The document discusses the normal distribution and its key properties: bell-shaped and symmetrical around the mean, extending from negative to positive infinity with an area under the curve of 1. Approximately 95% and 99.9% of the distribution lies within 2 and 3 standard deviations of the mean, respectively. It also discusses how to calculate probabilities using the standard normal distribution where the mean is 0 and standard deviation is 1, and how to standardize other normal distributions.

1. Normal distributionUnit 8 strand 1

3. It has the following features:bell-shaped

4. symmetrical about the mean

5. it extends from –infinity to + infinity

6. The area under the curve is 1Approximately 95% of the distribution lies between 2 SDs of the mean

7. Approximately 99.9% of the distribution lies between 3 SDs of the mean

8. The probability that X lies between a and b is written as:P(a<X<b).To find the probability we need to find the area under the normal curvebetween a and b.We can integrate or use tables.

9. The probability that X lies between a and b is written as:P(a<X<b).To find the probability we need to find the area under the normal curvebetween a and b.We can integrate or use tables.To simplify the tables for all possible values of mean and s² the variable X is standardised so that the mean is zero and the standard deviation is 1. The standardised normal variable is Z and Z (0, 1)

11. So, the probability of Z < 0.16 is 0.5636.

12. Find the probability of Z < 0.429

13. Now try theseDraw sketches to illustrate your answersIf Z ~N (0, 1), find1. P (Z <0.87) 2. P (Z > 0.87) 3. P (Z < 0.544) 4. P (Z > 0.544)

16. ExampleFind P (Z < 0.411) P (Z > - 0.411) P (Z > 0.411) P (Z < - 0.411)SolutionP(Z < 0.411) = [from tables 6591 + 4] = 0.6595P(Z > - 0.411) = P(Z < 0.411) = 0.6595 (from (above))P(Z > 0.411) = 1 – Φ(0.411) = 1 – 0.6595 = 0.3405P(Z < - 0.411) = P(Z > 0.411) = 0.3405

17. Now try these1. P (Z > - 0.314) 2. P (Z < - 0.314) 3. P (Z > 0.111) 4. P (Z > - 0.111)

18. What if we don’t have a distribution where the mean is 0 and the standard deviation is 1?

19. Standardising a normal distributionTo standardise X where X ~ N(μ, σ²)subtract the mean and then divide by the standard deviation: where Z ~ N(0,1)