🎯⭐ INTERACTIVE LESSON

Confidence Intervals for Means

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Confidence Intervals for Means - Complete Interactive Lesson

Part 1: Inference for Means Basics

📊 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 $df \\to \\infty$ , $t t o N ($

Conditions for t-Procedures

Random: Data from random sample or experiment
Normal/Large Sample: Population is Normal OR $n \\geq 30$ (CLT)
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

Part 2: T-Distribution

📏 Confidence Intervals for Means

Part 2 of 7 — One-Sample t Interval

Formula

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

Part 3: Confidence Intervals for Means

⚖️ Hypothesis Tests for Means

Part 3 of 7 — One-Sample t Test

Test Statistic

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

Part 4: Hypothesis Tests for Means

📊 Two-Sample t-Procedures

Part 4 of 7 — Comparing Two Means

Two-Sample t-Interval

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

Part 5: Matched Pairs

🤝 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

Compute differences $d = x_1 - x_2$ for each pair

Part 6: Problem-Solving Workshop

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

Choosing the Right Procedure

Scenario	Procedure
One mean, $\\sigma$ unknown	One-sample t
Two independent means	Two-sample t
Paired data	Matched pairs t
One proportion	One-sample z
Two proportions	Two-sample z

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 ( $\\bar{x}$ , , , etc.)

Part 7: Mixed Review

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Summary Table

Procedure	Statistic	SE	df
One-sample t	$\\bar{x}$	$s/\\sqrt{n}$

= s / s q r t n = ?

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

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

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

$n = 20$ , $\\bar{x} = 45.2$ , $s = 6.8$ , 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?

Steps (4-Step Process)

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

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

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

$df = 15$ . 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) $df = ?$

sqrtfracs12n1+fracs22n2

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

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

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

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)

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.

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) $df = ?$

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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).

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)

Confidence Intervals for Means

Interpretation

Example

Steps (4-Step Process)

Example

Two-Sample t-Test

Degrees of Freedom

Key Point

Example

Key Reminders