Study Guides (260,923)
AUS (6,766)
ANU (252)
STAT (12)
STAT1008 (12)
Bronwyn (12)
Final

STAT1008 Final: Normal Distribution

2 Pages
75 Views

Department
Statistics
Course Code
STAT1008
Professor
Bronwyn

This preview shows half of the first page. Sign up to view the full 2 pages of the document.
Normal Distribution
Density Curve
A density curve is a theoretical model to describe a variable's distribution.
Think of a density curve as an idealised histogram, where:
1. The total area under the curve is equal to 1.
2. The area over any interval is the proportion of the distribution in that interval.
Normal Distribution
A normal distribution has a symmetric bell-shaped density curve.
Two features distinguish one normal density from another:
1. The mean is its centre of symmetry (μ).
2. The standard deviation controls it spread (σ).
Notation: X~N(μ,σ)
Normal Density Curve
A normal distribution follows a bell-shaped curve.
We use the two parameters mean, μ, ad stadard deiatio, σ, to
distinguish one normal curve from another.
For a shorthand we often use the otatio N μ, σ to speify that a distriutio is oral N.
Graph of a Normal Density Curve
The graph of a oral desity ure N μ, σ is a ell-shaped curve which:
o Is etred at the ea μ
o Has a horizontal scale such that 95% of the area under the curve falls within two SDs
of the ea ithi μ ± σ.
Standard Normal N(0, 1)
Stadard Noral: μ =  ad σ =  Z ~ N (0, 1).
To convert any X~N(μ, σ to )~N,, use the z-score: Z = (X-μ/σ.
To oert fro ) ~ N , to ay X ~ N μ, σ, e reerse the stadardisatio ith: X = μ + )σ.
Standardising
The process of converting any Normal random variable to a Standard Normal Random Variable.
If X~N(μ,σ²), then use the linear transformation below: Z= (X μ/ σ ~ N,.
For Z~N(0,1), values for Pr(Z<z) are available in tables.
Symmetry P(Z<-a) = P(Z>a).
Total area under curve is 1, total area under each half of curve is 0.5, i.e. P(Z<0) = P(Z>0) = 0.5.
If we are interested in a point so large that it is not in our tables, we consider the tail
probability to be 0.
Eg: P(Z>1.08) = 1 P(Z<1.08) = 1 0.8599) (from tables) = 0.1401
Eg: P(Z<-1.08) = P(Z>1.08) by symmetry = 0.1401
Eg: P(-1.51<Z<1.08) = P(Z<1.08) P(Z<-1.51) = 0.8599 0.0655 = 0.7944
Eg: P(1.08<Z<1.51) = P(Z<1.51) P(Z<1.08) = 0.9345 0.8599 = 0.0746
5.2 Confidence Intervals and P-values using Normal Distributions
Central Limit Theorem
For random samples with a sufficiently large sample size, the distribution of sample statistics
for a mean or a proportion is normally distributed.
The central limit theorem holds for any original distribution, although "sufficiently large
sample size" varies.
The more skewed the original distribution is, the larger n has to be for the central limit
theorem to work.
find more resources at oneclass.com
find more resources at oneclass.com

Loved by over 2.2 million students

Over 90% improved by at least one letter grade.

Leah — University of Toronto

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto
Saarim — University of Michigan

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan
Jenna — University of Wisconsin

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin
Anne — University of California

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California
Description
Normal Distribution Density Curve A density curve is a theoretical model to describe a variable's distribution. Think of a density curve as an idealised histogram, where: 1. The total area under the curve is equal to 1. 2. The area over any interval is the proportion of the distribution in that interval. Normal Distribution A normal distribution has a symmetric bell-shaped density curve. Two features distinguish one normal density from another: 1. The mean is its centre of symmetry (). 2. The standard deviation controls it spread (). Notation: X~N(,) Normal Density Curve A normal distribution follows a bell-shaped curve. We use the two parameters mean, , and standard deviation, , to distinguish one normal curve from another. For a shorthand we often use the notation N (, ) to specify that a distribution is normal (N). Graph of a Normal Density Curve The graph of a normal density curve N (, ) is a bell-shaped curve which: o Is centred at the mean o Has a horizontal scale such that 95% of the area under the curve falls within two SDs of the mean (within 2). Standard Normal N(0, 1) Standard Normal: = 0 and = 1 Z ~ N (0, 1). To convert any X~N(, ) to Z~N(0,1), use the z-score: Z = (X-)/. To convert from Z ~ N (0,1) to any X ~ N (, ), we reverse the standardisation with: X = + Z. Standardising The process of converting any Normal random variable to a Standard Normal Random Variable. If X~N(,), then use the linear transformation below: Z= (X )/ ~ N(0,1). For Z~N(0,1), values for Pr(Za). Total area under curve is 1, total area under each half of curve is 0.5, i.e. P(Z<0) = P(Z>0) = 0.5. If we are interested in a point so large that it is not in our tables, we consider the tail probability to be 0. Eg: P(Z>1.08) = 1 P(Z<1.08) = 1 0.8599) (from tables) = 0.1401 Eg: P(Z1.08) by symmetry = 0.1401 Eg: P(-1.51
More Less
Unlock Document


Only half of the first page are available for preview. Some parts have been intentionally blurred.

Unlock Document
You're Reading a Preview

Unlock to view full version

Unlock Document

Log In


OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit