03 Normal Distribution

Question 1

What is a probability density function?

Answer 1

• An expression giving the frequencies of continuous random variables.
• The area under a probability density function curve is equal to the probability; not the height

Question 2

What is the total area under a probability density curve?

Answer 2

1, because the sum of the probabilities of all possible outcomes must equal 1.

Question 3

The normal distribution curve is … and … ?

Answer 3

Symmetric and unimodal (= bell-shaped)

Question 4

What are the parameters of the normal distribution?

Answer 4

• μ and σ, the mean and standard deviation
• N(μ, σ)

Question 5

What are the 5 important properties of the normal distribution?

Answer 5

1. It is symmetric about its mean, μ
2. The maximum value is at x = μ
3. The area under the curve is 1
4. The area on either side of x = μ is 0.5
5. The Empirical Rule:
   ● μ±σ = 0.68 of the area under the curve
   ● μ±2σ = 0.95 of the area under the curve
   ● μ±3σ = 0.997 of the area under the curve

Question 6

If a variable has a normal distribution, the probabilities depend only on one thing. What is it?

Answer 6

The probablities depend only on how many standard deviations away from the mean the data is

Question 7

How do we measure how many standard deviations away from the mean the data is?

Answer 7

• By calculating a z-score
• z = (x-μ)/σ
• z-scores measure distance above the mean

Question 8

What is the step-by-step process you should go through to find a probability using the normal distribution?

Answer 8

The probability only depends on the number of standard deviations above the mean, z.
So

1. find the z-score:
   z = (x-μ)/σ
2. Sketch the Z distribution, N(0, 1)
3. Shade the areas you need to find
4. Use the sketch and the symmetry of the normal distribution to work out what inequalities you need to find
5. Look up values in the tables

Question 9

What is the connection between the binomial distribution and the normal distribution?

Answer 9

The normal distribution can be used to approximate the binomial distribution if n (the number of trials) is big enough

Question 10

What are the conditions for using the normal distribution to approximate the binomial distribution?

Answer 10

• When np and √np(1-p) are both greater than 10
• The approximation is more accurate when p is close to 0.5 and least accurate when p is close to 0 or 1

Question 11

Tables of normal distribution values only show…

Answer 11

…areas to the left of the z-score; ie probabilities that Z-score is less than the z-score P(Z < z)

Question 12

What are the 3 key things to know about the normal distribution?

Answer 12

1. A probability density function is an expression that gives the frequencies of continuous random variables
2. The Empirical Rule states that, in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean
3. The z-score measures the distance an observation is from the mean in standard deviations