7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Unit 8 (Chapter 7): Probability Distributions with Continuous Variables
7
7.1 ­ Continuous Random Variables
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Continuous Probability Distributions
• So far, we have seen probability distributions that deal with discrete data. (i.e. binomial distribution, hypergeometric distribution, etc.)
• For these kinds of distributions, the random variable, X, is always a natural number (usually it represents the number of successes or failures)
• However, many of the things we study involve continuous data, where the random variable X is a rational number (i.e. can include decimals), such as heights, weights, time, etc.
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
• Probability distributions from the previous chapter can be graphed using a bar graph , because they have discrete outcomes.
• However, in this chapter, probabiity distributions will be represented as smooth curves, because they have continuous outcomes
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
• Graph B is called the "normal distribution"
• The normal distribution is called a "unimodal" distribution, because it has one hump.
• Types of unimodal distributions:
ex. the normal distribution
ex. X represents the number of children in Canadian families
ex. The grade of elementary students attending an overnight trip. • There are also bimodal distributions. (ex. adult shoe sizes)
7.1 Continuous Random Variables with ink.notebook
7.1
December 16, 2016
Continuous Random Variables
Investigate Comparing Discrete and Continuous Random Variables
2. Some students were asked for the number of siblings in their families. The table shows the results.
a) Classify the number of siblings as a discrete or a continuous variable.
Explain your reasoning.
b) Represent the data using a histogram.
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7.1 Continuous Random Variables with ink.notebook
7.1
December 16, 2016
Continuous Random Variables
3. Students recorded the time, to the nearest minute, spent on math homework one evening. The table shows the results.
a) Classify time as a discrete or a continuous variable. Explain your reasoning.
b) Why is the time shown in intervals?
c) Draw a scatter plot of these data. For the time value, use the midpoint of each interval. Sketch a smooth curve through the points on the scatter plot. d) Reflect Does the shape of the curve make sense? Explain.
e) Extend Your Understanding Consider the choice of intervals in the table. Why must you be careful not to have too few or too many intervals?
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7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Solutions
Investigate Comparing Discrete and Continuous Random Variables
1. Consider attributes of students in your class, such as number of siblings or height.
a) List several attributes that are counted using discrete values.
Examples: number of siblings, number of coins in pockets, number of books in your bag
b) List several attributes that are measured using continuous values.
Examples: weight, height, longest distance you can jump, reaction time
2. Some students were asked for the number of siblings in their families. The table shows the results.
a) Classify the number of siblings as a discrete or a continuous variable.
Explain your reasoning.
Number of siblings is a discrete variable, it can only take on whole number values.
b) Represent the data using a histogram.
Go back to the question
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Solutions
Investigate Comparing Discrete and Continuous Random Variables
(continued)
3. Students recorded the time, to the nearest minute, spent on math homework one evening. The table shows the results.
a) Classify time as a discrete or a continuous variable. Explain your reasoning.
Time is a continuous variable. It can take on any real value.
b) Why is the time shown in intervals?
Since time is a continuous variable, there are an infinite number of possible values. It must, therefore be grouped into intervals.
c) Draw a scatter plot of these data. For the time value, use the midpoint of each interval. Sketch a smooth curve through the points on the scatter plot. d) Reflect Does the shape of the curve make sense? Explain.
Example: The shape of the curve makes sense because most people in the class will spend an average amount of time (35 min) on homework, and the number of students who spend more or less time will gradually decrease as you go farther from that average time.
e) Extend Your Understanding Consider the choice of intervals in the table. Why must you be careful not to have too few or too many intervals?
Example: If there are too few intervals, the shape of the distribution will not be apparent, because most of the entries will be in only a small number of intervals. If there are too many intervals, there will be few or no entries in each interval, and this will also obscure the shape of the distribution.
Go back to the question
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Calculating Probabilities for Continuous Variables
• The probability that a variable falls within a range of values is equal to the area under the probability density graph for that range of values.
• You can represent a sample of values for a continuous random variable using a frequency table, a frequency histogram, or a frequency polygon.
• The frequency polygon approximates the shape of the probability density distribution.
7.1 Continuous Random Variables with ink.notebook
7.1
December 16, 2016
Continuous Random Variables
Example 1
Determine a Probability Using a Uniform Distribution
At a local supermarket, a new checkout lane is opened whenever the wait time is more than 6 min. As a result, the time required for a customer to wait at the checkout lanes varies from 0 to 6 min, with all times in between being equally likely.
a) What kind of distribution is this? How do you know?
b) Sketch a graph that illustrates this distribution.
c) What is the probability that a customer will wait between 3 min and 6 min to check out?
d) How many values are possible for the time required to be served at the checkout? Explain your answer.
e) Is it possible to determine the probability that a customer will need to wait exactly 3 min at the checkout lane, using the area under the graph? Explain your answer.
Click here for the solution.
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Solutions
Example 1
Determine a Probability Using a Uniform Distribution
a) What kind of distribution is this? How do you know?
Since all outcomes are equally likely, this is a uniform distribution.
b) Sketch a graph that illustrates this distribution.
Since all values from 0 min to 6 min are equally probable, the graph is a horizontal line from 0 min to 6 min. The area under the graph represents the total of all of the probabilities. Therefore the area must equal 1. The base of the rectangle has a
length of 6 min.
c) What is the probability that a customer will wait between 3 min and 6 min
to check out?
The probability that a customer will wait between 3 min and 6 min is equal to the shaded area under the graph from 3 min to 6 min.
The probability that a customer will wait between 3 min and 6 min to check out is 0.5.
d) How many values are possible for the time required to be served at the checkout?
Explain your answer.
Since this is a continuous distribution, any real number between 0 and 6 min is a possible value. An infinite number of possible values exist for the time required to be served at the checkout.
e) Is it possible to determine the probability that a customer will need to wait exactly 3 min at the checkout lane, using the area under the graph? Explain your answer.
If you pick a single value, such as 3 min, the rectangle under the graph will have a width of 0 min. The probability for a single value of a continuous distribution is 0. The area cannot be used for single values of a continuous variable, only for a range of values.
Go back to the question
7.1 Continuous Random Variables with ink.notebook
7.1
December 16, 2016
Continuous Random Variables
Example 2
Frequency Table, Frequency Histogram, Frequency Polygon
The heights of all students in a mathematics of data management class are measured and recorded in the table below, to the nearest centimetre. b) Use a table like the one below to determine the frequency for each interval. Hint
a) Can you use the data in the table to determine whether the data seem to follow a uniform distribution? Can you make a reasonable estimate of the mean height in this class? If a data value falls on the bo
between two intervals, it is u
the lower interval. For example, the first data v
is recorded in the 150 cm ­ 1
c) Using the completed frequency table, can you now answer part a) more easily?
d)
In what ways can a frequency table help you to analyse the raw data from a sample like this one?
e) Use the frequency table to draw a frequency histogram. Then add a frequency polygon to the histogram. f)
How is the shape of the frequency polygon related to the shape of the probability density distribution for height? Can you use the area under the frequency polygon to calculate probabilities for any range of values?
Click here for the solution.
7.1 Continuous Random Variables with ink.notebook
December 16, 2016
Solutions
Example 2
Frequency Table, Frequency Histogram, Frequency Polygon
a) Can you use the data in the table to determine whether the data seem to follow a uniform distribution? Can you make a reasonable estimate of the mean height in this class? The data are difficult to analyse in this form. It is not obvious whether the distribution is b) Use a table like the one below to determine the frequency for each interval. Hint
uniform or not. Similarly, it is difficult to estimate the value of the mean with any accuracy.
If a data value falls on the between two intervals, it is
the lower interval. For example, the first data
is recorded in the 150 cm interval.
c) Using the completed frequency table, can you now answer part a) more easily?
From the frequency table, it appears that the frequencies vary from 0 to 8. The distribution is not uniform. The mean height appears to be around 165 cm.
d) In what ways can a frequency table help you to analyse the raw data from a sample like this one?
The frequency table groups the raw data into intervals. The frequency in each interval makes the shape of the distribution more obvious (if you turn your head sideways, the tally column looks like a rudimentary histogram) and gives an indication of the location of the mean.
e) Use the frequency table to draw a frequency histogram. Then add a frequency polygon to the histogram. f) How is the shape of the frequency polygon related to the shape of the probability density distribution for height? Can you use the area under the frequency polygon to calculate probabilities for any range of values?
The shape of the frequency polygon gives an indication of the shape of the probability distribution for height, but it represents a small sample, relative to the overall population that it represents. Also, the total area under the frequency polygon is not equal to 1. You cannot calculate probabilities using areas under the frequency polygon. You need to use a probability distribution to determine probabilities.
Go back to the question
7.1 Continuous Random Variables with ink.notebook
Work: p.327 #1­3,7,9 December 16, 2016