# IB Math HL – Normal Distribution

A Normal distribution is a continuous probability distribution for a random variable X. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties.

- The mean, median, and mode are equal.
- The normal curve is bell-shaped and is symmetric about the mean.
- The total area under the normal curve is equal to one.
- The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean.

Approximately 68% of the area under the normal curve is between and .

Approximately 95% of the area under the normal curve is between and .

Approximately 99.7% of the area under the normal curve is between and .

The **standard normal distribution** is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

**Q**uestions and **A**nswers

### 1. IB Maths HL -Continuous Probability Distribution-Normal Distribution

How can we find the standard deviation of the weight of a population of dogs which is found to be normally distributed with mean 7.4 Kg and the 30% of the dogs weigh at least 8 Kg.

Solution

Let the random variable W denote the weight of the dogs, so that

We know that

Since we don’t know the standard deviation, we cannot use the inverse normal. Therefore we have to transform the random variable to that of , using the transformation we have the following

Using GDC Casio fx-9860G SD

MAIN MENU > STAT>DIST(F5)>NORM(F1)>InvN>

Setting Tail: right

Area: 0.3

:1

:0

We find that the standardized value is 0.5244

Therefore,

### 2. IB Mathematics HL – Statistics Probability, Normal Distribution Continuous Probability Distribution

How can we find the mean

of the weight of a population of students which is found to be normally distributed with standard deviation 2 Kg and the 30% of the students weigh at least 53 Kg.

Solution

Let the random variable W denote the weight of the students, so that

We know that

Since we don’t know the mean, we cannot use the inverse normal. Therefore we have to transform the random variable to that of

, using the transformation , we have the following

Using GDC Casio fx-9860G SD

MAIN MENU > STAT>DIST(F5)>NORM(F1)>InvN>

Setting Tail: right

Area: 0.1

:1

:0

We find that the standardized value is 0.5244

Therefore,

### 3. IB Mathematics HL – Normal Distribution Continuous Probability Distribution

How can we find the percentage of the population will benefit from a new tax law expected to benefit families with income between $40,000 and $50,000 given that the family income follows a normal distribution with mean $45,000 and standard deviation $ 12,000?

Solution

Let the random variable I denote the family income, so that

the probability is

Using GDC Casio fx-9860G SD

MAIN MENU > STAT>DIST(F5)>NORM(F1)>Ncd>

Setting Lower: 40,000

Upper: 50,000

: 12,000

: 45,000

We find that the probability is 0.323

So 32.3% of the families will benefit from this new tax law.