Confidence Intervals 95% Confidence Interval
Chapter 7 7.1
An alternative to reporting a single If after observing X1 = x1,…, Xn = xn, we
value for the parameter being estimated compute the observed sample mean x, then
Statistical Intervals is to calculate and report an entire Basic Properties of a 95% confidence interval for µ can be
Based on a interval of plausible values – a Confidence Intervals expressed as
confidence interval (CI). A confidence σ σ
Single Sample level is a measure of the degree of x − 1.96 ⋅
n
, x + 1.96 ⋅
n
reliability of the interval.
Deriving a Confidence Interval
Other Levels of Confidence Other Levels of Confidence Sample Size
Let X1,…, Xn denote the sample on which
z curve A 100(1 − α )% confidence interval for The general formula for the sample size the CI for the parameter θ is to be based.
shaded area = α / 2 the mean µ of a normal population Suppose a random variable satisfying the
n necessary to ensure an interval width
1−α when the value of α is known is given w is following properties can be found:
by 1. The variable depends functionally on
− zα / 2 0 zα / 2 2
σ σ σ both X1,…,Xn and θ .
x − zα / 2 ⋅ , x + zα / 2 ⋅ n = 2zα / 2 ⋅
P ( − zα / 2 ≤ Z ≤ zα / 2 ) = 1 − α n n w
2. The probability distribution of the
variable does not depend on θ or any
other unknown parameters.
Deriving a Confidence Interval Deriving a Confidence Interval Large-Sample Confidence Interval
Let h( X1,..., X n ;θ ) denote this random 7.2 If n is sufficiently large, the standardized
Now suppose that the inequalities can be
variable. In general, the form of h is variable
manipulated to isolate θ : X −µ
usually suggested by examining the
distribution of an appropriate estimator
Large-Sample Z=
S/ n
P (l ( X1,..., X n )) < θ < u ( X1,..., X n ))
θˆ. For any α between 0 and 1, constants Confidence Intervals has approximately a standard normal
distribution. This implies that
a and b can be found to satisfy lower confidence
limit
upper confidence
limit for a Population Mean x ± zα / 2 ⋅
s
n
P (a < h( X1,..., X n ;θ ) < b) = 1 − α
For a 100(1 − α )% CI.
and Proportion is a large-sample confidence interval
P (l ( X1,..., X n )) < θ < u ( X1,..., X n )) for µ with level 100(1 − α )%.
Confidence Interval for a Population Large-Sample Confidence Bounds
Proportion p with level 100(1 − α )% for µ Normal Distribution
7.3
Upper Confidence Bound:
Lower(–) and upper(+) limits: s The population of interest is normal, so that
µ < x + zα ⋅
n
Intervals Based on a X1,…, Xn constitutes a random sample from
zα2 / 2 ˆ ˆ zα2 / 2
pq Lower Confidence Bound: Normal Population a normal distribution with both
pˆ + zα / 2 + 2 µ and σ unknown.
2n n 4n s
= µ > x − zα ⋅ Distribution
( 2
1 + zα / 2 ) /n n
t Distribution Properties of t Distributions Properties of t Distributions
Let tv denote the density function curve t Critical Value
When X is the mean of a random sample 3. As v increases, the spread of the
of size n from a normal distribution with for v df. corresponding tv curve decreases.
mean µ , the rv Let tα ,v = the number on the measurement
1. Each tv curve is bell-shaped and 4. As v → ∞, the sequence of tv
X −µ axis for which the area under the t curve
T= centered at 0.
S/ n curves approaches the standard with v df to the right of
2. Each tv curve is spread out more normal curve (the z curve is called a tα ,v is α ; tα ,v is called a t critical value.
has a probability distribution called a t t curve with df = ∞.
distribution with n – 1 degrees of freedom than the standard normal (z) curve.
(df).
Pictorial Definition of tα ,v Confidence Interval
Let x and s be the sample mean and
tv curve standard deviation computed from the
shaded area = α results of a random sample from a
normal population with mean µ .The
100(1 − α )% confidence interval is
0
s s
tα ,v x − tα / 2, n−1 ⋅ , x + tα / 2 , n −1 ⋅
n n
