false

Class Notes
(835,674)

Canada
(509,327)

University of Alberta
(13,389)

Statistics
(248)

STAT141
(21)

Paul Cartledge
(13)

Lecture

Unlock Document

Statistics

STAT141

Paul Cartledge

Winter

Description

Ch. 19 - Statistical Inference
Def’n: Estimation is the assignment of value(s) to a population parameter based on a
value of the corresponding sample statistic.
An estimator is a rule used to calculate an estimate.
An estimate is a specific value of an estimator.
Note: in this chapter, always assuming an SRS.
- Notation:
Let - θ be a generic parameter.
ˆ
Let - θ be an estimator – a statistic calculated from a random sample
Cons-equently, θ is an r.v. with mean E( θˆ) = µ θ and std. dev. σ θ
Def’n: A point estimate is a single number that is our “best guess” for the parameter.
Æ like a statistic, but more precise towards parameter estimation.
An interval estimate is an interval of numbers within which the parameter value is
believed to fall.
Generic large sample confidence intervals:
Def’n: A confidence interval (CI) for a parameter θ is an interval estimate of plausible
values for θ. With a chosen degree of confidence, the CI’s construction is such that the
value of θ is captured between the statistics L and U, the lower and upper endpoints of the
interval, respectively.
The confidence level of a CI estimate is the success rate of the method used to
construct the interval (as opposed to confidence in any particular interval). The generic
notation is 100(1 – α)%. Typical values are 90%, 95%, and 99%.
Ex19.1) Using 95% and the upcoming method to construct a CI, the method is
“successful” 95% of the time. That is, if this method was used to generate an interval
estimate over and over again with different samples, in the long run, 95% of the resulting
intervals would capture the true value of θ.
Many large-sample CIs have the form:
point estimate ± (critical value) × (standard error)
where “point estimate” is a statistic θˆ used to estimate parameter θ,
“standard error” is a statisticσˆ used to estimate std. dev. of estimator θˆ,
θ
“critical value” is a fixed number z defined so that if Z has std. norm. dist’n, then
P(-z ≤ Z ≤ z) = 1 – α = confidence level
The product of the “standard error” and “critical value” is the margin of error.
Note: critical value z often denoted by z α/2here the notation reflects P(Z > z) = α/2.
Ex19.2) if the confidence level is 95%, then α/2 = 0.025 and z 0.025= 1.96.
(diagram drawn in class) Table 19X0 – Critical values for usual confidence levels
100(1 – α)% α α/2 zα/2
90% 0.10 0.050 1.645
95% 0.05 0.025 1.96
99% 0.01 0.005 2.58
ˆ ˆ
The estimator θ and its standard error σθˆare defined so that, when the sample size n is
sufficiently large, the sampling distribution of
ˆ
θθ− ▯ N(0,1)
σˆˆ
θ
Thus,
⎛ θ −θ ⎞
P − z ≤ ≤ z ≈1−α
⎝ σθˆ ⎠
Algebraic manipulation yields
Pθ − zσ ≤ˆ ≤θ + zσ ˆ ˆ)≈1−α
θ θ
Large Sample CI for Population Proportion
Recall the 3 rules regarding

More
Less
Related notes for STAT141

Join OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

Continue

Continue
OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.