CI - mean (variance known) of a normal distribution

Saturday, 17 August 2019

4:50 PM

Standardise𝑍=﷐﷐﷐﷐𝑋﷯−μ﷯﷮σ﷯﷮﷐﷮𝑛﷯﷯=﷐﷮𝑛﷯﷐﷐𝑋﷯−μ﷮𝜎﷯~𝑁﷐0,1﷯
﷐𝑋﷯=﷐1﷮𝑛﷯﷐𝑖=1﷮𝑛﷮﷐𝑋﷮𝑖﷯﷯~𝑁﷐μ,﷐σ﷮﷐﷮𝑛﷯﷯﷯

𝑃﷐𝐿≤μ≤𝑈﷯=1−𝛼

Can rewrite as
𝑃﷐−﷐𝑍﷮1−﷐α﷮2﷯﷯≤𝑍≤﷐𝑍﷮1−﷐α﷮2﷯﷯﷯=1−𝛼

﷐𝑍﷮1−﷐α﷮2﷯﷯ is the quantile of level 1−﷐α﷮2﷯

Now, when isolating 𝜇,
Untitled picture.png Machine generated alternative text:
P X— Zl_a/2
X + Zl_a/2

So, confidence interval of level 100×﷐1−α﷯% for 𝜇 is
[﷐𝑋﷯−﷐﷐𝑍﷮1−﷐α﷮2﷯﷯σ﷮﷐﷮𝑛﷯﷯, ﷐𝑋﷯+﷐﷐𝑍﷮1−﷐α﷮2﷯﷯σ﷮﷐﷮𝑛﷯﷯]

Untitled picture.png Machine generated alternative text:
7. Inferences concerning a mean
7.4 Cl on the mean (normal, variance known)
Confidence interval on the mean of a normal
distribution, variance known
From Slide 58,
Lecture 4, zo.95 =
1.645,
= 1.96 and
= 2.575
4.995
confidence interval for of level 90% is
R- 1.645
+ 1 .645
confidence interval for of level 95% is
R -1.96
R + 1.96
X
confidence interval for of level 99% is
R - 2.575
R + 2.575
X
We see that the respective lengths of these intervals are
3.29 x
3.92 x
and 5.15
MATH2099/2859 (Statistics)
Dr Jia Deng
Term 2, 2019 - Lecture 5
36/ 56

Untitled picture.png Machine generated alternative text:
1
—Zl_u/2
Zl_u/2

Created with Microsoft OneNote 2016.