October 31 – Lecture 7
Confidence Intervals IV (IMPORTNAT FOR EXAM)
1-a a a/2 Za/2
.90 .10 .05 1.645
.95 .05 .025 1.96
.98 .02 .01 2.33
.99 .01 .005 2.58
(a – isAlpha)
Example – to say
We are 95% or watever confident that our sample will fall in the following range
– For 95% confidnent there is 5% chance it wont be
– If we undertook a sample, --> 95% will contain population Parameter and 5% of them wont
When dealing with Sampling distribution – the numbers above are the ones that are always needed
and be used.
If we have a 95% confidence then there is 5% significance
Choosing Correct Sample Size
– The only way we could have it all – high level of confidence and high level of precision is to
increase the number of sample size – it gets steaper , the small distribution around the mean
– therefore, the smaller the number of error .
– Therefore we could be more precise in our
– How prices/close
– accuracy means --> does our sample actually reflect the
– how close our smaple really easy to our population sample. ??? not sure
- Total amount of information is due to:
– Sampling design used
– Sample size n
– the bigger the sample – the more information about the population it will tell
– But, how many observations should be included in the sample?
– Greater or equal to 30 – the more the better
– Have to consider the relationship between the width of the interval and the level of
– the width increases and we have to be concerned about it
– increasing of interval width, increases confidence
– but decreases precision
– Only way to increase confidence without increasing the width, is to increase sample
– to have a small interval and have a high confidence, increase the SAMPLE***
Chosing Correct Sample Size II
– Taking a sample larger than necessary wastes time and effort (very costly) – Factors to consider are:
– Type of sample i.e. Random, Stratified etc.
– Population parameter being estimated
– is it the mean or std deviation ??
– Degree of Precision (width of confidence interval)
– Level of Confidence
– what level of confidence do i require and what level i will be testing
– the only way to increase to confidence and decrese the width is to increase sample
– At a particular confidence level, increasing the sample size provides greater precision, and
narrows the confidence interval
Choosing Correct Sample Size III (CHECK BOOK)
n = ( E )2
Where E is the amount of sampling error the researcher is wiling to tolerate
For Data measured in proportions, the formula is ...
n= ( E )2
1. Urban planner wants to estimate the mean number of people per household
2. Using 90% confidence level, and stating that their estimate will be within .3 persons of the
true population mean.
3. S is determined through pre-sampling to be 1.25
4. What is the minimum number of households that must be sampled?
Using formula Za/2S (1.65) (1.25)
n = ( E )2 ---> n= ( .3 )2
= 47.26 or 48 households
Therfore, he/she needs a random sample of, at least 48 households to ensure that level of precision
at the 90% confidence level.
Estimation – Small Samples
– When n