Empirical Probabilities

Chapter 6

[6.1, 6.2]

STAT 109 MSU SPRING 2022

A probability distribution is similar to a relative frequency distribution, except that they can be based on theoretical or empirical probabilities.

​

A probability distribution always shows:

- All possible outcomes of a probability experiment (the sample space)​

- The probability of each of these outcomes.​

​

Note: The sum of all probabilities in a distribution must be 1. Otherwise, that distribution must not show all outcomes and is not a probability distribution.​

6.1 Probability Distributions Are Models of Random Experiments

6.1 Probability Distributions Are Models of Random Experiments

6.1 Probability Distributions Are Models of Random Experiments

- There are no gaps between possible values. For example, between 1 and 1.1 is 1.15.

- Every real number (even really long obnoxious decimals) between the min and max values are possible values.

- Anything that is measured or has units that can always be broken into smaller units is continuous.​

- Possible values leave gaps. Either no decimals are included or not all decimals are included.

- ​Anything that is counted is discrete.

- There is some base smallest unit which can't be broken down into smaller units.​

​

​Discrete Variables

​Continuous Variables

Multiple Choice

The number of students in a room.

• Discrete

• Continuous

Multiple Choice

The height of a student in the room.

• Discrete

• Continuous

Multiple Choice

The length of a song.

• Discrete

• Continuous

Multiple Choice

The theoretical probability of getting some number of heads in several coin flips.

• Discrete

• Continuous

6.1 Probability Distributions Are Models of Random Experiments

- Outcomes in the sample space are represented along the x-axis

- Probabilities are represented along the y-axis.

- Use the height of a bar (or the heights of several bars added together) to calculate probabilities.

​

​P(0) = 0.5

P(Not losing money) = ​0.5+0.33 = 0.83

Using a Discrete Probability Distribution

6.1 Probability Distributions Are Models of Random Experiments

- Outcomes in the sample space are represented along the x-axis

- Probabilities are represented along the y-axis.

- Calculate or determine the area under the curve that overlaps the outcomes for which you want to calculate a probability

- Depending on the shape of the curve, this calculation can be very complicated!​

Using a Continuous Probability Distribution

6.1 Probability Distributions Are Models of Random Experiments

​A probability distribution is uniform if all outcomes in the sample space are equally likely. Since the uniform distribution is a rectangle, we can calculate areas under it using Area = base x height.

- The base is determined by the range of outcomes for which you want to calculate a probability

- The height is given on the y-axis.​

Uniform Probability Distributions

Fill in the Blank

The probability distribution represents the amount of time that someone waits at a bus stop for the bus to arrive.

What is the probability that this person waits more than 5 minutes?

(Give your answer as a decimal, rounded to the nearest hundredth.)

We have already seen that many distributions fall on a Normal (or approximately Normal) distribution. Thus, we can use the Normal curve to calculate probabilities.

​

Calculating areas under this curve requires some Calc II, so we use a table of pre-calculated values based on z-scores. Since data that falls on a close-to-Normal distribution can be standardized, these pre-calculated values apply universally to all normal models.​

​

Note: You need z-scores to use the table!​

6.2 The Normal Model

6.2 The Normal Model

6.2 The Normal Model

Suppose you have calculated a z-score: z. The Z-Tables give the area under the normal curve to the left of z. This is the probability that a randomly selected z-score will be less than z.

- Find the ones and tenths digit of z in the first column, and the hundredths digit of z in the first row.

- The area to the left of z is given in the box where the row and column associated with z intersect. ​

Using the Z-Tables

​The area under the curve to the left of z=1.21 is 0.8869.

6.2 The Normal Model

Let A be the area under the normal curve to the left of z. This number is found in the table.

​

- The area to left of z is A.

- The area to the right of z is 1-A

- ​The area between z₁ and z₂ is A₂-A₁

Using the Z-Tables

6.2 The Normal Model

If you know that ​a distribution is approximately normal:

1. Sketch and shade a normal distribution to represent the probability you are trying to calculate.

2. Calculate the z-score (or z-scores) associate with the probability you are trying to calculate.

3. Use the normal tables to determine the area under the normal curve that represents that probability. Use the appropriate rules for (Left, Right, and Between).

Using the Normal Model to Calculate Probabilities

6.2 The Normal Model

Using the Normal Model to Calculate Probabilities

​ Suppose the heights of adult women in the US are normally distributed with a mean of 64 inches and a standard deviation

of 3 inches. What is the probability that a randomly selected woman will be...

​ (a) Less than 60 inches tall?

(b) More than 60 inches tall?

(c) Between 60 and 70 inches tall?​

​

​1. Sketch. 2. Calculate a z-score.​ 3. Find the area under the curve (to the

​ left) of z=-1.33

​

​ A = 0.0918

​

​

6.2 The Normal Model

Using the Normal Model to Calculate Probabilities

​ Suppose the heights of adult women in the US are normally distributed with a mean of 64 inches and a standard deviation

of 3 inches. What is the probability that a randomly selected woman will be...

​ (a) Less than 60 inches tall?

(b) More than 60 inches tall?

(c) Between 60 and 70 inches tall?​

​

​ 1. Sketch. 2. Calculate a z-score.​ 3. Find the area under the curve (to the

​ right) of z=-1.33

​

​ A = 0.0918

​ 1-A = 0.9082

​

6.2 The Normal Model

Using the Normal Model to Calculate Probabilities

​ Suppose the heights of adult women in the US are normally distributed with a mean of 64 inches and a standard deviation

of 3 inches. What is the probability that a randomly selected woman will be...

​ (a) Less than 60 inches tall?

(b) More than 60 inches tall?

(c) Between 60 and 70 inches tall?​

​

​ 1. Sketch. 2. Calculate a z-score.​ 3. Find the area under the curve

between z₁=-1.33 and z₂=

​

​ A₁ = 0.0918

​ A₂ = 0.9772

​ A₂ - A₁ = 0.8854

Multiple Choice

Suppose IQ scores for American adults are normally distributed with a mean IQ score of 100 and a standard deviation in IQ scores of 15 points.

A person with an IQ score of 140 or above is considered a genius. What is the probability that a randomly selected American adult is a genius?

• 0.9962

• 0.0038

• 0.9923

• 0.0077

6.2 The Normal Model

The data given is or is assumed to be normal. You are given a mean and standard deviation.

1. You are asked to find the area under the normal curve to the left of some value, right of some value, or between two different values.

2. You are asked to find the probability that a randomly selected subject/participant/etc. will have a value less than some value, greater than some value, or between two different values.

3. You are asked what percentage of values in a distribution fall below some value, above some value, or between two different values.

How do you know you're looking at a question that requires these calculations?