MAT prob&stat1 the normal distribution

Question 1

What is the Normal distribution?

Answer 1

The Normal distribution is a continuous probability distribution with a bell-shaped curve, symmetrical about the mean.

Question 2

What are the properties of a Normal curve?

Answer 2

Symmetry around the mean (μ)
Mean = Mode = Median
Points of inflection are μ ± σ
68% of values lie within ±1σ, 95% within ±2σ, and 99.75% within ±3σ.

Question 3

How is a Normal distribution denoted?

Answer 3

X ~ N(μ, σ²), where μ is the mean and σ² is the variance.

Question 4

What is the standard Normal distribution?

Answer 4

A Normal distribution with a mean of 0 and a standard deviation of 1. Denoted as Z ~ N(0, 1).

Question 5

What is the formula to standardize a Normal variable?

Answer 5

The formula is:
z = (x - μ) / σ
Where:
- z is the standard score (z-score),
- x is the value to be standardised,
- μ is the mean of the distribution,
- σ is the standard deviation.

Question 6

How are probabilities calculated in a Normal distribution?

Answer 6

Probabilities are found for ranges, not specific values. For continuous distributions, P(Z < a) = P(Z ≤ a).

Question 7

What does P(Z < 1) = 0.8413 represent?

normal distribuition

Answer 7

It means 84.13% of values in a standard Normal distribution are less than 1 standard deviation above the mean.

Question 8

How do you use the Normal approximation for a binomial distribution?

Answer 8

A binomial B(n, p) can be approximated by N(np, npq) if n is large and np, nq ≥ 5. Use a continuity correction, e.g., P(X ≤ 3) → P(X < 3.5).

Question 9

What are the steps to find P(X > a) in a standard Normal distribution?

Answer 9

Standardize X: z = (x - μ) / σ
Use P(Z > a) = 1 - P(Z < a)

Question 10

How do you find the mean and variance for a Normal approximation of B(100, 0.4)?

Answer 10

Mean: np = 100 × 0.4 = 40
Variance: npq = 100 × 0.4 × 0.6 = 24

Question 11

What is the z-score formula for a given probability?

normal

Answer 11

To find z for P(Z < z) = p, use the inverse Normal distribution or Normal tables.

Question 12

How is P(X = 30) approximated for a binomial using a Normal distribution?

Answer 12

Use continuity correction:
P(X = 30) = P(29.5 < X < 30.5)
Standardize each boundary and find the difference between the probabilities.

Question 13

What is the formula to find σ if P(X > a) = p?

Answer 13

σ = (a - μ) / z
where z is the standard Normal score corresponding to P(Z < z) = 1 - p.

Question 14

What are the conditions for using a Normal approximation for B(n, p)?

Answer 14

n is large.
np ≥ 5 and nq ≥ 5.
p is close to 0.5 for symmetry

Question 15

What does the z-score represent?

Answer 15

The z-score represents how many standard deviations a value (x) is from the mean (μ). A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.

Question 16

Why is standardising a variable useful in Normal distribution problems?

Answer 16

Standardising allows you to compare values from different Normal distributions or to use standard Normal distribution tables (with μ = 0 and σ = 1) to calculate probabilities.

Question 17

Example: If X ~ N(65, 10²), what is the z-score for x = 85?

Answer 17

Using z = (x - μ) / σ:
z = (85 - 65) / 10 = 2.
This means x = 85 is 2 standard deviations above the mean.

Question 18

How do you interpret P(Z > z) in a Normal distribution?

Answer 18

P(Z > z) is the probability that the standard Normal variable (Z) is greater than the given z-score. It represents the area under the curve to the right of z.

Question 19

How is the z-score used in probability calculations?

Answer 19

1. Convert x to z using z = (x - μ) / σ.
2. Use z to find probabilities using standard Normal tables or software.
   Example: P(X > 85) = P(Z > 2).

Question 20

Example: For X ~ N(65, 10²), find P(70 < X < 85).

Answer 20

1. Standardize 70 and 85:
   z₁ = (70 - 65) / 10 = 0.5,
   z₂ = (85 - 65) / 10 = 2.
2. Use standard Normal tables:
   P(70 < X < 85) = P(Z < 2) - P(Z < 0.5).