stat 515 fa 2021 lec 12 slides confidence interval for the
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STAT 515 fa 2021 Lec 12 slides
Confidence interval for the mean when variance unknown
Karl B. Gregory
University of South Carolina
These slides are an instructional aid; their sole purpose is to display, during the lecture,definitions, plots, results, etc. which take too much time to write by hand on the blackboard.
They are not intended to explain or expound on any material.
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 1 / 10
Recall: If X1, . . . ,Xnind⇠ Normal(µ,�2), then
X̄n ± z↵/2�pn
is a (1 � ↵)⇥ 100% CI for µ.
But what if we don’t know �?
Using X̄n ± z↵/2Sn/pn is okay if n is large, but not if n is small. . .
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 2 / 10
Twith Sn
SE E E x
OI estimator of E
The t-distributionsThe probability distribution with pdf given by
f (x) =�( ⌫+1
2 )
⌫⇡�( ⌫2 )
✓1 +
x2
⌫
◆� ⌫+12
, where �(z) =
Z 1
0uz�1e�udu,
for ⌫ > 0 is called the t-distribution with degrees of freedom ⌫.
We write T ⇠ t⌫ if a rv T has this distribution.
⌫ = 1, same as Normal(0, 1)
⌫ = 1⌫ = 2
t⌫ pdf with ⌫ = 10
0�1 1 2�2 3�3 4�4 x
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 3 / 10
g
yayt T.nu
Sampling distribution of studentized mean
If X1, . . . ,Xnind⇠ Normal(µ,�2), then
X̄n � µ
Sn/pn⇠ tn�1.
Exercise: Show above using this: If Z ⇠ Normal(0, 1) and W ⇠ �2⌫ are ind. then
T =ZpW /⌫
⇠ t⌫ .
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 4 / 10
Fj N 0,1Willi
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Oh
i30,000 bees
15 30,000 4500 dran
Scoop 294
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plx n.li E294
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X 40
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0 tn�1,↵/2
Area = ↵/2
�tn�1,↵/2
Area = ↵/2
pdf of t⌫ distribution
x
Can use function qt() or a t-table to look up the values, e.g.
t19,0.025 = qt(.975,19) = 2.093024t19,0.005 = qt(.995,19) = 2.860935.
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 5 / 10
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Confidence interval for mean of a Normal population with � unknown
Let X1, . . . ,Xnind⇠ Normal(µ,�2). Then a (1 � ↵)⇥ 100% CI for µ is
X̄n ± tn�1,↵/2Snpn.
Show where this CI comes from.
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 6 / 10
In Zu
According to Gossett
Etna FEEin
Exercise: These are the commute times (sec) to class of a sample of students.
1832 1440 1620 1362 577 934 928 998 1062 900
1380 913 654 878 172 773 1171 1574 900 900
1 Make a Q-Q plot to check Normality of the population.
2 Construct a 95% confidence interval for the mean commute time of all students.
3 Construct a 99% confidence interval for the mean commute time of all students.
4 What if the intervals X̄n ± z↵/2 · Sn/pn are used? How are they different?
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 7 / 10
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n O
In 1048 4 5 394.6
4.974 tuj.E CZ
863.7 1233.11048.4
Mean weight of 2 cup up flow scooped
In 153.2857 n 21
Sn 18.892
A sst I tuµ
is
153.2857 tz 1 0.02s HÄ27086 flo 975,20
44.69 161.89
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z α2S
nn
5 15 25 35 45 55 65 75 85 95
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 8 / 10
In Zu Er
ß
I
CI for mean of non-Normal population with � unknownLet X1, . . . ,Xn be a rs from a pop. with mean µ, and with µ4 < 1, then
X̄n ± z↵/2 ·Snpn
is an approximate (1 � ↵)⇥ 100% CI for µ when n is large (� 30, say).
In the above µ4 = E|X1|4. This limits the heavy-tailedness of the population.
Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 12 slides 9 / 10
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