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Confidence interval on the mean of a normal distribution, variance known

N
1 ∑
4
(ˆν 4 ) y = (y i − ¯y) = 27732.5
N
i=1
Substituting these values into (2.9), we obtain
( ) 1/2
N
ˆ
ˆ σ x = s x ± (V [ˆσ x ]) 1/2 = 12.2 ± 6.7, (2.14)
N − 1
( ) 1/2
N
ˆ
ˆ σ y = s y ± (V [ˆσ y ]) 1/2 = 11.2 ± 3.6, (2.15)
N − 1
Finally, we estimate the population correlation Corr[x, y], which we shall denote by ρ. From
(2.11) we have
N
ˆ ρ = r xy = 0.60.
N − 1
Under the assumption that the sample was drawn from a two-dimensional Gaussian population
P(x, y), the variance of our estimator is given by (2.13). Since we do not know the true value
of ρ, we must use our estimate ˆρ. Thus, we find that the standard error ∆ρ in our estimate is
given approximately by
( )
10 1 2 2
∆ρ ≈ [1 − (0.60) ] = 0.05.
9 10

Confidence interval on the mean of a normal distribution, variance known

The basic ideas of a confidence interval (CI) are most easily understood by initially considering a
simple situation. Suppose that we have a normal population with unknown mean µ and known
2
variance σ . This is a somewhat unrealistic scenario because typically both the mean and
variance are unknown. However, in subsequent sections we will present confidence intervals for
more general situations.

Development of the Confidence Interval and Its Basic Properties. Suppose that X 1 , X 2 , . . . ,
2
X n is a random sample from a normal distribution with unknown mean µ and known variance σ .
From the results of previous section 5 we know that the sample mean X is normally distributed
¯
2
with mean µ and variance σ /n. We may standardize X by subtracting the mean and dividing by
the standard deviation, which results in the variable
¯
X − µ
Z = √ . (2.16)
σ/ n
The random variable Z has a standard normal distribution.
A confidence interval estimate for µ is an interval of the form l ≤ µ ≤ u, where the end-points l
anduarecomputedfromthesampledata. Becausedifferentsampleswillproducedifferentvalues
of l and u, these end-points are values of random variables L and U, respectively. Suppose that
we can determine values of L and U such that the following probability statement is true:

P{L ≤ µ ≤ U} = 1 − α (2.17)

where 0 ≤ α ≤ 1. There is a probability of 1 − α of selecting a sample for which the CI will contain
the true value of µ. Once we have selected the sample, so that X 1 = x 1 , X 2 = x 2 , . . . , X n = x n ,
and computed l and u, the resulting confidence interval for µ is

l ≤ µ ≤ u. (2.18)

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