mean weight of..."). |
| A | Assess Conditions: 1. Random: Random sample or randomized experiment. 2. 10%: n ≤ 10% of the population. 3. Normal/Large Sample: Population is Normal, OR n ≥ 30 (CLT), OR a graph of the sample data shows no strong skew or outliers. |
| N | Name Procedure: "1-Sample t-Interval for μ". |
| I & C | Interval & Conclude: Calculate interval. "We are C% confident that the interval from [lower] to [upper] captures the true mean..." |
⚠️ The Normal/Large Sample Condition: If n < 30, you must sketch a quick dotplot or boxplot of the data on your FRQ to show there is no strong skewness or outliers. AP Graders look for this specifically!
Calculator Commands (TI-83/84)
STAT ➔ TESTS ➔ 8: TIntervalinvT(area_to_left, df) to find t* critical values.
7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval
Just like with proportions, we can use our confidence interval for a mean to justify or refute a claim.
Example: A manufacturer claims their magic wand cat toys weigh an average of 45 grams. HighFiveAP takes a random sample of 20 wands and constructs a 95% confidence interval for the true mean weight: (41.2g, 44.1g).
Does the interval support the manufacturer's claim?
Conclusion: No. Since 45 grams is not included in the 95% confidence interval, we have convincing evidence that the true mean weight of the wands is actually less than the claimed 45 grams.
7.4 Setting Up a Test for a Population Mean
When testing a claim about a mean, we follow the same logic as proportions but use the t-statistic.
H₀: μ = μ₀ (The null hypothesized mean)
Hₐ: μ > μ₀ (or <, or ≠)
Test Statistic Formula
7.5 Carrying Out a Test for a Population Mean
We use the PHANTOMS acronym to execute a 1-Sample t-Test.
- Parameter: Define μ in context.
- Hypotheses: State H₀ and Hₐ.
- Assumptions: Random, 10%, Normal/Large Sample (n ≥ 30 or plot data).
- Name the Test: 1-Sample t-Test for μ.
- Test Statistic: Calculate t and state df = n − 1.
- Obtain p-value: Find the area under the t-curve using
tcdf(lower, upper, df). - Make a Decision: Reject H₀ (if p ≤ α) or Fail to Reject H₀ (if p > α).
- State Conclusion: Contextualize your decision regarding Hₐ.
7.6 Confidence Intervals for the Difference of Two Means
When comparing two separate populations or two treatments in an experiment, we use a 2-Sample t-Interval.
Formula: 2-Sample t-Interval for μ₁ − μ₂
🎯 To Pool or Not to Pool?
In AP Statistics, when doing two-sample procedures for means, NEVER POOL the variances. Always select "No" for pooling on your calculator. We only pool for proportions (pC), never for means!
7.7 Justifying a Claim About the Difference of Two Means
As with proportions, analyzing the difference between two means boils down to checking if zero is in the interval.
Interval Does Not Contain 0
Example: (2.5, 6.1)
Because 0 is not a plausible value, we have convincing evidence that there is a difference between the two population means.
Interval Contains 0
Example: (-1.2, 4.3)
Because 0 is in the interval, it is plausible that μ₁ − μ₂ = 0. We do not have convincing evidence of a difference.
7.8 Setting Up a Test for the Difference of Two Population Means
When running a 2-Sample t-Test, we assume the means are equal under the null hypothesis.
H₀: μ₁ = μ₂ (or μ₁ − μ₂ = 0)
Hₐ: μ₁ > μ₂ (or <, or ≠)
You must check conditions for BOTH samples (Random, 10%, Normal/Large Sample for both n₁ and n₂).
7.9 Carrying Out a Test for the Difference of Two Population Means
2-Sample t-Test Statistic
⚠️ Calculating Degrees of Freedom: The df formula for a 2-sample t-test is incredibly ugly (Welch's approximation). You do not need to calculate this by hand. Simply use 2-SampTTest on your TI-84, copy the df it provides, and write it on your paper!
Calculator Commands (TI-83/84)
STAT ➔ TESTS ➔ 4: 2-SampTTest
Remember: Set "Pooled" to NO.
7.10 Skills Focus: Selecting Inference Procedures
The AP Exam will mix all these questions together. You must be able to read a prompt and immediately identify the correct procedure.
| Data Type | Number of Samples | Goal: Estimate | Goal: Test a Claim |
|---|---|---|---|
| Categorical (Proportions) | 1 Sample | 1-Prop Z-Interval | 1-Prop Z-Test |
| Categorical (Proportions) | 2 Samples | 2-Prop Z-Interval | 2-Prop Z-Test |
| Quantitative (Means) | 1 Sample | 1-Sample t-Interval | 1-Sample t-Test |
| Quantitative (Means) | 2 Independent Samples | 2-Sample t-Interval | 2-Sample t-Test |
| Quantitative (Means) | Paired Data (1 group, 2 measurements) | Matched Pairs t-Interval | Matched Pairs t-Test |
Unit 7 Key Takeaways
Use t-distribution for means because σ is unknown.
Degrees of freedom: df = n − 1 (for 1-sample).
Check Normal/Large Sample: CLT (n ≥ 30) OR graph data to check for skew/outliers.
For 2-Sample Means: NEVER pool the variances.
Let the calculator find the messy 2-sample degrees of freedom.
End of Unit 7 Study Guide. Almost to the finish line!