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, ad stadard deiatio, σ, to

distinguish one normal curve from another.

• For a shorthand we often use the otatio N μ, σ to speify that a distriutio is oral N.

Graph of a Normal Density Curve

• The graph of a oral desity ure N μ, σ is a ell-shaped curve which:

o Is etred 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)

• Stadard Noral: μ = ad σ = → Z ~ N (0, 1).

• To convert any X~N(μ, σ to )~N,, use the z-score: Z = (X-μ/σ.

• To oert fro ) ~ N , to ay X ~ N μ, σ, e reerse the stadardisatio 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.

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