Download Unit 4 Study Guide: Probability and Statistics What makes a sample

Unit 4 Study Guide: Probability and Statistics
What makes a sample valid? A sample is valid if it is chosen at random and representative of the population.
Using samples to make inferences about the population: a proportional relationship can be used to make
inferences about a population based on the sample
Example: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a
town with a population of 1500?
315
=
600
𝑥
1500
x = 788 votes
Example: Make two inferences regarding food preferences based on the information provided.
Most students prefer pizza.
Student Sample Hamburgers Tacos
Pizza
Total
More people prefer pizza than hamburgers and tacos combined.
#1
12
14
74
100
#2
12
11
77
100
*Note: You cannot compare hamburgers to tacos because the
numbers are too close together and only 100 people were
sampled.
Representative Samples: A sample must be representative of the population. Therefore if 62% of the population
is sixth graders and 48% of the population is seventh graders, then the sample must have the same characteristics.
The sample must be 62% sixth graders and 48% seventh graders.
Example: There are 400 sixth graders and 600 seventh graders at Lakeside Middle School. If 80 students are
chosen at random, how many sixth graders and how many seventh graders would need to be chosen to have a
representative sample?
6𝑡ℎ 𝑔𝑟𝑎𝑑𝑒𝑟𝑠
𝑥
400
Create a proportion: 𝑡𝑜𝑡𝑎𝑙 : 120 = 1000
x= 48 sixth graders
120-48 = 72 seventh graders
th
th
Notice that the ratio of 6 to 7 graders is the same for the sample and the population
48
2
400
2
=
and
=
so the sample is representative of the population.
72
3
600
3
Probability


When there is no chance of an event occurring, the probability of the
event is zero (0). When it is certain that an event will occur, the
probability of the event is one (1).
Theoretical Probability: probability based on reasoning written as
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Experimental probability: probability based on trials written as
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑐𝑐𝑢𝑟𝑒𝑑
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Probability model: a mathematical model that defines the probability of an event occurring
Example: A die is rolled.
Roll
1
2
3
4
5
6
Theoretical Probability
1/6
1/6
1/6
1/6
1/6
1/6
Note: The sum of all possible
outcomes is always equal to 1.
Expected value: how many times you would expect an event to happen over a given number of trials (Remember:
expected value is an estimate)
Determined by multiplying the probability of the event occurring by the number of trials
Example: How many times would you expect a die to land on a 5 if the die is rolled 300 times?
Multiply the probability of landing on a 6 by 300: 1/6 * 300 = 50 times
Probability of compound events
Independent events: the outcome of one event does not affect the outcome of the other event (with replacement)
Probability (A and B) = Probability(A) * Probability(B)
Dependent events: the outcome of one event does affect the outcome of the other event (without replacement)
Probability (A and B) = Probability(A) * Probability(B given A)
Tree diagrams: tree diagrams can be used to show all of the possible outcomes and determine the probability of
compound events
Example: Flipping three coins simultaneously
The probability of all three coins landing on heads is 1/8.
The probability of only exactly two coins landing on tails is 3/8
Mean: the average value of a data set
Finding the mean of a data set: Add all of the numbers in the data set and
the sum by the amount of numbers in the set.
Mean Absolute Deviation: the average distance between each data value
mean
Finding the mean absolute deviation of a data set: Determine the mean of
set. Then determine the distance of each data point from the mean.
(Remember that distance must be positive.) Finally, find the mean of the
distances.
Comparing two sets of data:
Compare the means to determine whether one has a higher mean or they are the same
Compare the MAD. A larger MAD means that there is a wider variability in the data set.
Reading a box-and-whisker plot/ Stem-and-leaf Plot
divide
and the
the data