Inference: Two-Sample Hypothesis Tests (Chapter 11) I. Hypothesis Tests for Two Population Proportions (Two-Sample z-test for Proportions) Assumptions: Independent samples where n1 p1 ≥ 10 , n1 (1− p1 ) ≥ 10 , n2 p2 ≥ 10 , and n2 (1− p2 ) ≥ 10 . Step 1: State the null and alternative hypotheses. Left-Tailed Right-Tailed Null H0 : π1 − π 2 = δ 0 H0 : π1 − π 2 = δ 0 Alternative Ha : π 1 − π 2 < δ0 H a : π 1 − π 2 > δ0
Two-Tailed H0 : π1 − π 2 = δ 0 H a : π 1 − π 2 ≠ δ0
Step 2: Decide on the significance level, α. α = _______ Step 3: Determine the critical value(s) and rejection region(s). Classical Approach: Left-Tailed Right-Tailed Critical Value(s) − zα zα Rejection Region(s)
Z ≤ −zα
Z ≥ zα
Two-Tailed − zα and zα 2
Z ≤ −zα
2
2
or Z ≥ zα
2
P-value Approach: Reject the null hypothesis if p-value ≤ α. Step 4: Compute the value of the test statistic and p-value (if necessary). Left-Tailed Right-Tailed Two-Tailed Test Statistic ( p1 − p2 ) − 0 ( p1 − p2 ) − 0 ( p1 − p2 ) − 0 Z= Z= Z= when δ0 = 0 ⎛1 1⎞ ⎛1 1⎞ ⎛1 1⎞ (pooled) pc (1− pc )⎜ + ⎟ pc (1− pc )⎜ + ⎟ pc (1− pc )⎜ + ⎟ ⎝ n1 n2 ⎠ ⎝ n1 n2 ⎠ ⎝ n1 n2 ⎠ Test Statistic ( p1 − p2) − δ0 ( p1 − p2) − δ0 ( p1 − p2) − δ0 Z= Z= Z= when δ0 ≠ 0 p1(1− p1 ) p2 (1− p2 ) p1(1− p1 ) p2 (1− p2 ) p1(1− p1 ) p2 (1− p2 ) (non-pooled) + + + n n n1 n2 n1 n2 1 2 p-value
P( z ≤ Z )
P( z ≥ Z )
P( z ≤ − Z or z ≥ Z )
n1 p1 + n 2 p2 y1 + y 2 = is the combined (pooled) sample proportion. n1 + n2 n1 + n2 Note 2: P( z ≤ − Z or z ≥ Z ) = 2⋅ P(z ≥ Z ) = 2 ⋅ P( z ≤ − Z ) Note 1: pc =
Step 5: Make your decision. If the value of the test statistic falls in the rejection region(s) or if p-value ≤ α, then reject H0 ; otherwise, fail to reject H0 . Step 6: State the conclusion in words. Reject H0 : Fail to Reject H0 :
“At the α = ____ level of significance, there is enough evidence to conclude that ( Ha in words).” “At the α = ____ level of significance, there is not enough evidence to conclude that ( Ha in words).”
II.
Hypothesis Tests for Two Population Means (Independent Samples, Variances Known) Assumption: Independent samples (from approx. normal pops.) where the population variances are known. Step 1: State the null and alternative hypotheses. Left-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 < δ 0
Null Alternative
Right-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 > δ 0
Two-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 ≠ δ 0
Step 2: Decide on the significance level, α. α = _______
Step 3: Determine the critical value(s) and rejection region(s). Classical Approach: Left-Tailed Critical Value(s) − zα
Right-Tailed zα
Two-Tailed − zα and zα
Z ≥ zα
Z ≤ −zα
Z ≤ −zα
Rejection Region(s)
2
2
2
or Z ≥ zα
2
P-value Approach: Reject the null hypothesis if p-value ≤ α. Step 4: Compute the value of the test statistic and p-value (if necessary). Left-Tailed Test Statistic
p-value
Z=
Right-Tailed
( y1 − y 2 ) − δ 0
(σ
2 1
P( z ≤ Z )
n1 ) + (σ 22 n2 )
Z=
( y1 − y 2 ) − δ 0
(σ
2 1
P( z ≥ Z )
n1 ) + (σ 22 n2 )
Two-Tailed Z=
( y1 − y 2 ) − δ 0
(σ
2 1
n1 ) + (σ 22 n2 )
P( z ≤ − Z or z ≥ Z )
Note: P( z ≤ − Z or z ≥ Z ) = 2⋅ P(z ≥ Z ) = 2 ⋅ P( z ≤ − Z ) Step 5: Make your decision. If the value of the test statistic falls in the rejection region(s) or if p-value ≤ α, then reject H0 ; otherwise, fail to reject H0 . Step 6: State the conclusion in words. Reject H0 : Fail to Reject H0 :
“At the α = ____ level of significance, there is enough evidence to conclude that ( Ha in words).” “At the α = ____ level of significance, there is not enough evidence to conclude that ( Ha in words).”
III. Hypothesis Tests for Two Normal Population Means Using Non-Pooled Variances (Independent Samples, Unequal Unknown Variances) Assumption: Independent samples from normal populations with unequal, unknown variances. Step 1: State the null and alternative hypotheses. Null Alternative
Left-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 < δ 0
Right-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 > δ 0
Two-Tailed H0 : µ1 − µ2 = δ 0 H a : µ1 − µ2 ≠ δ 0
Step 2: Decide on the significance level, α. α = _______
Step 3: Determine the critical value(s) and rejection region(s). Classical Approach: Left-Tailed Critical Value(s) −tα T ≤ −tα
Rejection Region(s)
Note: df = υ , where υ =
(
s12 n1
+
( s12 n1 )2 n1 −1
+
s 22 n2
)
( s22
Right-Tailed tα
Two-Tailed −tα and tα
T ≥ tα
T ≤ −tα
2
2
2
or T ≥ tα
2
2
n2
)2
rounded down to the nearest integer.
n 2 −1
P-value Approach: Reject the null hypothesis if p-value ≤ α. Step 4: Compute the value of the test statistic and p-value (if necessary). Left-Tailed Test Statistic
p-value
T=
Right-Tailed
( y1 − y 2) − δ0
(s
2 1
P(t ≤ T )
n1 ) + ( s n2 ) 2 2
T=
Two-Tailed
( y1 − y 2) − δ0
(s
2 1
P(t ≥ T )
n1 ) + ( s n2 ) 2 2
T=
( y1 − y 2) − δ0
(s
2 1
n1 ) + ( s22 n2 )
P(t ≤ − T or t ≥ T )
Note 1: df = υ (see Step 3) Note 2: P(t ≤ − T or t ≥ T ) = 2 ⋅ P (t ≥ T ) Note 3: T-test p-values can be estimated using Table IV of the textbook. Step 5: Make your decision. If the value of the test statistic falls in the rejection region(s) or if p-value ≤ α, then reject H0 ; otherwise, fail to reject H0 . Step 6: State the conclusion in words. Reject H0 : Fail to Reject H0 :
“At the α = ____ level of significance, there is enough evidence to conclude that ( Ha in words).” “At the α = ____ level of significance, there is not enough evidence to conclude that ( Ha in words).”
IV. Paired-Sample Hypothesis Tests for Two Population Means with Variance Known (Paired z-test) Assumptions: Paired samples; differences are from a normal population with known variance. NOTE: This is simply the one-sample z-test procedure performed on the differences. Step 1: State the null and alternative hypotheses. Null Alternative
Left-Tailed H0 : µd = δ 0 H a : µd < δ 0
Right-Tailed H0 : µd = δ 0 H a : µd > δ 0
Two-Tailed H0 : µd = δ 0 H a : µd ≠ δ 0
Step 2: Decide on the significance level, α. α = _______
Step 3: Determine the critical value(s) and rejection region(s). Classical Approach: Left-Tailed Critical Value(s) − zα Rejection Region(s)
Right-Tailed zα
Two-Tailed − zα and zα
Z ≥ zα
Z ≤ −zα
Z ≤ −zα
2
2
2
or Z ≥ zα
2
P-value Approach: Reject the null hypothesis if p-value ≤ α. Step 4: Compute the value of the test statistic and p-value (if necessary). First, calculate the paired differences, y d = y1 − y 2 , of the sample pairs. Test Statistic
Left-Tailed y − δ0 Z= d σd n d
p-value
P( z ≤ Z )
Right-Tailed y − δ0 Z= d σd n d P( z ≥ Z )
Two-Tailed y − δ0 Z= d σd n d
P( z ≤ − Z or z ≥ Z )
Note 1: P( z ≤ − Z or z ≥ Z ) = 2⋅ P(z ≥ Z ) = 2 ⋅ P( z ≤ − Z ) Note 2: σ d2 = σ y21 + σ 2y2 − 2σ y1 y 2 , where σ y1 y 2 is the covariance for the two populations Step 5: Make your decision. If the value of the test statistic falls in the rejection region(s) or if p-value ≤ α, then reject H0 ; otherwise, fail to reject H0 . Step 6: State the conclusion in words. Reject H0 : Fail to Reject H0 :
“At the α = ____ level of significance, there is enough evidence to conclude that ( Ha in words).” “At the α = ____ level of significance, there is not enough evidence to conclude that ( Ha in words).”
V. Paired-Sample Hypothesis Tests for Two Pop. Means with Variance Unknown (Paired t-test) Assumptions: Paired samples; differences are from a normal population with unknown variance NOTE: This is simply the one-sample t-test procedure performed on the differences. Step 1: State the null and alternative hypotheses. Null Alternative
Left-Tailed H0 : µd = δ 0 H a : µd < δ 0
Right-Tailed H0 : µd = δ 0 H a : µd > δ 0
Two-Tailed H0 : µd = δ 0 H a : µd ≠ δ 0
Step 2: Decide on the significance level, α. α = _______
Step 3: Determine the critical value(s) and rejection region(s). Classical Approach: Left-Tailed Critical Value(s) −tα T ≤ −tα
Rejection Region(s)
Right-Tailed tα
Two-Tailed −tα and tα
T ≥ tα
T ≤ −tα
2
2
2
or T ≥ tα
2
Note: df = nd − 1 P-value Approach: Reject the null hypothesis if p-value ≤ α. Step 4: Compute the value of the test statistic and p-value (if necessary). First, calculate the paired differences, y d = y1 − y 2 , of the sample pairs. Test Statistic
Left-Tailed y − δ0 T= d sd nd
p-value
P(t ≤ T )
Right-Tailed y − δ0 T= d sd nd P(t ≥ T )
Two-Tailed y − δ0 T= d sd nd
P(t ≤ − T or t ≥ T )
Note 1: df = nd − 1 Note 2: P(t ≤ − T or t ≥ T ) = 2 ⋅ P (t ≥ T ) Note 3: T-test p-values can be estimated using Table IV of the textbook.
Note 4: sd =
∑y
2 d
(∑ y ) −
2
d
nd nd − 1
Step 5: Make your decision. If the value of the test statistic falls in the rejection region(s) or if p-value ≤ α, then reject H0 ; otherwise, fail to reject H0 . Step 6: State the conclusion in words. Reject H0 : Fail to Reject H0 :
“At the α = ____ level of significance, there is enough evidence to conclude that ( Ha in words).” “At the α = ____ level of significance, there is not enough evidence to conclude that ( Ha in words).”