5 Basic Steps in Any Hypothesis Test

Step 1: Determine hypotheses (H0 and Ha).

Step 2: Verify necessary conditions, compute an appropriate

test statistic.

Step 3: Assuming H0 is true, find Decision Rule

Step 4: Decide whether or not Reject H0.

Step 5: Report the conclusion in the context of the problem.

H0: μd = 0 (μ1 –μ2=0)

Ha: μd > 0 (μ1 –μ2>0)

upper-sided

Ha: : μd ≠ 0 (μ1 –μ2≠0)

two-sided

Ha: μd <0 (μ1 –μ2<0)

lower-sided

Hypothesis Test for Population

Mean Difference (H0: μd =μd0)

Required assumptions:

Data are 2 dependent random samples (on the

same set of objects)

Large n(≥30) or Normal Distribution of

differences X1 - X2

Test Statistics:

Population Difference in Means

Test

Scenario

Data Population

Parameter

Sample

Statistics

Response Explanatory

Variable

Population

Difference

in Means

2

Independent

Samples

Numerical

(GPA,

Weight)

Group

(Gender)

Do students who live on campus tend to spend more time studying

in the library than students who live off-campus, on average?

Do women spend more money than men, on average?

Do women have more shoes than men, on average?

Name Scenario

A researcher is studying the effect of a new teaching

technique for middle school students.

A class of 30 students is taught using the new technique

and their mean score on a standardized test is compared to

the mean score of a class of 27 students who were taught

using the old technique.

One-sample test

Two independent sample test

Paired test

Name Scenario

A company claims that the economy size version of their

product contains 32 ounces.

A consumer group decides to test the claim by examining a

random sample of 100 economy size boxes of the product,

since they have received reports that the boxes contain less

than the 32 ounces claimed.

One-sample test

Two independent sample test

Paired test

Name Scenario

At some universities, an athletic director decides to test

whether or not athletes do in fact have lower GPAs. If

there is evidence of lower GPAs, additional academic

support for athletes will be instituted.

A random sample of 200 student athletes and a random

sample of 500 non-athlete students is taken and their

GPAs are recorded.

One-sample test

Two independent sample test

Paired test

Name Scenario

As part of a biology project, some high school students

compare heart rates before and after running a mile for 40

of their classmates.

They want to see if the after heart rate of student’s their

age is faster than the before heart rate, on average.

One-sample test

Two independent sample test

Paired test

Name Scenario

A hospital is studying patient costs and they decide to

follow 500 surgery patients’ hospital and medical bills for

a year after surgery and compare them to the estimated

costs provided to the patients before surgery.

They want to see if the estimated and actual costs are

comparable on average.

One-sample test

Two independent sample test

Paired test

Name Scenario

A group of students is interested if BU students in West

campus exercise more regularly than those in East, on

average.

One-sample test

Two independent sample test

Paired test

Name Scenario

A group of students is interested if BU female students

spend more time studying than BU male students, on

average.

One-sample test

Two independent sample test

Paired test

Name Scenario

Can we conclude that students tend to sleep more at the

beginning of the semester than at the end of the semester,

on average?

One-sample test

Two independent sample test

Paired test

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Required assumptions:

Data are 2 independent random samples

Large samples (n1≥30 and n2 ≥30) or Normal

Distribution of X1 and X2

Test Statistics: Z or t

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Case 1: Two independent Populations –

Population Variances Known

Attributes Test Statistic Confidence Interval

Example

Estimating Difference in Mean Walking Distance

between Two Physical Therapy Programs.

Response: the number of feet patients can walk

independently

Patients Undergoing

Total Knee Replacement

Randomize

Physical Therapy 1

4 Sessions

(1 hour/day)

Physical Therapy 2

2 Sessions

(2 hour/day)

Example

Estimating Difference in Mean Walking Distance

between Two Physical Therapy Programs.

Response: the number of feet patients can walk

independently

Summary:

4-day program: n1=18

2-day program: n2=16

Question: How can we compare physical therapy

programs?

Example

Question: How can we compare physical therapy

programs?

4-day program: n1=18

2-day program: n2=16

Idea 1: Compute CIs for mean # feet for both

therapies:

Example

Question: How can we compare physical therapy

programs?

4-day program: n1=18

2-day program: n2=16

Idea 1: Compute CIs for mean # feet for both

therapies:

4-day program: 95% CI (46.03 feet, 48.37 feet)

2 –day program: 95%CI (23.6 feet, 25.6 feet)

Example

Question: How can we compare physical therapy

programs?

4-day program: n1=18

2-day program: n2=16

Idea 2: Compute 95% CI for difference in population

means:

Example

Question: How can we compare physical therapy

programs?

4-day program: n1=18

2-day program: n2=16

Idea 2: 95% CI for difference in population means:

(21.06 feet, 24.14 feet)

Does this 95% CI contain 0?

What does it mean?

Example

Question: How can we compare physical therapy

programs?

4-day program: n1=18

2-day program: n2=16

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

Example

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

4-day program: n1=18

2-day program: n2=16

Step1:

Parameter:

H0: Ha:

Significance level α =_______

Example

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

4-day program: n1=18

2-day program: n2=16

Step2:

Assumptions:

Test Statistic:

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Case 1: Two independent Populations –

Population Variances Known

Attributes Test Statistic Confidence Interval

Example

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

4-day program: n1=18

2-day program: n2=16

Step3: Decision Rule:

Reject H0 if __________________

Rejection Rule

H0: μ=μ0(p=p0)

Ha: μ>μ0(p>p0)

upper-sided

Reject H0

if Z≥ Z1-α

(t≥ t1-α, df)

Ha: μ≠μ0(p≠p0)

two-sided

Reject H0

if Z≥ Z1-α/2 (t≥ t1-α/2,df)

or Z≤- Z1-α/2 (t≤- t1-α/2,df )

Ha: μ<μ0(p<p0)

lower-sided

Reject H0

if Z≤- Z1-α

(t≤- t1-α,df)

Example

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

4-day program: n1=18

2-day program: n2=16

Step4: Decision:

Reject H0 Fail to Reject H0

Example

Idea 3: Hypotheses testing(HT): Is there a

significant difference between two physical therapy

programs (at 5% sign.)?

4-day program: n1=18

2-day program: n2=16

Step5: Conclusion:

Based on two independent samples of n1=___ and n2=___, there

____significant evidence to conclude, at α = _____, that Physical

therapy 4-day program is different than 2-day program, on

average.

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Case 2:

Attributes Test Statistic Confidence Interval

n1≥30, n2 ≥30

n1<30 or n2 <30

df=n1+n2-2

Common Standard

Deviation Estimate

Example

To improve medical students' analytic skills a

new curriculum implemented across medical

school.

New curriculum may be differentially effective

among male and female students.

To evaluate, random sample of male and female

students who completed the new curriculum are

selected and given a test to assess their analytic

skills.

Example

Question: Is there a significant Difference in Mean

Number of problems Solved Between Males and

Females?

Male: n1=15

Female: n2=12

Step1:

Parameter:

H0: Ha:

Significance level α =_______

Example

Question: Is there a significant Difference in Mean

Number of problems Solved Between Males and

Females?

Male: n1=15

Female: n2=12

Step 2:

Assumptions:

Test Statistics:

Hypothesis Test for Population

Difference in Means (H0: μ1 – μ2 =0)

Case 2:

Attributes Test Statistic Confidence Interval

n1≥30, n2 ≥30

n1<30 or n2 <30

df=n1+n2-2

Common Standard

Deviation Estimate

Example

Question: Is there a significant Difference in Mean

Number of problems Solved Between Males and

Females?

Male: n1=15

Female: n2=12

Test Statistics: t = 4.43

Step 4: Decision Rule:

Reject H0 if _______________

Example

Question: Is there a significant Difference in Mean

Number of problems Solved Between Males and

Females?

Male: n1=15

Female: n2=12

Step 3: Decision:

Reject H0 Fail to Reject

Example

Question: Is there a significant Difference in Mean

Number of problems Solved Between Males and

Females?

Male: n1=15

Female: n2=12

Step 5: Conclusion:

Based on two independent samples of n1=___ and

n2=___, there ____significant evidence , at α = _____,

to conclude that ______________________________