Continuous Probability Distributions Flashcards

Continuous Probability Distributions Flashcards
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Find the area that falls between z = 1 and z = -1.

1. z = 1: 0.84134; z = -1: 0.15866

2. 0.84134 - 0.500 = 0.34134; 0.15866 - 0.500 = -0.34134

3. 0.34134 - -0.34134 = 0.68268

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Determining the Area Outside Two Z-Scores

1. Find the area between the two z-scores.

2. Subtract the area from 1.

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Determining the Area Between Two Z-Scores

1. Locate z-scores on 'Z-Scores and Normal Curve Areas' table

2. Subtract values by 0.500 if table measures from 0 (if area for 1 = 0.8413)

3. Subtract larger z-score area by smaller z-score area

Got it
A population of tropical birds has a mean wing span of 18 inches with a standard deviation of 0.6. Find the wing span of a bird with a z-score of 2.4.

Complete the steps in reverse:

2. Divide the difference by the standard deviation: x / 0.6 = 2.4; x = 1.44

1. Subtract the data point by the mean: x - 18 = 1.44; x = 19.44 inches

Got it
A long jumper has a mean jump distance of 18 feet with a standard deviation of 2.5. Find the z-score for a jump distance of 23 feet.

1. Subtract the data point by the mean: 23 - 18 = 5

2. Divide the difference by the standard deviation: 5 / 2.5 = 2

The jump is 2 standard deviations to the right of the mean.

Got it
Calculating the Z-Score

1. Subtract the data point by the mean.

2. Divide the difference by the standard deviation.

If result is negative, it is to the left of the mean on the graph of normal distribution.

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Graph of Z-Scores

Normally distributed data

Mean = 0 (if z-score = 0, then data point = mean)

Standard deviation = 1

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Z-Score

This describes the number of standard deviations a data point is from the mean. It's useful for quickly and accurately determining normal distribution probabilities.

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Characteristics of Normal Distribution Graph

Symmetrical bell curve

Centered on the mean, which is equal to the median and the mode

Width dependent on size of standard deviation (larger standard deviation = larger spread)

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Normal Distribution

Also known as Gaussian Distribution, this is a continuous distribution of data. The graph takes the shape of a bell curve, centered on the mean with both sides as mirror images of each other.

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Empirical Rule (68-95-99.7 Rule)

For normally distributed data:

68% of data within 1 standard deviation of the mean

95% of data within 2 standard deviations of the mean

99.7% of data within 3 standard deviations of the mean

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Graph of Normal Distribution

Graph

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25 cards in set

Flashcard Content Overview

If you randomly selected 100 people from anywhere in the world, approximately how many of them would be over 6 feet tall? If you flipped a coin 50 times, what is the probability you would get at least 40 heads? These questions can be easily answered if you use calculations involving the normal distribution.

Use this flashcard set to learn more about normally distributed data and calculations that can be done with the data.

Front
Back
Graph of Normal Distribution

Graph

Empirical Rule (68-95-99.7 Rule)

For normally distributed data:

68% of data within 1 standard deviation of the mean

95% of data within 2 standard deviations of the mean

99.7% of data within 3 standard deviations of the mean

Normal Distribution

Also known as Gaussian Distribution, this is a continuous distribution of data. The graph takes the shape of a bell curve, centered on the mean with both sides as mirror images of each other.

Characteristics of Normal Distribution Graph

Symmetrical bell curve

Centered on the mean, which is equal to the median and the mode

Width dependent on size of standard deviation (larger standard deviation = larger spread)

Z-Score

This describes the number of standard deviations a data point is from the mean. It's useful for quickly and accurately determining normal distribution probabilities.

Graph of Z-Scores

Normally distributed data

Mean = 0 (if z-score = 0, then data point = mean)

Standard deviation = 1

Calculating the Z-Score

1. Subtract the data point by the mean.

2. Divide the difference by the standard deviation.

If result is negative, it is to the left of the mean on the graph of normal distribution.

A long jumper has a mean jump distance of 18 feet with a standard deviation of 2.5. Find the z-score for a jump distance of 23 feet.

1. Subtract the data point by the mean: 23 - 18 = 5

2. Divide the difference by the standard deviation: 5 / 2.5 = 2

The jump is 2 standard deviations to the right of the mean.

A population of tropical birds has a mean wing span of 18 inches with a standard deviation of 0.6. Find the wing span of a bird with a z-score of 2.4.

Complete the steps in reverse:

2. Divide the difference by the standard deviation: x / 0.6 = 2.4; x = 1.44

1. Subtract the data point by the mean: x - 18 = 1.44; x = 19.44 inches

Determining the Area Between Two Z-Scores

1. Locate z-scores on 'Z-Scores and Normal Curve Areas' table

2. Subtract values by 0.500 if table measures from 0 (if area for 1 = 0.8413)

3. Subtract larger z-score area by smaller z-score area

Determining the Area Outside Two Z-Scores

1. Find the area between the two z-scores.

2. Subtract the area from 1.

Find the area that falls between z = 1 and z = -1.

1. z = 1: 0.84134; z = -1: 0.15866

2. 0.84134 - 0.500 = 0.34134; 0.15866 - 0.500 = -0.34134

3. 0.34134 - -0.34134 = 0.68268

Find the area that falls outside z = 1 and z = -1.

1. Area between z = 1 and z = -1: 0.68268

2. 1 - 0.68268 = 0.31732

Variance

Found by subtracting each point from a data set by the mean, squaring each answer, and calculating the average.

Square root of variance = standard deviation

The average number of stripes on a tabby cat's tail is 8 with a variance of 16. Determine the number of cats that will have fewer than 10 stripes in a population of 100.

Standard deviation = square root of variance = 4

Z-score for 10 stripes: (10 - 8) / 4 = 0.5

0.5 on table: 0.69146

0.69146 x 100 = 69.146 = 69 cats

The average tail length of tabby cats is 12.5 inches with a standard deviation of 0.5. Determine the percentage of cats with a tail length greater than 13 inches.

Z-score for 13 inches: (13 - 12.5) / 0.5 = 1

1.0 on table: 0.84134

Subtract from 1 to find percent greater (to the right): 1 - 0.84134 = 0.15866 = 15.9% = 16%

The average whisker length of tabby cats is 4.4 inches with a standard deviation of 0.2. Determine the percentage of cats with a whisker length between 4.2 and 4.8 inches.

Z-score for 4.2 inches: (4.2 - 4.4) / 0.2 = -1

Z-score for 4.8 inches: (4.8 - 4.4) / 0.2 = 2

-1.0 on table: 0.15866

2.0 on table: 0.97725

0.97725 - 0.15866 = 0.81859 = 81.9% = 82%

The average body length of tabby cats is 18 inches with a standard deviation of 1.0. Determine the percentage of cats with a body length between 16 and 19 inches.

Z-score for 16 inches: (16 - 18) / 1.0 = -2

Z-score for 19 inches: (19 - 18) / 1.0 = 1

-2.0 on table: 0.02275

1.0 on table: 0.84134

0.84134 - 0.02275 = 0.81859 = 81.9% = 82%

The average tail length of tabby cats is 12.5 inches with a standard deviation of 0.5. Determine the number of tabby cats that will have a tail less than than 11 inches in a population of 1,000.

Z-score for 11 inches: (11 - 12.5) / 0.5 = -3

-3.0 on table: 0.00135

0.00135 x 1000 = 1.35 = 1 cat

The average life span of tabby cats is 18 years with a standard deviation of 2. Determine the percentage of tabby cats that will live longer than 22 years.

Z-score for 20 years: (22 - 18) / 2 = 2

2.0 on table: 0.97725

Subtract from 1 to find percent greater (to the right): 1 - 0.97725 = 0.02275 = 2.275%

How to use the normal distribution to approximate binomial distribution

n = number of trials, p = probability of success

Mean = np

Standard deviation = √(np(1-p))

To solve: calculate z-score(s), find value(s) on normal distribution table, calculate probability

Using the normal distribution to approximate the binomial distribution, calculate the probability of getting at least 12 heads when flipping a fair coin 40 times.

Mean = np = 40 x 0.5 = 20

Standard deviation = √(40 x 0.5 x (1 - 0.5)) = 3.2

Z-score of 12: (12 - 20) / 3.2 = -2.5

-2.5 on table: 0.0062

1 - 0.0062 = 0.9938 = 99%

Using the normal distribution to approximate the binomial distribution, calculate the probability of getting between 25 and 30 heads when flipping a fair coin 40 times.

Mean = 40*0.5 = 20

Std. dev. = √(40*0.5 * (1-0.5) = 3.2

Z-score of 25: (25-20)/3.2 = 1.56

Z-score of 30: (30-20)/3.2 = 3.13

1.56 on table: 0.941

3.13 on table: 0.999

0.999 - 0.941 = 0.058 = 5.8%

Binomial Probability Distribution

This is used to calculate probabilities of processes that have success and failure as the two possible outcomes. It's used frequently in real-world problem-solving.

Continuous Probability Distribution

You use this process model when dealing with a number of possible outcomes that you cannot count. If you can count outcomes, you should use discrete probability distribution.

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