Statistical Inference STAT 431
Lecture 8: Inferences for Single Samples (III)
Inferences with Normal Population Distribution
1. Inferences for μ with small sample, with known
2. Inferences for μ with small sample, with unknown
3. Inferences for 2
• We are mainly interested in confidence interval and hypothesis testing problems
• For all the above problems, we assume that the population distribution is normal
i.i.d. 2
X1,…,Xn ⇠ N(μ, )
STAT 431
Chi-Square Distribution
• Connection to normal distribution:
i.i.d. Pn 2 2 IfZ1,…Zn ⇠ N(0,1),then i=1Zi ⇠ n.
• Shape of the Chi-square distributions
– Positively skewed
– The level of skewness decreases as
the degrees of freedom (d.f.) increase
– The curves shift to the right
as the d.f. increase
– For large d.f., the density curve has
an approximate bell shape
Chi−Square Densities
STAT 431
0 10 20 30 40 50 x
n = 5, 10, 15, 25
Density
0.00 0.05 0.10 0.15
Student’s t-Distribution • Connection to normal and chi-square distributions:
IfZ⇠N(0,1)isindependentofU⇠ 2n,thenT=pZ ⇠tn . U/n
t−Distribution Densities
Density
0.0 0.1 0.2 0.3 0.4
• Shape of the t distributions
– Symmetric around 0
– Have heavier tails than N (0, 1)
– Larger d.f.èlighter tails
– As d.f. tends to infinity, the density
curve converges to that of N (0, 1)
STAT 431
−4 −2 0 2 4
n = 1, 2, 5; N(0,1)
Key Properties of Sample Mean and Sample Variance Assumption: i.i.d. 2
• Sample mean satisfies
• Sample variance S2 =
̄ 2 X ⇠ N(μ, n )
1 Pn (Xi X ̄)2 satisfies n 1 i=1
(n 1)S2 2 2 ⇠ n 1
X1,…,Xn ⇠ N(μ, )
• They are mutually independent.
• The assumption of a normal population distribution is critical.
STAT 431
Small Sample Inferences for Mean (Unknown ) i.i.d. 2
• The new pivotal r.v.:
• By its pivotality, we have
X1,…,Xn ⇠ N(μ, ) X ̄ μ
T = S/pn ⇠ tn 1
✓ X ̄ μ ◆
Pμ tn 1,↵/2 T = S/pn tn 1,↵/2 • So a two-sided 100(1 ↵)% confidence interval for μ is
̄S ̄S X tn 1,↵/2 pn , X + tn 1,↵/2 pn
=1 ↵
Two-sided t-interval
STAT 431
• Dataset: 29 measurements of earth • density made by Henry Cavendish in
1798 (Textbook Example 7.7)
Normal Q−Q Plot
95% two-sided confidence interval for earth density μ : t28,0.025 = 2.048
Example: Earth Density
[5.367, 5.542]
t28,0.025 = 2.048 > z0.025 = 1.960
• Compare z-interval X ̄ ± z0.025 / n witht-interval X ̄±tn 1,0.025S/ n:
èWe need a larger multiplier to accommodate the extra uncertainty
in S
• The difference between tn 1, /2
and z /2 becomes smaller as the sample size increases
• Note that
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−2 −1 0 1 2 Theoretical Quantiles
x ̄ = 5.454, s = 0.230
STAT 431
Sample Quantiles
−2 −1 0 1
Normal distribution vs. t-distribution
P(-2