confidence intervals Flashcards
what is an estimator of a population parameter?
An estimator of a population parameter is a random variable that uses sample information to provide an approximation of this unknown parameter
what is an estimate?
A specific value of that random variable is called an estimate
what is a point estimator?
A point estimator πΜ is an unbiased estimator of the parameter if its expected value (or mean) is equal to that parameter:
πΈ(πΜ )=π_πΜ =π
what are examples of point estimators?
The sample mean is an unbiased estimator of the population mean,
β The sample proportion is an unbiased estimator of the population proportion,
what is the bias?
The bias in is defined as the difference between its expected value and π
π΅πππ (πΜ )=πΈ(πΜ )βπ
what is the bias of an unbiased estimator?
0
what is the most efficient estimator?
The most efficient estimator is the unbiased estimator with the smallest variance
what is the relative efficiency of πΜ_1 with respect to πΜ_2?
The relative efficiency of πΜ_1 with respect to πΜ_2 is the ratio of their variances:
π
ππππ‘ππ£π πΈπππππππππ¦= πππ(πΜ_2 )/πππ(πΜ_1 )
when is πΜ_1 is said to be more efficient than πΜ_2?
πΜ_1 is said to be more efficient than πΜ_2 if πππ(πΜ_1 )<πππ(πΜ_2 )
what is the general form for all confidence intervals?
The general form for all confidence intervals is:
πππππ‘ πΈπ π‘ππππ‘πΒ±ππππππ ππ πΈππππ
what are the necessary assumptions to find the confidence interval estimate for the mean when varience is known?
Population variance π^2 is known
Population is normally distributed or, if population is not normal, sample is large
what is the formula for the confidence interval estiamate for the mean when the population variance is known?
π₯Μ
Β± [π§_(πΌβ2) π/βπ]
where π§_(πΌβ2) is the value of the standard normal distribution, above which lies 100(πΌβ2)% of the distribution
what is the formula for the marginal of error?
The margin of error, ππΈ= [π§_(πΌβ2) π/βπ]
when does the margin of error fall?
ππΈ=π§_(πΌβ2) π/βπ
1. The population standard deviation decreases (πβ)
2. The sample size increases (πβ)
3. The confidence level decreases (πΌβ)
where do the intervals extend to ?
UCL=π₯Μ
+π§_(πΌβ2) π/βπ
to
LCL=π₯Μ
βπ§_(πΌβ2) π/βπ
what intervals constructed contain π and which do not?
100(1βπΌ)% of intervals constructed contain π;
100(πΌ)% do not
what are the degrees of freedom (df)?
This is the number of observations that are free to vary after the sample mean has been calculated
what is the difference and similiarity between the t distribution and the normal distribution?
π‘ distributions are bell-shaped and symmetric, but have βfatterβ tails than the normal
when is the t distribution the standard normal distribution?
the t distribution is equal to the standard normal distribution when the degrees of freedom is equal to infinity
what is the formula for the t distribution with (n-1) degrees of freedome
π‘=(π₯Μ βπ)/(π ββπ)
what are the necessary assumptions needed to find the confidence interval estimation of the mean when the population variance is unknown?
- Population variance π^2 is unknown - We can substitute the sample standard deviation, π , for the population standard deviation, π, in the confidence interval formula used earlier but this introduces extra uncertainty, since π is variable from sample to sample. This is why we use the π‘ distribution instead of the normal distribution
- Population is normally distributed or, if population is not normal, sample is large
what is the formula for the confidence interval estimate when the population varience is unknown?
π₯Μ Β±π‘_(πβ1,πΌβ2) π /βπ
where π‘_(πβ1,πΌβ2) is the relevant value of the π‘ distribution with πβ1 df
when is the sample proportion approximately normal?
the distribution of the sample proportion is approximately normal if ππ(1βπ)>5, with standard deviation:
π_πΜ =β(π(1βπ)/π)
what is the confidence interval for the population proportion?
πΜΒ±π§_(πΌβ2) β((πΜ(1βπΜ ))/π)
where
π§_(πΌβ2) is the standard normal value for the level of confidence desired
πΜ is the sample proportion
π is the sample size
what is an example of a confidence intervals with population means and a dependent sample?
the same group before vs after the treatment
what is an example of a confidence intervals with population means and a independent sample?
group 1 vs an independent group 2
what is an example of confidence intervals with population proportions?
proportions of 1 vs proportion of 2
what is the necessary assumption for finding the difference in means between dependent samples?
both populations are normally distributed
what is the formula to find the mean difference between two dependent samples?
π_π=π_π₯βπ_π¦
what is the point estimate for the population mean diference?
πΜ = (β(π₯βπ¦ ) )/π
what is the sample standard deviation of d?
π _π=β((β(π_πβπΜ )^2 )/(πβ1))
what is the formula for the confidence interval for the difference between the two population means?
πΜ Β±[π‘_(πβ1,πΌβ2) x π _π/βπ]
what is the point estimate for the difference betweent the two sample means in an independent sample?
π₯Μ β π¦Μ
what is the formula for the confidence interval for the difference in sample means in an independent sample when varience of both samples are known?
(π₯Μ βπ¦Μ ) Β± π§_(πΌβ2) β((π_π₯^2)/π_π₯ +(π_π¦^2)/π_π¦ )
what is the formula for the varience of the difference in means when both variences are known and the samples are independent?
π_(π₯Μ βπ¦Μ )^2=(π_π₯^2)/π_π₯ + (π_π¦^2)/π_π¦