STAT 1350 Lecture Notes - Lecture 16: Seat Belt, Fair Coin, Statistic
27 Apr 2018
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3/6/18
Chapter 18: More on Probability Models
● Review
○ There is something wrong in each of the following statements. Please explain
what is wrong.
■ The probability a randomly selected driver will be wearing a seat belt
is .75, whereas the probability the driver will not be wearing a seat belt
is .30. ⇒ Probability must sum to 1
■ The probability that a randomly selected car is red is 1.20. ⇒ Probability
cannot be larger than 1
● More Review
○ A bag of Reese’s Pieces contains 50 candies: 20 orange, 12 brown, and 18
yellow.
○ If we were to write this as a probability model, what would it look like?
■
Outcome
Orange
Brown
Yellow
Probability
0.4 (20/50)
0.24
0.36
■ What is the probability that you will reach into the bag and obtain a
yellow candy or a brown candy? → 0.60
■ What is the probability that you will not choose a brown candy if you
randomly reach into the bag for a single candy? → 1-0.24 = 0.76
● What do you think?
○ If I toss a fair coin, the probability of the coin landing on “heads” is 0.5.
○ If I toss a fair coin 100 times, what is the probability it will land on “heads” in 65
or more tosses?
■ This question is more complex and requires us to think about probability
models for samples.
● What happens when we focus on samples?
○ Let’s imagine that rather than taking one Reese’s Pieces candy from a sample and
determining the probability it will be orange, we want to know the probability of
obtaining a certain percentage of orange candies from a sample of size 50.
■ If a bag of 50 candies is a sample, the proportion of orange candies in that
bag is a sample statistic. It is a sample proportion.
● P = p-hat - sample proportion (P = population proportion)
■ Although we might have only one single sample to work with, we need to
understand how sample statistics can vary if we want to use our
sample statistic to answer a question about the population.
● Sampling Distributions
○ Imagine choosing a random sample from a population and calculating a statistic,
such as the sample proportion.
○ If we repeated this process many times and attempted to look at the distribution of
sample statistics, this would give us a probability model called a sampling
distribution.
○ A sampling distribution shows us how sample statistics can vary. It assigns
probabilities to the values the sample statistic can take.
● Example: Reese’s Pieces
○ Suppose we are given a bag (or a sample) of 50 candies, and we want to
determine the proportion of candies that are orange. Out of 50 candies, suppose
that exactly 20 are orange. Our sample proportion is then: (P = 20/50 = 0.4)
○ What if we repeat this process many times? In other words, what if we take
thousands of samples of size 50 from the population and determine the proportion
of orange candies in each sample?
○ It turns out that in the population, approximately 50% of the candies in a bag of
Reese’s Pieces should be orange.
○ When we take many samples and then plot or graph the sample statistics, a
regular pattern begins to emerge
● More About the Sampling Distribution
