Confidence Intervals & Hypothesis Testing
| Confidence Intervals | |||||
|---|---|---|---|---|---|
| Mean | Proportion | Two Means | |||
| σ Known | σ Unknown | ||||
| Formula | (x̄ − E, x̄ + E) | (x̄ − E, x̄ + E) | (p̂ − E, p̂ + E) |
((x̄₁ − x̄₂) − E, (x̄₁ − x̄₂) + E) |
|
|
Calculator Function |
ZInterval | TInterval | 1-PropZInt | 2-SampZInt | |
| Margin of Error | E = z α 2 σ √n | E = t α 2 s √n | E = z α 2 √ p̂(1 − p̂) n | E = z α 2 √ σ₁² ₁ + σ₂² ₂ | |
| Critical Value | zα/2 | tα/2 | zα/2 | zα/2 | |
| Requirements |
• n ≥ 30 • Approximately Normal |
• n ≥ 30 • Approximately Normal |
• np ≥ 10 • n(1 − p̂) ≥ 10 • Approximately Normal |
• ₁ ≥ 30 & ₂ ≥ 30 • Both Populations Approximately Normal |
|
| Key Phrases |
• Interval • Endpoints • Mean • Population Standard Deviation Known |
• Interval • Endpoints • Mean • Population Standard Deviation Unknown • Sample Standard Deviation |
• Interval • Endpoints • Proportion |
• Interval • Endpoints • Difference Between Two Means • Population Standard Deviation Known |
|
| Extra Equations |
• E =
upper endpoint − lower endpoint
2
• μ = upper endpoint + lower endpoint 2 |
N/A |
• E =
upper − lower
2
• p = upper + lower 2 • x = p̂ × n |
• E =
upper − lower
2
• μ₁ − μ₂ = upper + lower 2 |
|
| Critical Value Calculations | |||||
|---|---|---|---|---|---|
| Z Interval | T Interval | Proportion Interval | 2 Sample Z Interval | ||
| Symbol | zα/2 | tα/2 | zα/2 | zα/2 | |
| General |
• 2nd > Vars • invNorm |
• Stat > Tests • 8: TInterval • Stat |
• 2nd > Vars • invT |
Same as Z Interval | Same as Z Interval |
| TI-83 | 1 − c 2 , 0, 1 |
• x̄: 0 • Sx: √n • n: Sample Size • C-Level: c |
N/A | ||
| TI-84 |
• Area:
1 − c
2
• Mean: 0 • SD: 1 |
• x̄: 0 • Sx: √n • n: Sample Size • C-Level: c |
• Area:
1 − c
2
• DF: n − 1 |
||
| Endpoint Calculations | ||||
|---|---|---|---|---|
| Z Interval | T Interval | Proportion Interval | 2 Sample Z Interval | |
| General |
• Stat > Tests • 7: ZInterval • Stat if Numbers • Data if List (Stat > Edit) |
• Stat > Tests • 8: TInterval • Stat if Numbers • Data if List (Stat > Edit) |
• Stat > Tests • A: 1-PropZInt |
• Stat > Tests • 9: 2-SampZInt • Stat if Numbers • Data if Lists (Stat > Edit) |
| Calculator Input |
• σ: Population SD • x̄: Sample Mean • n: Sample Size • C-Level: c |
• x̄: Sample Mean • Sx: Sample SD • n: Sample Size • C-Level: c |
• x: Number of Successes • n: Sample Size • C-Level: c |
• σ1: Population SD Group 1 • σ2: Population SD Group 2 • x̄1: Sample Mean Group 1 • n1: Sample Size Group 1 • x̄2: Sample Mean Group 2 • n2: Sample Size Group 2 • C-Level: c |
| Hypothesis Tests | |||||
|---|---|---|---|---|---|
| Mean | Proportion | Two Means | |||
| σ Known | σ Unknown | ||||
| Calculator Function | Z-Test | T-Test | 1-PropZTest | 2-SampZTest | |
| Alternative Hypotheses |
• μ ≠ μ • μ < μ • μ > μ |
• μ ≠ μ • μ < μ • μ > μ |
• p ≠ p • p < p • p > p |
• μ₁ ≠ μ₂ • μ₁ < μ₂ • μ₁ > μ₂ |
|
| Test Statistic | z = x̄ − μ ( σ √n ) | t = x̄ − μ ( S √n ) | z = p̂ − p √ (1−p) n | z = (x̄₁ − x̄₂) − (μ₁ − μ₂) √ σ₁² ₁ + σ₂² ₂ | |
| Requirements |
• n ≥ 30 • Approximately Normal |
• n ≥ 30 • Approximately Normal |
• np ≥ 10 • n(1-p) ≥ 10 •Approximately Normal |
• ₁ ≥ 30 & ₂ ≥ 30 • Both Populations Approximately Normal |
|
| Key Phrases |
• Test • Claim • Sufficient Evidence • Mean • Population Standard Deviation Known |
• Test • Claim • Sufficient Evidence • Mean • Population Standard Deviation Unknown • Sample Standard Deviation |
• Test • Claim • Sufficient Evidence • Proportion |
• Test • Claim • Sufficient Evidence • Difference Between Two Means • Population Standard Deviation Known |
|
| Conclusions | p-value ≤ α |
Reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis. |
|||
| p-value > α |
Do not reject the null hypothesis. There is not sufficient evidence to support the alternative hypothesis. |
||||
| Test Calculations | ||||
|---|---|---|---|---|
| Z Test | T Test | Proportion Test | 2 Sample Z Test | |
| General |
• Stat > Tests • 1: Z-Test • Stat if Numbers • Data if List (Stat > Edit) |
• Stat > Tests • 2: T-Test • Stat if Numbers • Data if List (Stat > Edit) |
• Stat > Tests • 5: 1-PropZTest |
• Stat > Tests • 3: 2-SampZTest • Stat if Numbers • Data if Lists (Stat > Edit) |
| Calculator Input |
• μ₀: Null Hypothesis • σ: Population SD • x̄: Sample Mean • n: Sample Size • μ: ≠ μ₀ < μ₀ > μ₀ Alternative Hypothesis |
• μ₀: Null Hypothesis • x̄: Sample Mean • Sx: Sample SD • n: Sample Size • μ: ≠ μ₀ < μ₀ > μ₀ Alternative Hypothesis |
• p₀: Null Hypothesis • x: Number of Successes • n: Sample Size • prop: ≠ p₀ < p₀ > p₀ Alternative Hypothesis |
• σ1: Population SD Group 1 • σ2: Population SD Group 2 • x̄1: Sample Mean Group 1 • n1: Sample Size Group 1 • x̄2: Sample Mean Group 2 • n2: Sample Size Group 2 • μ1: ≠ μ2 < μ2 > μ2 Alternative Hypothesis |
