13 - CLT, Standard Normal, Z-tables, Kurtosis

Question 1

Normal distributions

Answer 1

• characterized by mean (µ), and variance σ²
• Empirical Rule
• both Binomial & Poisson distributions start to look normally distributed
  
  • Binomial = when n gets large
  • Poisson = when lamda (the rate) gets large
• convergence to normality can be explained with CLT

Question 2

How to calculate probabilities for a normal random variable?

Answer 2

calculate by finding the area under the curve (integration, use JMP calculator or z-table)

Question 3

Location shift of distribution

Answer 3

mean, µ controls location of the center of the distribution

can take on any value between -infinity & +infinity

Question 4

Scale changes of distribution

Answer 4

variance, σ², controls how spread out the distribution is

• greater σ² = greater spread
• σ is always greater than or equal to 0

Question 5

difference between normal and binomial/poisson

Answer 5

normal: no relationship between µ & σ

binomial & poisson = σ² is linked to µ

Question 6

Central Limit Theorem (CLT)

Answer 6

Under appropriate conditions, as the sample size (n) gets large, the sample mean tends to a normal distribution

*we can use normal probability calculations for inferences about the sample mean*

*the CLT is true, regardless of the distribution of X itself*

Xbar_n → N(µ, σ²/n)

Xbar_n = sample mean

µ = E(X)

σ² = Var(X)

Question 7

Standard Normal

Answer 7

Z ~ N(0,1)

a random variable standardized to have a mean of 0 and a variance of 1

Question 8

How to turn a normal into a standard normal

Ex. X ~ N(5, 15)

find P(X<0)

Answer 8

take the z-score so we can use the z-tables!

P(X<0) = P [(X-5)/sqr(15) < (0-5)/sqr(15)]

= P [Z < (0-5)/sqr(15)]

= P (Z < -1.291)

⇒ look @z-tables, round to -1.3, find 0.0968

Question 9

Using the z-tables for P(-1.3 <= Z <= -.06)

Answer 9

area of interest can be expressed as area to left of -0.6 minus the area to the left of -1.3

P(Z <= -0.6) - P(Z<= -1.3)

= 0.2743 - 0.0968 = 0.1775

Question 10

Finding the x percentile of the distribution

Ex. Identifying the value x, associated with the event P(X<= x) = 0.8

(by how many days into the pregnancy have 80% of women given birth)

(80% from the left of the distribution)

Answer 10

1. solve the question for z, such that P(Z<=z) = 0.8
2. undo the z-score using the formula x = zσ + µ
3. Use right-hand z-table, find 0.8 in 3rd column, this corresponds to 0.8416 in first column
4. x = (0.8416)*(10) + 268 = 276.4
5. A little over a week past the average due date (268 days)

Question 11

Quantiles of the distribution

Answer 11

p^th quantile of the distribution is the value x such that P(X<=x) = p

*for the standard normal, the quantiles are exactly the quantities in the first column of the RHS z-score table*

******for quantiles less than 0.5 (50^th percentile), report the negative of the number in the first column of the table******

Question 12

Normal Quantile Plot (NQP)

Answer 12

displays observed quantiles plotted against the expected quantiles from a standard normal distribution

departures from normality = departures from 45 degree reference line (when observed = expected)

departures from normality = R/L skew, heavy tails

Question 13

NQP axes

Answer 13

x-axis = theoretical quantiles from a standard normal

y-axis = observed quantiles

any given point:

x-coord = theoretical normal quantile

y-coord = observed (actual) quantile

Question 14

When is there a departure from normality on a NQP?

Answer 14

if there are points outside the confidence bands (95% of normal quantile plots have points within the bands)

Question 15

NQP: symptom of…

right skewed

left skewed

heavy tails

Answer 15

right skewed = convexity

left skewed = concavity

heavy tails = s-shape (below ref line on LHS, above ref line on RHS)

Question 16

Skewness

Answer 16

measures lack of symmetry

K₃ = (z³₁ +…z³_n)/n

*for a normal random variable, K₃ = 0*

• * - skewness = left skew**
• * + skewness = right skew**

Question 17

Kurtosis

Answer 17

measures tail weight of the distribution (how much of the variance is coming form extreme observations)

K₄ = [(z⁴₁ +…+ z⁴_n)/n] - 3

• *for a normal distribution, K₄= 0**
• * + Kurtosis = heavy tails with respect to a normal distribution**