Greenwich University
Asef Khan
Economics : Confidence Intervals 2
Tutor: Alexander Guschanski
27.2.2023
INTRO
Any questions from your side re the module assessment etc?
How do we complete meeting protocol? See solution on the last sheet = submission
template. Check example of this
Fill in : ids of people attending and then write in next column what you did. Very
brief, spend only 5 minutes on this= everyone needs to fill in the same or looks
suspicious. Is a means of ensuring everyone has contributed to the work. Not
testing your writing skills
Easy 10 marks- any research project has protocol element> helps to see progress,
errors etc .
What type of questions can we answer following last week’s session?
ADVANCED METHOD OF CALCULATING CONFIDENCE INTERVALS
Today- about advanced way of implementing confidence intervals for population mean
and proportion in more realistic scenarios where we don’t know the standard deviation
of population.
SEE SESSION GOALS; AT END OF SESSION YOU WILL BE ABLE TO ANSWER THREE TYPES OF
QUESTIONS
1)First type of question- you can already answer. Way to estimate confidence estimate
based on formula>confidence interval for population mean based on sample mean and
known standard deviation of population
Recap
Process produces bags of sugar; weights of contents of these bags normally
distributed with standard deviation of 1.2kg
Bags are produced with certain average weight but weight varies from bag to bag
You want to work out confidence interval for average weight of production process-
see if it’s changed over time
Collect 25 bags as sample with mean weight of 19.8 kg, ask to construct 99%
confidence interval for true mean weight for all bags of sugar produced by process
Production process= population
Working
Formula- confidence interval for population mean based on sample mean and known
standard deviation of population ; x bar= mean of sample mean= 19.8kg plus/ minus
1
, Greenwich University
Asef Khan
Economics : Confidence Intervals 2
Tutor: Alexander Guschanski
27.2.2023
Formula
19.8 (mean) +- 2.58( z alpha half) then 1.2( standard deviation)/sq. root of
25( sample)
o We know something about the population- know about production
process based on data. Way in which bags vary around average weight is
described by standard deviation of 1.2 kg
o Looking for confidence interval of random sample- If you did not know
standard deviation you would be unable to calculate confidence
interval based on the formula
How to find z alpha half? If you want to construct confidence interval around pop
mean, you need to capture 99% of distribution > need to cut off 0.5% both sides
Go to standard normal distribution table- what does table tell us?
o Table divided into 2 parts
First row and first column = z value and are separate from rest of
table. From 0.5000- 0.997 onwards displays blue shaded areas to left
of z value that is associated with each of the areas
Looking for z value
What should blue shaded area to left be?
Needs to cover 99% including the red shaded areas of 0.5 . Take off
half a percentage point so = 99.5
So z value =blue shaded area of 99.5%
Blue shaded range of numbers= so value is between 0.9949 and 0.9951 in table.
Best convention= choose the larger number when between two values
So 0.9951 =z value = 2.58 cuts off 99.5 % to left and – 2.58 to right
So 99% confidence interval = 19.18( lower confidence limit)- to
20.42( higher confidence limit) based on production process
How to interpret? What does it tell us? Gives us certain range for the population when
we don’t know the mean – provides a confidence interval around the mean.
If we continue to construct these intervals in same way based on
different samples- e.g. second one might have 20.1 then 99% of them
would contain the population mean
It allows us to say something about the range of the population
mean that reflects spread and uncertainty in data we analyse.
If no confidence interval- say pop mean is somewhere around
19.18kg
If standard deviation of population is larger i.e. 5kg – how could this affect our
confidence interval- smaller or larger?
2
Asef Khan
Economics : Confidence Intervals 2
Tutor: Alexander Guschanski
27.2.2023
INTRO
Any questions from your side re the module assessment etc?
How do we complete meeting protocol? See solution on the last sheet = submission
template. Check example of this
Fill in : ids of people attending and then write in next column what you did. Very
brief, spend only 5 minutes on this= everyone needs to fill in the same or looks
suspicious. Is a means of ensuring everyone has contributed to the work. Not
testing your writing skills
Easy 10 marks- any research project has protocol element> helps to see progress,
errors etc .
What type of questions can we answer following last week’s session?
ADVANCED METHOD OF CALCULATING CONFIDENCE INTERVALS
Today- about advanced way of implementing confidence intervals for population mean
and proportion in more realistic scenarios where we don’t know the standard deviation
of population.
SEE SESSION GOALS; AT END OF SESSION YOU WILL BE ABLE TO ANSWER THREE TYPES OF
QUESTIONS
1)First type of question- you can already answer. Way to estimate confidence estimate
based on formula>confidence interval for population mean based on sample mean and
known standard deviation of population
Recap
Process produces bags of sugar; weights of contents of these bags normally
distributed with standard deviation of 1.2kg
Bags are produced with certain average weight but weight varies from bag to bag
You want to work out confidence interval for average weight of production process-
see if it’s changed over time
Collect 25 bags as sample with mean weight of 19.8 kg, ask to construct 99%
confidence interval for true mean weight for all bags of sugar produced by process
Production process= population
Working
Formula- confidence interval for population mean based on sample mean and known
standard deviation of population ; x bar= mean of sample mean= 19.8kg plus/ minus
1
, Greenwich University
Asef Khan
Economics : Confidence Intervals 2
Tutor: Alexander Guschanski
27.2.2023
Formula
19.8 (mean) +- 2.58( z alpha half) then 1.2( standard deviation)/sq. root of
25( sample)
o We know something about the population- know about production
process based on data. Way in which bags vary around average weight is
described by standard deviation of 1.2 kg
o Looking for confidence interval of random sample- If you did not know
standard deviation you would be unable to calculate confidence
interval based on the formula
How to find z alpha half? If you want to construct confidence interval around pop
mean, you need to capture 99% of distribution > need to cut off 0.5% both sides
Go to standard normal distribution table- what does table tell us?
o Table divided into 2 parts
First row and first column = z value and are separate from rest of
table. From 0.5000- 0.997 onwards displays blue shaded areas to left
of z value that is associated with each of the areas
Looking for z value
What should blue shaded area to left be?
Needs to cover 99% including the red shaded areas of 0.5 . Take off
half a percentage point so = 99.5
So z value =blue shaded area of 99.5%
Blue shaded range of numbers= so value is between 0.9949 and 0.9951 in table.
Best convention= choose the larger number when between two values
So 0.9951 =z value = 2.58 cuts off 99.5 % to left and – 2.58 to right
So 99% confidence interval = 19.18( lower confidence limit)- to
20.42( higher confidence limit) based on production process
How to interpret? What does it tell us? Gives us certain range for the population when
we don’t know the mean – provides a confidence interval around the mean.
If we continue to construct these intervals in same way based on
different samples- e.g. second one might have 20.1 then 99% of them
would contain the population mean
It allows us to say something about the range of the population
mean that reflects spread and uncertainty in data we analyse.
If no confidence interval- say pop mean is somewhere around
19.18kg
If standard deviation of population is larger i.e. 5kg – how could this affect our
confidence interval- smaller or larger?
2
