Comparing means

Norhafizah Ab Manan

After class, you should

• Understand independent t test, paired t test and ANOVA

• Know how to calculate the t statistics• Find the t tabulated from t

distribution table

Comparing two means

• How can we get a mean?• What data?- categorical or

numerical?

Independent t test

• Is cholesterol level differ between male and female students?

• What is the null hypothesis for this study?

male female

Independent t test

• Measure –Compare two means

• Assumptions 1. In each group of the dependent

variables, the distribution is normal

2.Random sample• How to test the assumption?

Steps in hypothesis testing

1. Define the null and alternative hypothesis Ho= The population means in two groups

are equal Ha= The population means in the groups are

not equal

2. Calculate the t statistics/ t calculated3. Compare the t statistics to the value from

t-distribution4. Interpret the results

Example;

• A researcher interested to compare cholesterol level between male and female students. There are 12 males and 7 females. The data was:

Group Sample size

Mean (mmol/L)

SD (mmol/L)

Male 12 6.192 0.3919

Female 7 5.414 0.6492

1. Define null and alternative hypothesis.

• Ho= The cholesterol means in male and female students are equal

• Ha= The cholesterol means in male and female students are not equal

2. Calculate the t statistics

2. Calculate the t statistics

Group Sample size

Mean (mmol/L

)

Variance (mmol/L

)

Male 12 6.192 0.3919

Female 7 5.414 0.6492

= 0.483

2. Calculate the t statistics

Group Sample size

Mean (mmol/L

)

SD (mmol/L

)

Male 12 6.192 0.3919

Female 7 5.414 0.6492

= 0.778/0.3305=2.36

3. Compare the t statistics to the value from t-distribution

• If t calculated > t tabulated (from table)- we reject the null hypothesis• If t calculated < t tabulated (from table) –we fail to reject the null hypothesis

One tailed

•Right-tailed •Sign of Ha is > •Key word: More than

•Left-tailed •Sign of Ha is < •Key word: Less than

Rejection area

Two tailed

• The sign of HA is ≠

• Key word: no different

• Rejection area

3. Compare the t statistics to the value from t-distribution

• Find the t tabulated from t distribution table

• =2.45 (from table) with alpha error= 95%, Upper tailed = 2.5%.

• Degree of freedom= the smaller of (n1-1) or (n2-1)

• 6?• T statistics=2.36

2,45-2,45

0

4. Interpret the results

• The t calculated value is in the critical region

• Reject the null hypothesis• There is different of cholesterol

between gender

Paired t test

• Measure –Compare two dependent means (before and after)

• Assumptions 1.Distribution of the different is

normal2.Random sample

• How to test the assumption?

Example

• A researcher interested to determine the effectiveness of an intervention towards BP. The BP of the subjects were measured twice; before and after the intervention.

Measure Sample size

Mean of d

Sd

Pre-interV

15 6.4 8.48

1. Define null and alternative hypothesis.

• Ho= there is no different of BP before and after the intervention

• Ha= there is a different before and after the intervention

2. Calculate the t statistics

• The formula for t statistics:

t=test statisticsḋ= mean of the differenceSd =Sd of the differencen= sample size

2. Calculate the t statistics

Measure Sample size

Mean of d

Sd

Pre-interV

15 6.4 8.458

15

458.804.6

t=2.930

Note: A hypothesized mean difference (μd) can be any specified value. The most common value specified is zero.

3. Compare the t statistics to the value from t-distribution

• T calculated= 2.930• T tabulated with df (14) and

alpha(0.05)= 2.14

4. Interpret the results

• The t calculated value is in the critical region

• Reject the null hypothesis• There is a different between before

and after the intervention

ANOVA

• To compare means between more than two groups

• Variable:– Independent variable: Categorical– Dependent variable: Numerical

• Assumptions:– Data is normal distributed– Equal variance

Examples

• To determine whether BMI is different between age groups or not

• To study the effect of 3 different types of anti hypertensive drug on 120 patients.

• To compare the mean different of IQ scores among 3 classes

References• Basic Biostatistics statistics for public health

practice. 2008. B Burt Genstman. Jones and Batlett Publisher Inc.

• Medical statistics at a glance. 3rd Edition. 2009. Aviva Petrie & Caroline Sabin. Wiley-Blackwell.