Ap Statistics Test Review
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Inference Formulas & Procedures Situation/Tes t 1 sample z test
2 sample z test 1 sample t test
2 sample t test
Matched pairs t test
Conditions
Null Hypothesis
1. SRS And either: 2. Pop is normal, or large sample size ( n ≥ ), or if n<30, the sample dist is not severely skewed and does not have extreme outliers. Same as above for BOTH samples
H 0 :µ = µ
1. SRS and either: 2. Pop is normal, or n>40 (even if skewed & outliers), or 15
H 0 : µ = µ or H 0 : µ − µ = H 0 :µ = µ
H 0 :µ = µ
H 0 :µ = where µ d is the difference of dependent
Test Statistic z=
z=
t=
t=
t=
Confidence Interval
− µ σ
σ x ±
− − µ − µ
σ σ (x1 − ± +
σ σ + − µ ; df = n-1
x ± ; df = n-1
− − µ − µ
(x1 − ± + ; df = n-1
+
− µ ; df = n-1
x ± ; df = n-1
1 proportion z test
2 proportion z test
Chi Square Goodness of Fit
Chi Square Homogeneit y
1. SRS 2. Population ≥ 10n 3. CI: npö ≥ , n(1 − ≥ HT: np0 ≥ , n(1 − ≥ (Verify use of normal approx) 1. Populations are independent 2. SRS (both samples) 3. Populations ≥ 10n 4. CI: n1 pö1 ≥ , n1 (1 − ≥ n2 pö2 ≥ , n2 (1 − ≥ HT: n1 pö ≥ , n1 (1 − ≥ n2 pö ≥ , n2 (1 − ≥ (Verify use of normal approx) 1. All expected counts ≥ 1 2. 80% of expected counts ≥ 5
Same as above.
Chi Square Same as above. Independenc e
Linear Regression; regression slope
1. Independent ordered pairs 2. Roughly linear scatterplot
sample means H 0 : p =
H 0 : p1 = or H 0 : p1 − =
H 0 : Actual proportions are equal to hypothesized proportions H 0 : Proportions are all equal or a fixed value H 0 : No associations between variables or the variables are independent of each other H 0 :β = no linear relationship between the two variables
z=
z=
− −
− − −
− + + where pö = +
pö ±
−
( pö1 − ±
− − +
(O - E)2 E df = k-1 k = # of categories
One-way table Single SRS, one categorical variable
(O - E)2 E df = (r-1)(c-1) (O - E)2 χ2 = å E df = (r-1)(c-1)
Two-way table Two SRS, two categorical variables
χ2 = å
χ2 = å
t=
, df = n-2
Two way table Single SRS, two categorical variables b ± ; df = n-2
3. Standard deviation of response is constant (scatter around LSRL is consistent) 4. Response variable varies normally (residuals are approx normal)
2 sample z test 1 sample t test
2 sample t test
Matched pairs t test
Conditions
Null Hypothesis
1. SRS And either: 2. Pop is normal, or large sample size ( n ≥ ), or if n<30, the sample dist is not severely skewed and does not have extreme outliers. Same as above for BOTH samples
H 0 :µ = µ
1. SRS and either: 2. Pop is normal, or n>40 (even if skewed & outliers), or 15
H 0 : µ = µ or H 0 : µ − µ = H 0 :µ = µ
H 0 :µ = µ
H 0 :µ = where µ d is the difference of dependent
Test Statistic z=
z=
t=
t=
t=
Confidence Interval
− µ σ
σ x ±
− − µ − µ
σ σ (x1 − ± +
σ σ + − µ ; df = n-1
x ± ; df = n-1
− − µ − µ
(x1 − ± + ; df = n-1
+
− µ ; df = n-1
x ± ; df = n-1
1 proportion z test
2 proportion z test
Chi Square Goodness of Fit
Chi Square Homogeneit y
1. SRS 2. Population ≥ 10n 3. CI: npö ≥ , n(1 − ≥ HT: np0 ≥ , n(1 − ≥ (Verify use of normal approx) 1. Populations are independent 2. SRS (both samples) 3. Populations ≥ 10n 4. CI: n1 pö1 ≥ , n1 (1 − ≥ n2 pö2 ≥ , n2 (1 − ≥ HT: n1 pö ≥ , n1 (1 − ≥ n2 pö ≥ , n2 (1 − ≥ (Verify use of normal approx) 1. All expected counts ≥ 1 2. 80% of expected counts ≥ 5
Same as above.
Chi Square Same as above. Independenc e
Linear Regression; regression slope
1. Independent ordered pairs 2. Roughly linear scatterplot
sample means H 0 : p =
H 0 : p1 = or H 0 : p1 − =
H 0 : Actual proportions are equal to hypothesized proportions H 0 : Proportions are all equal or a fixed value H 0 : No associations between variables or the variables are independent of each other H 0 :β = no linear relationship between the two variables
z=
z=
− −
− − −
− + + where pö = +
pö ±
−
( pö1 − ±
− − +
(O - E)2 E df = k-1 k = # of categories
One-way table Single SRS, one categorical variable
(O - E)2 E df = (r-1)(c-1) (O - E)2 χ2 = å E df = (r-1)(c-1)
Two-way table Two SRS, two categorical variables
χ2 = å
χ2 = å
t=
, df = n-2
Two way table Single SRS, two categorical variables b ± ; df = n-2
3. Standard deviation of response is constant (scatter around LSRL is consistent) 4. Response variable varies normally (residuals are approx normal)
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