Ch. 8 - 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.
Let - θ be a generic parameter.
Let - θ be an estimator – a statistic calculated from a random sample
Cons-equently, θˆ is an r.v. with mean E( θˆ) = µ θˆand std. dev. σ θ
8.1 Point and Interval Estimates
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
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%.
Ex8.1) 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 θˆused to estimate parameter θ,
“standard error” is a statistic σθˆused to estimate std. dev. of estimator 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/2ere the notation reflects P(Z > z) = α/2.
Ex8.2) if the confidence level is 95%, then α/2 = 0.025 and z = 1.96.
(diagram drawn in class) Table 8X0 – Critical values for usual confidence levels
100(1 – α)% α α/2 z
90% 0.10 0.050 1.645
95% 0.05 0.025 1.96
99% 0.01 0.005 2.58
The estimator θˆ and its standard error σθ are defined so that, when the sample size n is
sufficiently large, the sampling distribution of
is well approximated by the standard normal distribution. Consequently,
⎜ θ −θ ⎟
P⎜− z σˆ ≤ z⎟≈ −α
⎝ θˆ ⎠
Algebraic manipulation yields
Pθ − zσ ≤θ ≤θ + zσˆ ˆ ) ≈1−α
8.2 Large Sample CI for Population Proportion
Recall the 3 rules regarding the general properties of the sampling distribution of .
Then, when n is large, a (1 – α)100% CI for p is
p(1 − p)
p ± zα /2
Note that n being large also allows for the standard error to use since p is unknown.
The interval can be used as long as
(1) np ≥ 15 and n(1− p) ≥ 15,
(2) the sample can be regarded as a random sample from the population of interest.
Ex8.3) A survey of 1356 random adults asked them to pick out the funniest city name in a
list. 923 chose “Keokuk”, 74 chose “Walla Walla”, and 359 chose “Seattle”. Let p be
the proportion of all adults who would have answered “Seattle” had they been polled.
Construct a 95% confidence interval for p.
: e z i s e l p - Parama eter = p - n = 1356
- Estimate: p = 359/1356 ≈ 0.265