6. Continuous probability Distributions: The Normal Distribution

Question 1

Rules for normal distribution

Answer 1

follows empirical rule

Exactly bell shaped (skewness= 0, mean = median)

Question 2

Features of the normal distribution:

Answer 2

Bell shaped density curve, location given by population mean, spread given by st dev, (random variable has infinite theoretical range of +- infinity)

Question 3

What’s a normal, guassian distribution?

Answer 3

It’s where you’re aiming for pretty much normal but not 100% exactly normal (pretty much impossible!)
About how close it is to being normal

Question 4

How does the shape of the normal distribution change?

Answer 4

population mean (mu) shifts population left or right
Changing standard deviation increases/decreases spread

Question 5

What are we interested in mostly for normal distributions?

Answer 5

since ND defined by expected values (mean), variables,, we’re interested mostly in the tails (what’s not expected)

Question 6

What’s a standardised normal distribution?

Answer 6

Any normal distribution can be transformed into standardised normal distribution Z.
(We translate X units into Z units by formula)

Question 7

What are the features of a standardised normal distribution?

Answer 7

Z has population mean (mu)=0
St dev =1
–> x values below the mean have -ve Z values
–> X values above the man have +ve Z values

Question 8

What does having a standardised normal distribution enable us to do ?

Answer 8

calculate how far away we are in units of standard deviation

Question 9

Formula in excel for cumulative standardised normal distirbution (Z scores)

Answer 9

1. Id mean & st dev
2. Find Z score from x value with formula:
   =STANDARDISE(x, mu, st dev)
3. Find cumulative probability:
   =NORM.DIST(x, mu, st.dev, TRUE)

Question 10

How to assess normality?

Answer 10

1. is sample mean approx= sample median? (symmetrical)
2. Empirical rule satisfied? (bell shaped, using Xbar, S)
3. Is IQR approx =1.33 st dev

Question 11

How to assess normality?

Answer 11

1. is sample mean approx= sample median? (symmetrical)
2. Empirical rule satisfied? (bell shaped, using Xbar, S)
3. Is IQR approx =1.33 st dev
4. Is boxplot/histogram close to symmetric?
   - —> Is histogram roughly bell shaped?
5. Absence of clear extreme–>, fat tails
6. Are sample skewness & kurtosis approx=0?

Question 12

What do we use boxplots for more when assessing normality?

Answer 12

smaller sample sizes

Question 13

What do we use histograms for more when assessing normality?

Answer 13

larger sample sizes

Question 14

Why does IQR have to be approx =1.33 for guassian distribution?

Answer 14

0.25 st devs away from mean is -0.665, 1/2 way btw 0.66 & 0.65. If you double it you get 1.33

Question 15

What is the uniform distribution?

Answer 15

Where values are evenly distributed in the range between smallest value & largest value

Question 16

What’s are continuous uniform distribution?

Answer 16

Has = density for all possible outcomes of the random variable –> rectangular distribution

Question 17

What’s the population mean of a uniform distribution

Answer 17

mu= a+b/2

where a & b are x values on the x-axis

Question 18

How to find probability for uniform distribution?

Answer 18

base*height

Question 19

Standard deviation of uniform distribution?

Answer 19

pop st dev = Sqrt [(b-a)^2/12]

Question 20

What’s the exponential distribution?

Answer 20

Continuous distribution that’s right skewed & ranges from 0 to infinity.
(Often used to model length of the time between 2 occurrences of an event)

Question 21

Features of an exponential distribution

Answer 21

• Always +ve values,
• Right skewed,
• mode

Question 22

Excel: exponential distribution

Answer 22

1. have table of data identifying mean, lambda, x value

2. =EXPON.DIST (x, lambda, TRUE/FALSE)