Transcript Probability Distributions

Probability Distributions
Statistics Chapter 4
Normal Distribution
Distribution
 The description of the possible values of
a random variable and the possible
occurrences of these values.
Normal Distribution Curve
 A symmetrical curve
that shows the
highest frequency in
the center with an
identical curve on
either side of that
center.
 Usually called a “bell
curve” from its
shape.
Discrete Values
Data where a finite number of
values exist between any two
other values. Data points are not
joined on a graph.
Continuous Variables
Variables that take on any value
within the limits of the variable.
Example of Variables
 If we are counting from 1 to 10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 would be
discrete values.
3.5 would be a continuous variable.
Other Examples
 Examples of discrete values:
Number of people in class, number of
correct answers on a test
 Examples of continuous values:
Lengths, temperatures, ages, heights
 Can you think of two other examples of
each?
Example 1
Does the histogram represent
normal distribution?
Example 2
Does the histogram represent
normal distribution?
Example 3
Does the histogram represent
normal distribution?
Probabilities
Binomial Distributions
Binomial Experiments
Experiments that involve only 2
choices as answers. (either a
success or a failure)
Binomial Distribution
Distribution of the observations in
a binomial experiment
Example 4
 Keith took a poll of the students in his
school to see if they agreed with a new
“no cell phone” policy. He found the
following results.
 Make a histogram
to show this data.
Age
Number
Responding
‘No’
 Is this data normally
distributed?
14
56
15
65
16
90
17
95
18
60
Example 5
 Jake sent out the same survey
as Keith, except he sent it to
every junior high school and high
school in the district. He found
these results.
 Make a histogram to show this
data.
 Is the data normally distributed?
Age
Number
Respondi
ng ‘No’
12
274
13
261
14
259
15
289
16
233
17
225
18
253
19
245
20
216
Example 6
A local food chain has determined
that 40% of the people who shop
in the store use an incentive card,
such as air miles. If 10 people
walk into the store, what is the
probability that AT MOST half of
these will be using an incentive
card?
Example 7
Karen and Denny want to have 5
children after they get married.
What is the probability that they
will have exactly 3 girls?
TECHNOLOGY
When using exactly:
Use Binompdf function
2nd  VARS  binompdf  n
value, probability of success,
number of successes
TECHNOLOGY
When using at least, more than,
less than, or at most:
Use Binompdf function
2nd  VARS  binomcdf  n
value, probability of success,
number of successes
Example 8
A fair coin is tossed 50 times.
What is the probability that you
will get heads in 30 of these
tosses?
Example 9
A fair coin is tossed 50 times.
What is the probability that you
will get heads in at most 30 of
these tosses?
Example 9
A fair coin is tossed 50 times.
What is the probability that you
will get heads in at least 30 of
these tosses?
Exponential Distributions
Exponential Distribution
 Exponential Distribution: A probability
distribution showing a relation in the form y =
𝑎𝑏 𝑥 .
 Continuous Data: Data where an infinite
number of values exist between any two other
values. Data points are joined on a graph.
 Standard Distribution: Normal distributions,
which are often referred to as bell curves.
 Continuous Random Variables: A variable that
can form an infinite number of groupings.
Three Types of Distribution
What are some similarities and
differences between the three
distribution curves?
Example 10
ABC Computer Company is doing
a quality control check on their
newest core chip. They randomly
chose 25 chips from a batch of
200 to test and examined them to
see how long they would
continuously run before failing.
The following results were
obtained.
Example 10 Continued
What kind of data is represented
in the table?
How do we know? Number of
Chips
Hours to Failure
8
1,000
6
2,000
4
3,000
3
4,000
2
5,000
2
6,000
Example 10 Continued
TECHNOLOGY
1. Enter the data
STAT  Edit  Enter data into L1
and L2
2. Turn on stat plots
2nd  Y =  1 enter Turn on stat
plots  Clear
3. Find values
STAT CALC  EXPREG
Example 10 Continued
TECHNOLOGY
MAKE SURE:
2nd  0  Scroll down to
DIAGNOSTICON
TECHNOLOGY
Coefficient of Determination
(r^2): A standard quantitative
measure of best fit. Has values
from 0 to 1, and the closer the
value is to 1, the better the fit is.
Example 11
Radioactive substance are
measure using a Geiger-Muller
counter. Robert was working in
his lab measuring the count rate
of a radioactive particle. He
obtained the following data.
Example 11 Continued
Is this data representative of an
exponential distribution? If so find
the equation. What would a count
be at 7.5 hours?
Time (hr) Count (atoms)
15
544
12
272
9
136
6
68
3
34
1
17