# Continuous Probability Distributions Flashcards

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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

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

2. Subtract the area from 1.

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

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

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.

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.

Normally distributed data

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

Standard deviation = 1

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.

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)

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.

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

Graph

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## 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.

Graph

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

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.

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)

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.

Normally distributed data

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

Standard deviation = 1

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.

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.

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

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

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

2. Subtract the area from 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

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

2. 1 - 0.68268 = 0.31732

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

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

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%

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%

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%

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

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%

*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

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%

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%

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.

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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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

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- Overview of Statistics Flashcards
- Summarizing Data Flashcards
- Tables and Plots Flashcards
- Probability Flashcards
- Discrete Probability Distributions Flashcards
- Continuous Probability Distributions Flashcards
- Statistical Estimation Flashcards
- Hypothesis Testing in Statistics Flashcards
- Z-Scores & Standard Normal Curve Areas Statistical Table
- Critical Values of the t-Distribution Statistical Table
- Binomial Probabilities Statistical Tables
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