Confidence Interval Estimation Flashcards
Point estimate
A point estimate is the value of a single sample statistic.
Confidence interval
A confidence interval provides a range of values constructed around the point estimate.
Confidence interval estimation
An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample. Based on observations from 1 sample.
Gives information about closeness to unknown population parameters.
Stated in terms of level of confidence. Can never be 100% confident.
Common confidence intervals
Common confidence levels = 90%, 95% or 99%: Also written (1 - ) = 0.90, 0.95 or 0.99
A relative frequency interpretation
In the long run, 90%, 95% or 99% of all the confidence intervals that can be constructed (in repeated samples) will contain the unknown true parameter.
Specific intervals
A specific interval will either contain or will not contain the true parameter. No probability involved in a specific interval.
Confidence Interval for μ (σ Known) assumptions
Assumptions:
Population standard deviation σ is known
Population is normally distributed
If population is not normal, use Central Limit Theorem.
Will the true average always be in the middle of the confidence interval
Not necessarily. PHOTO 9, A good but not perfect measure
Confidence interval for μ (σ Unknown)
If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S.
This introduces extra uncertainty, since S is variable from sample to sample.
So we use the Student t distribution instead of the normal distribution:
The t value depends on degrees of freedom denoted by sample size minus 1 i.e. (d.f = n - 1).
d.f are number of observations that are free to vary after sample mean has been calculated.
Degrees of freedom
: Number of observations that are free to
vary after sample mean has been calculated
Confidence interval example interpretation PHOTO
We are 95% confident that the true percentage of left-handers in the population is between 0.1651 and 0.3349 i.e.:
16.51% and 33.49%
Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from repeated samples of size 100 in this manner will contain the true proportion.
Sampling error
The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - alpha).
The margin of error is also called a sampling error:
The amount of imprecision in the estimate of the population parameter.
The amount added and subtracted to the point estimate to form the confidence interval.
Determining the sample size for the mean
To determine the required sample size for the mean, you must know:
The desired level of confidence (1 - ), which determines the critical Z value.
The acceptable sampling error, e.
The standard deviation, σ.
Rule for rounding confidence intervals
Always round up (sideways)
If σ is Unknown
If unknown, σ can be estimated:
From past data using that data’s standard deviation.
If population is normal, range is approx. 6σ so we can estimate σ by dividing the range by 6.
Conduct a pilot study and estimate σ with the sample standard deviation, s.
