Tests for Means

📊 Inference for Means

Part 1 of 7 — The t-Distribution

Why Not Z?

For means, we rarely know the population standard deviation $\sigma$. We estimate it with the sample standard deviation $s$, introducing extra uncertainty.

The t-Distribution

$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$

Properties:

• Bell-shaped and symmetric around 0
• Wider tails than Normal (more spread)
• Depends on degrees of freedom $df = n - 1$
• As $df \to \infty$, $t \to N(0,1)$

Conditions for t-Procedures

1. Random: Data from random sample or experiment
2. Normal/Large Sample: Population is approximately Normal, OR $n \geq 30$ (CLT), OR (if $n < 30$) the sample data are free from strong skewness and outliers
3. Independent: $n < 10\%$ of population

Concept Check 🎯

t-Distribution Basics 🧮

$n = 25$, $\bar{x} = 82$, $s = 10$.

1) Degrees of freedom?

2) Standard error $= s/\sqrt{n} = ?$

3) t-statistic for testing $\mu = 80$: $t = (82 - 80)/SE = ?$

📏 Confidence Intervals for Means

Part 2 of 7 — One-Sample t Interval

Formula

$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

where $t^*$ comes from the t-table with $df = n - 1$.

Interpretation

“We are [C]% confident that the true mean [context] is between [lower] and [upper].”

Example

$n = 20$, $\bar{x} = 45.2$, $s = 6.8$, 95% CI.

$df = 19$, $t^* = 2.093$ (from table)

$45.2 \pm 2.093 \times \frac{6.8}{\sqrt{20}} = 45.2 \pm 3.18$

CI: $(42.02, 48.38)$

Concept Check 🎯

t-Interval 🧮

$n = 36$, $\bar{x} = 110$, $s = 12$, 95% CI ($t^* \approx 2.030$ for $df = 35$).

1) $SE = s/\sqrt{n} = ?$

2) Margin of error $= t^* \times SE = ?$ (round to 1 place)

3) Lower bound of CI?

⚖️ Hypothesis Tests for Means

Part 3 of 7 — One-Sample t Test

Test Statistic

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

Steps (4-Step Process)

1. State: $H_0: \mu = \mu_0$ vs. $H_a: \mu \neq \mu_0$ (or $<$ or $>$)
2. Plan: Check Random, Normal, Independent conditions
3. Do: Calculate $t$ and find p-value using $t$-table with $df = n-1$
4. Conclude: Compare p-value to $\alpha$, interpret in context

Example

$H_0: \mu = 100$, $H_a: \mu > 100$. $n = 16$, $\bar{x} = 106$, $s = 12$.

$t = \frac{106 - 100}{12/\sqrt{16}} = \frac{6}{3} = 2.0$

$df = 15$. From the t-table, $P(t > 2.0) \approx 0.032$.

Since $0.032 < 0.05$, reject $H_0$.

Concept Check 🎯

t-Test 🧮

$H_0: \mu = 50$, $H_a: \mu \neq 50$. $n = 25$, $\bar{x} = 53$, $s = 5$.

1) $SE = ?$

2) $t = ?$

3) $df = ?$

📊 Two-Sample t-Procedures

Part 4 of 7 — Comparing Two Means

Two-Sample t-Interval

$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Two-Sample t-Test

$t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Degrees of Freedom

Use the calculator’s Welch’s approximation (complex formula), or the conservative approach:

$df = \min(n_1 - 1, n_2 - 1)$

Key Point

AP two-sample $t$ procedures never pool: each sample keeps its own $s$ in the SE. The technology df lies between $\min(n_1 - 1, n_2 - 1)$ and $n_1 + n_2 - 2$.

Concept Check 🎯

Two-Sample Comparison 🧮

Group A: $\bar{x}_1 = 78, s_1 = 10, n_1 = 30$. Group B: $\bar{x}_2 = 72, s_2 = 12, n_2 = 25$.

1) Point estimate for $\mu_1 - \mu_2$?

2) Conservative $df = ?$

3) $SE = \sqrt{10^2/30 + 12^2/25}$ = ? (round to 2 places)

🤝 Matched Pairs

Part 5 of 7 — Paired t-Procedures

When to Use Paired t

• Same subjects measured twice (before/after)
• Subjects matched in pairs
• Two measurements on the same item

Procedure

1. Compute differences $d = x_1 - x_2$ for each pair
2. Perform a one-sample t-test on the differences

$t = \frac{\bar{d} - 0}{s_d/\sqrt{n}}$

where $n$ = number of pairs, $\bar{d}$ = mean of differences, $s_d$ = SD of differences.

Example

10 patients’ blood pressure before and after a drug: $\bar{d} = -8.5$ (mean decrease), $s_d = 6.2$

$t = \frac{-8.5 - 0}{6.2/\sqrt{10}} = \frac{-8.5}{1.96} = -4.34$

$df = 9$. Strong evidence that the drug reduces blood pressure.

Concept Check 🎯

Matched Pairs 🧮

12 students take a test before and after tutoring. Mean difference $\bar{d} = 5.0$ (improvement), $s_d = 3.6$.

1) $SE = s_d/\sqrt{n} = ?$ (round to 2 places)

2) $t = \bar{d}/SE = ?$ (round to 2 places)

3) $df = ?$

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

Choosing the Right Procedure

AP Scoring Tips

• Name the procedure explicitly (“one-sample t-test”, not just “t-test”)
• Always identify the parameter in context
• State ALL conditions, not just assume them
• Use proper notation ($\bar{x}$, $s$, $t$, etc.)

Concept Check 🎯

Procedure Selection 🧮

Name the correct procedure for each:

1) Estimating the mean GPA of all students at a school based on a random sample of 50.

2) Comparing mean test scores between students who used a study app vs. those who didn’t.

3) Testing whether a training program improved employees’ productivity (measured before and after).

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Summary Table

Key Reminders

• t-procedures are robust against non-Normality for large $n$
• Check for outliers with small samples
• For $n < 30$, check a dotplot or boxplot of the sample for strong skewness or outliers
• Degrees of freedom determine the shape of the t-distribution

Concept Check 🎯

Final Challenge 🧮

$n = 50$, $\bar{x} = 25.3$, $s = 4.0$. Test $H_0: \mu = 24$ vs. $H_a: \mu \neq 24$ at $\alpha = 0.05$.

1) $t = ?$ (round to 2 places)

2) $df = ?$

3) With $|t| = 2.30$ and $df = 49$, is the result significant at $\alpha = 0.05$? (yes/no)