1. The Normal Distribution

2. DeMoivre-Laplace Theorem:
Let X be a binomial random variable defined on n independent trials each having success probability p. For any number c and d:
If this integral is a continuous probability density function, then:

3. Standard Normal Distribution:
A Random Variable Z is said to have a standard normal distribution given by the following continuous pdf:
A Standardized Normal Distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

4. Cumulative Distribution Function for a Standard Normal Distribution
There is a special notation used for a cumulative distribution of a standard normal distribution or

5. Let the random variable X have a normal distribution. Then:

6. Find the area between z = 0 and z = 1 in a standard normal curve.
Find the probability of getting a z-value less than 0.43 in a standard normal distribution.
Answer:
Find the area between z = 0 and z = 1 in a standard normal curve.
Answer:

7. Find the probability of getting a z-value in a standard normal distribution that is greater than 1.82.
Answer:
If the probability of getting less than a certain z-value is , what is the z-value?
Answer:

8. Find the probability of getting a z-value in a standard normal distribution that is between 0.46 and 1.75.
Answer:
Find the z-value for which the area under the standard normal curve to the right of that value is
Answer: 1.96

9. Properties of the Standard Normal Distribution:
1. The probability that a z-score falls within 1 standard deviation of the mean on either side, that is, between z = -1 and z = 1, is approximately 68%.
2. The probability that a z-score falls within 2 standard deviations of the mean on either side, that is, between z = -2 and z = 2, is approximately 95%.
3. The probability that a z-score falls within 3 standard deviations of the mean on either side, that is, between z = -3 and z = 3, is approximately 99.7%.

10. A Random Variable X is said to have a normal distribution with standard mean and variance is given by:
This function can be used for normal distributions with a mean different from 0 and a standard deviation different from 1.
We can convert this into a standardized normal variable by converting each of its scores into standard scores, given by:

11. In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value greater than 30?
Answer:
In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value less than 20?
Answer:

12. In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value greater than 120?
Answer:
In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value between -6 and 9?
Answer:

13. In a Normal Distribution, the mean and standard deviation are given below. Find the x-value for which the area to the left of that value is 0.04.
Answer: 72
In a Normal Distribution, the mean and standard deviation are given below. Find the x-value for which the area to the right of that value is 0.6.
Answer: 1.99

14. In a 1905 study, R. Pearl determined that the brain weights of Swedish men are approximately normally distributed with a mean and standard deviation given below. Determine the percentage of Swedish men with brain weights between 1.50 and 1.70 kg.
Answer: or 17.8%
The annual wages, excluding board, of US farm laborers in 1926 are normally distributed with a mean and standard deviation given below. In 1926, what is the probability that a US farm laborer will have an annual wage of at least $400?
Answer:

15. The masses of a certain species of bird have a normal distribution with mean 50 grams. If 10 percent of the birds weigh more than 60 grams, find the variance of their masses.
Answer: 61 grams
The annual rainfall in a town has a normal distribution with standard deviation of 5 inches. If the rainfall is over 20 inches for a third of the years, find the mean rainfall.
Answer: inches

16. The random variable X is normally distributed and
The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.12 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg.
Answer:
M02/HL1/11
The random variable X is normally distributed and
Answer: 9.19
M03/HL1/14
Find E(X).

17. Z is the standard normal variable with mean 0 and variance 1
Z is the standard normal variable with mean 0 and variance 1. Find the value of a such that:
Answer: a = 1.15
M01/HL1/13
The speed of cars at a certain point on a straight road are normally distributed with mean and standard deviation. 15% of the cars travelled at speeds greater than 90 km/hr and 12% of them at speeds less than 40 km/hr. Find the mean and standard deviation.
Answer:
Mean = 66.6
Standard Deviation = 22.6
SPEC06/HL1/8

18. b) Find the upper quartile of the distribution.
A certain type of vegetable has a weight which follows a normal distribution with mean 450 grams and standard deviation 50 grams.
a) In a load of 2000 of these vegetables, calculate the expected number with a weight greater than 525 grams.
b) Find the upper quartile of the distribution.
Answers: 134, 484
N06/HL1/9
The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 200 grams. The weights of 88% of the loaves are less than 220 grams. Find the standard deviation.
Answer:
Standard Deviation = 17.0
M06/HL1/8