Confidence Intervals for Means

Why t-Distribution?

Problem: Population σ usually unknown

Solution: Use sample standard deviation s, but this adds uncertainty

Result: Use t-distribution instead of normal

t-distribution:

• Similar to normal (symmetric, bell-shaped)
• Heavier tails (accounts for extra uncertainty from using s)
• Depends on degrees of freedom (df = n - 1)
• As df increases, approaches normal

One-Sample t-Interval for Mean

Formula:

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

Where:

• $\bar{x}$ = sample mean
• s = sample standard deviation
• n = sample size
• t* = critical value from t-distribution with df = n - 1

Conditions for t-Interval

Random: Random sample
Normal: Population approximately normal OR n ≥ 30 (CLT)
Independent: n < 10% of population (if sampling without replacement)

For normality:

• If n < 15: Data must be very close to normal (check with plot)
• If 15 ≤ n < 30: Data should be roughly symmetric, no outliers
• If n ≥ 30: Can proceed unless severe outliers or extreme skew

Finding t* Critical Value

Calculator: invT(area to left, df)

Example: 95% CI with n = 20 (df = 19)

• Area to left = (1 + 0.95)/2 = 0.975
• invT(0.975, 19) ≈ 2.093

Table: Look up df and confidence level

Example 1: Simple t-Interval

Test scores: n = 25, $\bar{x}$ = 78, s = 12

95% CI:

Conditions:

• Random: Assume ✓
• Normal: n = 25, assume roughly normal ✓
• Independent: 25 < 10% of students ✓

Calculate:

• df = 25 - 1 = 24
• t* = 2.064 (from table/calculator)
• SE = 12/√25 = 2.4

$CI = 78 \pm 2.064(2.4) = 78 \pm 4.95$

$(73.05, 82.95)$

Interpretation: We are 95% confident the true mean score is between 73.05 and 82.95.

t vs z

Use z when:

• Known population σ (rare!)
• Working with proportions

Use t when:

• Unknown σ, using sample s (almost always for means!)

Key difference: t has heavier tails → wider intervals (more conservative)

Sample Size for Desired ME

Challenge: ME depends on s, which we don't know in advance

Approach:

1. Estimate s from pilot study or similar data
2. Use conservative t* (larger than final value)
3. Calculate n
4. Round up

Formula:

$n = \left(\frac{t^* s}{m}\right)^2$

Example 2: Checking Normality

Small sample (n = 12):

• MUST check for approximate normality
• Use dotplot, boxplot, or normal probability plot
• Look for: symmetric shape, no outliers, no severe skew

If data skewed or has outliers with small n: t-procedures NOT appropriate

Two-Sample t-Interval

Comparing two means:

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

df: Use calculator (complex formula) or conservative: min(n₁-1, n₂-1)

Conditions: Both samples meet conditions

Interpretation: If interval contains 0, no significant difference

Paired Data

When data naturally paired:

• Before/after on same subjects
• Twins, matched pairs

Analyze differences:

1. Calculate difference for each pair: d = x₁ - x₂
2. One-sample t-interval on differences

$\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

Where n = number of pairs, df = n - 1

Example 3: Paired Data

Blood pressure before/after medication (n = 15 patients):

• $\bar{d}$ = 8.2 (average decrease)
• s_d = 5.1

90% CI for mean decrease:

• df = 14
• t* = 1.761
• SE = 5.1/√15 ≈ 1.317

$CI = 8.2 \pm 1.761(1.317) = 8.2 \pm 2.32$

$(5.88, 10.52)$

Interpretation: We are 90% confident medication reduces blood pressure by 5.88 to 10.52 points on average.

Interpreting Confidence Level

Same as for proportions:

95% means if we repeated sampling many times, about 95% of intervals would contain true μ

NOT: "95% of data in interval" or "95% chance μ in interval"

Effect of Sample Size

Larger n:

• Smaller SE (dividing by √n)
• More df → smaller t* (approaches z*)
• Result: Narrower CI (more precise)

Trade-off: Cost and time of collecting larger sample

Robustness of t-Procedures

t-procedures fairly robust to violations of normality if:

• n reasonably large (≥ 30)
• No extreme outliers

Less robust for:

• Small samples with skewness
• Extreme outliers (affect both $\bar{x}$ and s)

Calculator Commands (TI-83/84)

STAT → TESTS → 8:TInterval

Enter:

• Data or Stats
• If Stats: $\bar{x}$, s, n
• C-Level
• Calculate

For two-sample: 0:2-SampTInt

Common Mistakes

❌ Using z* instead of t*
❌ Using t* from wrong df
❌ Not checking normality with small samples
❌ Confusing paired with two-sample
❌ Misinterpreting confidence level

Quick Reference

Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$ with df = n - 1

Conditions: Random, approximately normal (or n ≥ 30), independent

Use t (not z) when σ unknown, using s

Paired data: Analyze differences with one-sample t

Remember: t-distribution accounts for extra uncertainty from estimating σ with s. Always check conditions, especially normality for small samples!