Confidence Intervals 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 Normal OR $n \\geq 30$ (CLT)
3. Independent: $n < 10\\%$ of population

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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: $(42.02, 48.38)$

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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 $H_0$.

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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

Do NOT pool variances unless told the populations have equal variance (which is rare on the AP exam).

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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.

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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.)

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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
• Use a Normal probability plot to assess Normality for small $n$
• Degrees of freedom determine the shape of the t-distribution

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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)