Chapter 5 [5.1, 5.2]

Chapter 5

[5.1, 5.2]

STAT 109 MSU SPRING 2022

5.1 What is Randomness

In inferential statistics, it is vital that our inferences are developed on representative samples. A representative sample will accurately reflect factual aspects of a population, perhaps on a smaller scale. There will be some inaccuracies due to sample size. Ideally, an appropriately sized sample should minimize these inaccuracies.

- We achieve this using random samples and random sampling techniques.​

5.1 What is Randomness?

Biased Sampling

- A shared characteristic of all or most members in a sample that is not shared by the population is called bias.

- Biases may or may not influence the accuracy with which the sample represents the population. There is no way to know which biases will negatively impact accuracy.

- A representative sample should eliminate bias.

- We eliminate bias using random sampling techniques, which give all members of the population an equal chance of being sampled.​ This does NOT mean that every characteristic appears in the sample in equal proportions, but that every INDIVIDUAL subject had the same chance of being selected.

5.1 What is Randomness?

Biased Sampling Examples

Intentionally Biased Sampling:

- A retail company's CEO wants to show that ​all employees are paid fair wages which greatly exceed the state minimum. He samples only salaried district and store managers.

- A teacher wants to show that a new assessment method more accurately reflects student achievement, and samples students whose overall performance and skill levels appear to match their scores.

Unintentionally Biased Sampling:

- A researcher wants to know the effects of sleep on children aged 10-16, but needs permission to gather data from children. They collect data from family and friends.​

5.1 What is Randomness?

Open Ended

A nationallycomputer and technology retailer hires a research team to collect data and determine an appropriate price range for their products. Since the company sells electronics, the research team collected data from the employees of 100 randomly selected corporate offices in New York City, NY. What biases may exist in this sampling method?

Inferential Statistics and Probabilities

Inferential statistics (when done responsibly and ethically) relies on the probability that random events can occur. This means that it is vital to use random samples. There are two differing ideologies for calculating ​the probability that some event occurs.

- Theoretical Probability bases its calculations on what should be true in a probability experiment.

- Empirical Probability bases its calculations ​on what actually occurs while performing a probability experiment.

- As the number of trials increases towards infinity, the empirical probability of an event gets closer to the theoretical probability of that event.​

5.1 What is Randomness?

Inferential Statistics and Probabilities

- The theoretical probability of an event is the relative frequency of that event across infinitely many trials of a probability experiment.

- The theoretical probability of a coin flip landing on heads is 0.5. Because half of the

possible outcomes are heads, the heads should land up half of the time.​

​

- The empirical probability of an event is the relative frequency of that event across a finite number of trials, n (we use the same symbol for number of trials as we do for sample size).

- If I flip a coin 10 times, and it shows heads 6 times, the empirical probability of the coin

landing on heads is 0.6.​

5.1 What is Randomness?

Multiple Choice

Determine whether the calculated probability is an example of theoretical or empirical probability.

New Jersey saw 278,413 crashes in 2018. Of those, 563 lead to the death of an involved person. The probability that a crash is ultimately fatal is 0.002.

• Theoretical

• Empirical

Multiple Choice

Determine whether the calculated probability is an example of theoretical or empirical probability.

In Dungeons and Dragons, you use a 20-sided die to determine the success or failure of a course of action. Since all sides have the same chance of showing when the die is rolled, the probability that I will roll a natural 20 on my next check is 1/20.

• Theoretical

• Empirical

5.2 Calculating Theoretical Probabilities

Inferential Statistics uses empirical evidence and theoretical probabilities to determine what is likely or likely not true. This requires the calculation of theoretical probabilities, which are compared to the empirical ones obtained in a sample.

The language:

- Probability Experiment: Some action or sequence of actions whose outcomes are studied.

- Outcome: One possible result of a probability experiment.

- Sample Space: The set of all possible outcomes, including repetitions if they exist.​

- Event: A desired outcome or characteristic that can be satisfied by an outcome.

- Event Space: The set of outcomes satisfying an event.​

- Success: The occurrence of the desired event.​

5.2 Calculating Theoretical Probabilities

Facts About Probabilities

Consider an event A. The following are known about the probability P(A):

- P(A) must be between 0 and 1.

- If P(A)=0, event A can't be satisfied by any outcome in the sample space. We say event A is impossible within the context of the probability experiment.

- If P(A)=1, event A is satisfied by every outcome in the sample space.

- If P(A) > 0.95, we consider this event "usual," expected, or likely.

- If P(A) < 0.05, we consider this event "unusual",​ unexpected, or unlikely.

5.2 Calculating Theoretical Probabilities

Multiple Choice

Consider event A = tomorrow is Sunday.

Choose the most accurate statement about P(A).

• P(A) = 0

• P(A) is close to 0

• P(A) is close to 1

• P(A) = 1

Multiple Choice

Consider event A = you pass this course (assuming you attend regularly, complete all work on time, and study for your quizzes and tests).

Choose the most accurate statement about P(A).

• P(A) = 0

• P(A) is close to 0

• P(A) is close to 1

• P(A) = 1

Complementary Events

5.2 Calculating Theoretical Probabilities

Multiple Choice

Consider the set of all atmospheric weather conditions. Suppose one such condition is randomly selected. Are the events "Raining" and "Sunny" complementary events?

• Yes

• No

Multiple Choice

Consider the set of all atmospheric weather conditions. Suppose one such condition is randomly selected. Are the events "Raining" and "Not Raining" complementary events?

• Yes

• No

Equally Likely Outcomes

5.2 Calculating Theoretical Probabilities

And/Or Events

5.2 Calculating Theoretical Probabilities

5.2 Calculating Theoretical Probabilities

​

P(High School AND Divorced) = 59/665 = 0.089

Only 59 people satisfy both events together out of the 665.​

5.2 Calculating Theoretical Probabilities

​P(High School OR Divorced) = (68+240+59+30+10+15)/665 = 0.635

There are several groups of people satisfying one event or the other, or both.

5.2 Calculating Theoretical Probabilities

​P(High School OR Divorced) = (397+84-59)/665 = 0.635

If we use totals, we have to correct double counting! The group of people who satisfy both events are included in both totals.

And/Or Probability Rules

If data is provided, you can always calculate probabilities using the probability formula introduced on slide 15.

​

We also have formulas specifically for AND and OR probabilities that save us time. AND will be covered in our next lecture.

​

5.2 Calculating Theoretical Probabilities

Fill in the Blanks

Mutually Exclusive Events

Mutually exclusive events are events that can't occur simultaneous. That is, if A and B are mutually exclusive, then P(A and B) = 0.

​

Whether two events can occur simultaneously is entirely dependent on the sample space, not just your logic. Two events are mutually exclusive if there are no outcomes in the sample space that satisfy both events.

​

For example, a student can be a biology and art major in reality. But if a sample of 100 students does not include any students that double-major, then the events "majors in biology" and "majors in art" are mutually exclusive in this context.

5.2 Calculating Theoretical Probabilities