STAT 101 - Module Four Page 1 of 17
Inference for Differences in Proportions or Means
In previous lectures we learned about Confidence Intervals and Hypothesis Testing for a single
proportion or a single mean. But there are a whole litany of questions that could involve
• Is there a difference in the proportion of men that die from prostate cancer for those
that undergo a surgery versus those that do not?
• How much taller, on average, are adult males than adult females?
• Are children who play violent video games more aggresive than children who don’t ...
Structure of the Data
_______
________
In the above table, there are now two groups and we can gather information on both.
• Categorical: Group One (Yes Surgery), Group Two (No Surgery)
• Assume these groups are independent of each other
Quantity of Interest
• If categorical:
Proportion in category of interest from both groups
• If quantitative:
Mean value in each group
1
,Notation Proportions--Categorical
Population 1
• p1 : Population proportion for Group One
• n1 : Sample size from Group One
• pˆ1 : Sample proportion from Group One
Population 2
• p2 : Population proportion for Group Two
• n2 : Sample size for Group Two
• pˆ2 : Sample proportion from Group Two
Population 1
• µ1 : Population mean from Group One Means--Quantitative
• n1 : Sample size from Group One
• y¯1 : Sample mean from Group One
• s1 : Sample standard deviation from Group One
Population 2
• µ2 :Population mean from Group Two
• n2 :
Sample size from Group Two
• y¯2 : Sample mean from Group Two
• s2 : Sample standard deviation from Group Two
Examp ne w ne group or two groups. If it
involves two independent groups, identify the groups.
1. An educator wants to determine the average reading comprehension scores of her stu-
dents One group
2. An educator assigns half the class to one reading activity and the other half of the class to
another reading activity. She wants to determine if the average reading comprehension
scores are different between the activities.Two groups - the activities (Activity A and Activity B)
Difference in means
3. We want to compare the proportion of in-state students who get financial aid to the
proportion of out-of-state students who get financial aid. Two groups (Instate v Out-of-state)
Difference in proportions
4. We want to determine if the proportion of students at a university that are in-state
students is higher than the national average. One group + Single proportion
2
, STAT 101 - Module Four Page 3 of 17
We will consider the following
In this situation, the parameter and statistic are:
Parameter: Difference in population proportions (P1-P2)
^ - P2
Statistic: P1 ^
Conditions for BOTH groups
1. Randomization condition: Sample must be random
2. 10% condition: Sample must be less than 10% of population
3. Success/Failure condition: Must have 10 success/failures
n(p) >= 10
n(1-P) >=10
4. Independent Groups: Values of one group does not affect values of the other group
Formula
If the above conditions are met, the C% confidence interval for p1 − p2 is:
^ ^
1) Attain estimate of the difference: P1-P2
2) Build an interval around this with z* value with standard error
3
Inference for Differences in Proportions or Means
In previous lectures we learned about Confidence Intervals and Hypothesis Testing for a single
proportion or a single mean. But there are a whole litany of questions that could involve
• Is there a difference in the proportion of men that die from prostate cancer for those
that undergo a surgery versus those that do not?
• How much taller, on average, are adult males than adult females?
• Are children who play violent video games more aggresive than children who don’t ...
Structure of the Data
_______
________
In the above table, there are now two groups and we can gather information on both.
• Categorical: Group One (Yes Surgery), Group Two (No Surgery)
• Assume these groups are independent of each other
Quantity of Interest
• If categorical:
Proportion in category of interest from both groups
• If quantitative:
Mean value in each group
1
,Notation Proportions--Categorical
Population 1
• p1 : Population proportion for Group One
• n1 : Sample size from Group One
• pˆ1 : Sample proportion from Group One
Population 2
• p2 : Population proportion for Group Two
• n2 : Sample size for Group Two
• pˆ2 : Sample proportion from Group Two
Population 1
• µ1 : Population mean from Group One Means--Quantitative
• n1 : Sample size from Group One
• y¯1 : Sample mean from Group One
• s1 : Sample standard deviation from Group One
Population 2
• µ2 :Population mean from Group Two
• n2 :
Sample size from Group Two
• y¯2 : Sample mean from Group Two
• s2 : Sample standard deviation from Group Two
Examp ne w ne group or two groups. If it
involves two independent groups, identify the groups.
1. An educator wants to determine the average reading comprehension scores of her stu-
dents One group
2. An educator assigns half the class to one reading activity and the other half of the class to
another reading activity. She wants to determine if the average reading comprehension
scores are different between the activities.Two groups - the activities (Activity A and Activity B)
Difference in means
3. We want to compare the proportion of in-state students who get financial aid to the
proportion of out-of-state students who get financial aid. Two groups (Instate v Out-of-state)
Difference in proportions
4. We want to determine if the proportion of students at a university that are in-state
students is higher than the national average. One group + Single proportion
2
, STAT 101 - Module Four Page 3 of 17
We will consider the following
In this situation, the parameter and statistic are:
Parameter: Difference in population proportions (P1-P2)
^ - P2
Statistic: P1 ^
Conditions for BOTH groups
1. Randomization condition: Sample must be random
2. 10% condition: Sample must be less than 10% of population
3. Success/Failure condition: Must have 10 success/failures
n(p) >= 10
n(1-P) >=10
4. Independent Groups: Values of one group does not affect values of the other group
Formula
If the above conditions are met, the C% confidence interval for p1 − p2 is:
^ ^
1) Attain estimate of the difference: P1-P2
2) Build an interval around this with z* value with standard error
3
