Two-Sample Hypothesis Testing

• Example:A professor has designed an experiment to test the effect of reading the textbook before attempting to complete a homework assignment. Four students who read the textbook before attempting the homework recorded the following times (in hours) to complete the assignment:

3.1, 2.8, 0.5, 1.9 hours

Five students who did not read the textbook before attempting the homework recorded the following times to complete the assignment:

0.9, 1.4, 2.1, 5.3, 4.6 hours

Two-Sample Hypothesis Testing• Define the difference in the two means as:

μ1 - μ2 = d0

• What are the Hypotheses?

H0: _______________

H1: _______________

or

H1: _______________

or

H1: _______________

Our Example

Reading:

n1 = 4 x1 = 2.075 s12 = 1.363

No reading:

n2 = 5 x2 = 2.860 s22 = 3.883

If we assume the population variances are “equal”, we can calculate sp

2 and conduct a __________.

= __________________

2

)1()1(

21

222

2112

nn

snsnsp

Your turn …

• Lower-tail test ((μ1 - μ2 < 0)– “Fixed α” approach (“Approach 1”) at α = 0.05 level.– “p-value” approach (“Approach 2”)

• Upper-tail test (μ2 – μ1 > 0)– “Fixed α” approach at α = 0.05 level.– “p-value” approach

• Two-tailed test (μ1 - μ2 ≠ 0)– “Fixed α” approach at α = 0.05 level.– “p-value” approach

Recall 21

021

/1/1

)(

nns

dxxt

p

calc

Lower-tail test ((μ1 - μ2 < 0)

• Draw the picture:

• Solution:

• Decision:

• Conclusion:

Upper-tail test (μ2 – μ1 > 0)

• Draw the picture:

• Solution:

• Decision:

• Conclusion:

Two-tailed test (μ1 - μ2 ≠ 0)

• Draw the picture:

• Solution:

• Decision:

• Conclusion:

Another Example

Suppose we want to test the difference in carbohydrate content between two “low-carb” meals. Random samples of the two meals are tested in the lab and the carbohydrate content per serving (in grams) is recorded, with the following results:

n1 = 15 x1 = 27.2 s12 = 11

n2 = 10 x2 = 23.9 s22 = 23

tcalc = ______________________

ν = ________________ (using equation in table 10.2)

Example (cont.)

• What are our options for hypotheses?

• At an α level of 0.05,

– One-tailed test, t0.05, 15 = ________

– Two-tailed test, t0.025, 15 = ________

• How are our conclusions affected?