## Inference: Two-Sample Hypothesis Tests

Inference: Two-Sample Hypothesis Tests (Chapter 11) I. Hypothesis Tests for Two Population Proportions (Two-Sample z-test for Proportions) Assumptions...
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

2 1

P( z ≤ Z )

n1 ) + (σ 22 n2 )

Z=

( y1 − y 2 ) − δ 0

2 1

P( z ≥ Z )

n1 ) + (σ 22 n2 )

Two-Tailed Z=

( y1 − y 2 ) − δ 0

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).”