Confidence Intervals Flashcards
Confidence interval
An interval that contains ‘plausible” values of a parameter, usually mean
Comes from the idea that there is a variation in values from a collected data sample of independent observations So a point estimate and estimated standard error is combined
Confidence level
( 1- alpha ) X 100 %
The percent confidence of a particular event
Typ used in combination with confidence intervals to give a range of values a mean is likley to fall within CL% of the time
What is the point of a confidence interval
To take variation into account when gathering statistics from a population
What is known about the intervals observed ( x1…xn)
All independent trials from some unknown continues probability distribution
Interpretation of intervals based on sample size
Infinite sampling → CL % CI’s will contain mean
Limited → approximately
Single → does contain
When to use standard z-dist vs t-dist
Standard deviation known → z-dist
Using sample standard deviation→ t-dist
Factors that affect precision of CI
Increase CI size (L) = decrease precision
CL: increase CL = increase CI size
n: increase n = decrease CI size
Standard deviation : increase SD = increase CI size
Central limit theorem
If there is a sequence of xi….xn independent, identically distributed RV’s with a common mean and variance, their average can be approximated by a normal distribution the with same mean and variance divided by n.
This approx better as n gets larger.
Even if the individual observations are not normally distributed, if his large enough, the average will still be approx normal
Why can populations without normal distributions still use the Ci formula
By central limit theorem, if n is large enough, the average of the independent trials has an approx normal distribution
Difference between one-sided and two-sided intervals
Two sides CI describes a probability where the mean lies in the Center, alpha is divided by 2 and distributed to either end
One sided CI describes an upper or lover bound that the mean could lake
When calculating a minimum sample size, do we round the wander
Round up
How do we deal with not knowing the standard deviation
Approximate the sample standard deviation is approximately equal to the standard deviation, this works when n is large as s is a good approx for o
Additional note, me don’t need the population to be normal due to central limit theorem
If n is not large, we use a t-dist and reed observation to be from a normal dist
Issues with approx s as o when n is not large
1) if measurement don’t follow a normal dish then X average is not well approx by a normal dist
2) s is not a good approx of s due to sample size
Solution
I) ensure observations are from normal / approx normal dist
2) account for inaccurate of s as an estimation of o by using a standard t-dist instead of z-dist
Rot for a large n
Greater than 30
Comparison between the t-dist and z-dis
Same single peak shape ( bell shaped and symmetric about mean)
t-dist has a smaller peak and longer tails → more variation
Has degrees of freedom
