Download Lesson 1 – Continuous Probability Distribution

Continuous Probability Distributions
MDM4U
Ms. Kueh
The probability distributions that were studied in the previous chapter dealt with discrete data.
The possible values of the random variable are natural numbers, because they arise from
counting processes.
Recall: Determine the probability distribution for the sum of two dice.
𝒙
1
2
3
4
5
6
7
8
9
10
11
12
𝑷(𝒙)
𝑷(𝒙)
𝒙
Many characteristics of a population such as height and mass are continuous in nature. The
values for height could be fractional or decimal values.
Normal Distribution:
One example of a continuous distribution is the bell curve. It shows up so often in physical,
social and psychological sciences that it is named the normal distribution. In normal
distributions, the data are distributed symmetrically about the mean.

A population that follows a normal
distribution can be described by its mean,
, and standard deviation, .

The mean, median, and mode are all equal.

The smaller the standard deviation, the
narrower the graph

Only 3 points out of 1000 will fall
outside the area 3 standard
deviations either side of the
center line
Recall:
𝑧=
𝑥−𝜇
𝜎
Z-scores tell you how many standard deviations a single point is away from the mean.
Example 1 Giselle is 168 cm tall. In her high school, boys’ heights are normally distributed with
a mean of 174 cm and a standard deviation of 6 cm. What is the probability that the first boy
Giselle meets at school tomorrow will be taller than she is?
Example 2 A cosmetic powder is machine packaged in a process that puts approximately 50 g of
powder in each package. The actual masses have a normal distribution with mean 50.5 and
standard deviation 0.6. The manufacturers want to ensure that each package contains at least
49.5 g of powder. What percent of packages do not contain this much powder?
*Use the normal distribution tables from pages 606 – 607.
Recall: Probabilities must have a value between ____ and _____
Example 3 The Fresha Tea Company pack tea in bags marked as 250 g. A large number of packs
of tea were weighed and the mean and standard deviation were calculated as 255 g and 2.3 g
respectively. Assuming this data is normally distributed, what percentage of packs are over the
weight listed on the package?
Distinguish between when you need to use the indirect method or the direct method:
Example 4 Students pass a quiz if they score 50% or more. The marks of a large number of
students were sampled and the mean and standard deviation were calculated as 48% and 8%
respectively. Assuming this data is normally distributed,
a) what percentage of students passed the quiz?
b) what percentage of students failed the quiz?
For more explanation go to: http://www.mathsisfun.com/data/standard-normal-distribution.html
Homework: pg. 430 #2-4, 7, 13